Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series 3031144589, 9783031144585

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Lars-Erik Persson George Tephnadze Ferenc Weisz

Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series

Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series

Lars-Erik Persson • George Tephnadze • Ferenc Weisz

Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series

Lars-Erik Persson UiT The Artic University of Norway Narvik, Norway Karlstad University Sweden

George Tephnadze School of Science and Technology University of Georgia Tbilisi, Georgia

Ferenc Weisz Department of Numerical Analysis Eötvös Loránd University Budapest, Hungary

ISBN 978-3-031-14458-5 ISBN 978-3-031-14459-2 https://doi.org/10.1007/978-3-031-14459-2

(eBook)

Mathematics Subject Classification: 40F05, 42A38, 42B25, 42B05, 42B08, 42B30, 42C10, 43A75, 60G42 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: A magic even under the flashing sky with the famous northern light. Photograph by Dr. Hana Turcinová. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The classical Fourier Analysis has been developed in an almost unbelievable way from the first fundamental discoveries by J. J. Fourier (1768–1830). Especially a number of wonderful results have been proved, and new directions of such research have been developed, e.g., concerning wavelet theory, Gabor theory, timefrequency analysis, fast Fourier transform, abstract harmonic analysis, etc. One important reason for this is that this development is not only important for improving the “state of the art”, but also for its importance in other areas of mathematics and also for several applications (e.g. theory of signal transmission, multiplexing, filtering, image enhancement, coding theory, digital signal processing and pattern recognition). The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves, the Vilenkin (Walsh) functions are rectangular waves. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series. Because of this, it is inevitable to compare results of Vilenkin series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group. The classical theory of Lebesgue Lp spaces started to be developed around 1910. The weak-Lp spaces are function spaces which are closely related to Lp spaces. One of the most important early applications of these spaces were made by J. Marcinkiewicz in 1939, where he proved Marcinkiewicz interpolation theorem. Concerning the importance of this theorem, the corresponding so-called weak-type estimates and further development to which nowadays is called real interpolation method we refer to [30]. The Hardy spaces Hp were introduced by Riesz [281] in 1923, who named them after G. H. Hardy, because of the paper [156] from 1915. Such classical investigations of Hardy spaces by G. H. Hardy, F. and M. Riesz, J.E. Littlewood and others of Hp employed complex methods and showed that, for several problems

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in Fourier Analysis, the use of the Hp (0 < p ≤ 1) scale of spaces is preferable over that of the Lp (0 < p ≤ 1) scale. During four decades, a powerful theory of Hp (0 < p ≤ 1) spaces, especially in the n-dimensional case, has been developed by means of real methods, and various new applications to Fourier Analysis and singular operators have been given, see e.g. Coifman and Weiss [70], Fefferman and Stein [93], Taibleson and Weiss [339] and the references given therein. The history of martingale theory goes back to the early fifties when Doob [85] pointed out the connection between martingales and analytic functions. On the basis of Burkholder’s scientific achievements (see [59–61]), the theory of continuous parameter martingales can perfectly well be applied in complex analysis and in the theory of classical Hardy spaces. This connection is the main point of Durrett’s book [87]. Attention was also drawn to martingale Hardy spaces when Burkholder and Gundy [151, 152] proved the inequality named after them ever since. This inequality states that the Lp norms of the maximal function and the quadratic variation of a one parameter martingale are equivalent for 1 < p < ∞. Some years later, this result was extended to p = 1 by Davis [75]. In 1973 the dual of the one-parameter martingale Hardy space generated by the maximal function was characterized by Garsia [105] and Herz [161] as the space of functions of bounded mean oscillation (BMO). Next, we mention that some important steps in the early development can be found in the book by Schipp et al. [295] from 1990. The research continued intensively also after this. Some of the most important steps in these developments are presented in the two books [400] and [423] by F. Weisz from 1994 and 2002, respectively. The aim of this book is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we present and prove new estimations of the Vilenkin-Fourier coefficients and prove some new results concerning boundedness of maximal operators of partial sums. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of the partial sums is valid and develop new methods to prove Hardy type inequalities for the partial sums with respect to the Vilenkin systems. We also do the similar investigation for the Fejér means. Furthermore, we investigate some Nörlund means but only in the case when their coefficients are monotone. Some well-known examples of Nörlund means are Fejér means, Cesàro means and Nörlund logarithmic means. In addition, we consider Riesz logarithmic means, which are not examples of Nörlund means. It is also proved that these results are the best possible in a special sense. As applications both some well-known and new results are pointed out. Finally, we want to pronounce that we prove and discuss the analogy of the famous result of Carleson-Hunt concerning almost everywhere convergence of partial sums for Fourier series in Lp spaces, p > 1 in the case of Vilenkin systems. The book contains nine chapters and one appendix, which contains some basic facts concerning Walsh and Kaczmarz systems. One reason for this is that it will be more convenient for the reader to compare with the classical theory, and another reason is that it gives us a possibility to raise new open questions. It is maybe surprising that some of these open questions concern the classical situation but are

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motivated by the results we proved in this new situation. In fact, we have especially pointed out a great number of open questions in this book. We hope that this can stimulate the further development of this fascinating area. We now continue by describing the main content of each of the chapters. In Chap. 1, we first define Vilenkin groups and functions and study basic properties. We include some necessary preliminaries, in particular some classical inequalities, and we even present proofs of these inequalities by using a convexity approach, which we think is not presented in this form in some book before. After that, we investigate the classical theory of Lebesgue spaces and weak-Lp spaces and we state and prove Marcinkiewicz interpolation theorem. We study Dirichlet kernels and Lebesgue constants and prove two-sided estimates for them, which are very crucial to prove important results in Chap. 6. We study some classical conditions in the space of integrable functions, which provide pointwise convergence and convergence in L1 norm of partial sums. We also give some equivalent definitions of the modulus of continuity of Lp functions and give necessary and sufficient conditions concerning norm convergence of partial sums in L1 . In Chap. 2, we define martingales, and by using technique of martingale theory, we prove boundedness of maximal functions on Lebesgue spaces, from which it follows almost everywhere convergence of subsequences of partial sums with respect to Vilenkin systems. We also give the proof of Calderon-Zygmund decomposition theorem. Moreover, we prove an analogy of the Carleson-Hunt theorem with respect to Vilenkin systems. Finally, we prove an analogy of the Kolmogorov theorem and construct the integrable function whose Vilenkin-Fourier series diverges everywhere. It is also proved that for any 1 ≤ p ≤ ∞ and for any set with measure 0, there exists a function f ∈ Lp , whose Vilenkin-Fourier series diverges on this set. In Chap. 3, we first define Fejér means. We investigate Fejér kernels and study two-sided estimates for them. After that we prove almost everywhere convergence of Fejér means. Moreover, we define Lebesgue and Vilenkin-Lebesgue points of integrable functions and prove convergence of Fejér means of integrable functions with respect to Vilenkin systems in Vilenkin-Lebesgue points, which are almost everywhere points for any integrable function. We also study convergence in Lp norm of Fejér means. Moreover, some important upper and lower estimates of Fejér kernels are studied, which are very crucial to prove important results in Chap. 7. Moreover, we define what an approximate identity is and study some conditions, which provide convergence of convolution operators by approximate identity and Lp functions. In Chap. 4, we first define some summability methods, which are called Nörlund and T means and derive necessary and sufficient conditions which provide regularity of these summability methods. We prove new estimates for the kernels of these summability methods, which are very important to prove our main results in Chap. 8. Moreover, we study some conditions for Nörlund and T means from which it follows convergence in Lp norms, when p ≥ 1. Finally, we study almost everywhere convergence of Nörlund and T means in Lp spaces, when p ≥ 1.

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In Chap. 5, we first define martingales and give some basic results, which we use in the proofs of our main results. Moreover, we define martingale Hardy spaces and prove that they are isomorphic to Lebesgue spaces for p > 1. In the case when 0 < p ≤ 1 we prove some important theorems for this theory and also a crucial atomic decomposition result for these spaces, which, in particular, simplify our proofs of boundedness of some classical operators in Fourier analysis. Finally, we construct concrete martingales, which help us to prove sharpness of our main results in the later chapters. This technique to prove sharpness is fairly new and hopefully useful also far beyond the main scope of this book. Chapter 6 is devoted to present and prove some new and known results about Vilenkin-Fourier coefficients and partial sums of martingales in Hardy spaces. First, we show that the Fourier coefficients of martingales f ∈ Hp are not uniformly bounded when 0 < p < 1. By applying these results, we can prove some known Hardy and Paley type inequalities with a new more simple method. After that, we investigate partial sums with respect to the Vilenkin system and prove boundedness of weighed maximal operators of partial sums on the martingale Hardy spaces. Moreover, we derive necessary and sufficient conditions for the modulus of continuity for which norm convergence of the partial sums hold, and we present a new proof of a Hardy type inequality for it. We also investigate convergence and divergence rate of subsequences of partial sums with respect to Vilenkin systems in the martingale Hardy spaces for 0 < p < 1. Finally, we prove strong convergence results of partial sums with respect to Vilenkin systems. In Chap. 7, we investigate some analogous problems concerning the partial sums of Fejér means. First, we consider maximal operator Fejér means. Moreover, we investigate some weighted maximal operators of Fejér means and prove some boundedness results for them. After that, we apply these results to find necessary and sufficient conditions for the modulus of continuity for which norm convergence of Fejér means holds. Finally, we prove some new Hardy type inequalities for Fejér means, which are also called strong convergence results of Fejér means. We also prove sharpness of all our main results in this Chapter. In Chap. 8, we consider boundedness of maximal operators and weighted maximal operators of Nörlund and T means. After that, we prove some strong convergence theorems for these summability methods. Since Fejér, Cesàro, Riesz and Nörlund logarithmic means are examples of Nörlund and T means, both some well-known and new results can be pointed out. We also investigate Riesz and Nörlund logarithmic means simultaneously at the end of this Chapter. In Chap. 9, we consider Hardy spaces with variable exponents. Let p(·) be a measurable function defined on [0, 1) satisfying p− =: infx∈ p(x) > 0, supx∈ p(x) =: p+ < ∞ and the log-Hölder continuity condition. We investigate the martingale Hardy spaces Hp(·) and prove their atomic decompositions. Similarly to the spaces with a constant p, we obtain that the Hardy spaces Hp(·) are equivalent to the Lebesgue spaces Lp(·) if p− > 1. We generalize the classical results and show that the partial sums of the Vilenkin-Fourier series converge to the function in norm if f ∈ Lp(·) and p− > 1. The boundedness of the maximal Fejér operator

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on Hp(·) is proved whenever p− > 1/2 and the condition p1− − p1+ < 1 hold. One of the key points of the proof is that we introduce two new maximal operators and prove their boundedness on Lp(·) with p− > 1. As a consequence, we obtain theorems about almost everywhere and norm convergence of the Fejér means of the Vilenkin-Fourier series. For the readers’ convenience, we have also included an Appendix (Chap. 10) with some basic information about the dyadic group and the Walsh and Kaczmarz systems. Moreover, we prove some interesting results concerning summability with respect to the Walsh system, which are not known for Vilenkin systems. We also give a proof concerning boundedness of the maximal operator of Fejér means with respect to the Kaczmarz system and sharpness of this result. In particular, this fairly new method of proof, even of sharpness, in the case of Kaczmarz system can be very useful for researchers looking for the results for similar Walsh system and related systems.

How to Read the Book? Each chapter is divided into sections, similarly as the numbers of the formulas. Hence, for example 4.5 means the fifth Section of Chap. 4 and (4.3.6) means the sixth formula in Sect. 4.3 of Chap. 4. For the convenience of the reader, we have also added a list of symbols and notations at the end of the book. The three first chapters are basic including many definitions, basic theorems, etc. This part can be used as an introduction, e.g., in a course for PhD students and researchers with interest to broaden the knowledge in Fourier Analysis with new problems and thinking. Moreover, this part is basic and the best is to read it before to read the other six chapters, which, in their turn, can be read independent of each other. However, it is also possible to start to read any chapter independently by just on some places going back to limited and well-described information from some previous chapters. At the end of each chapter, there is a section called “Final Comments and Open Questions”, where we have collected some historical and other remarks, pointed out some relations to the theory of classical Fourier Analysis and raised a number of open questions. Especially, we hope that this interplay between classical and “modern” Fourier Analysis and some corresponding open questions will be very useful for a broad audience of readers and serve as a source of inspiration for further research in this fascinating area. Luleå, Sweden May 2022

Lars-Erik Persson George Tephnadze Ferenc Weisz

Acknowledgements

The second author wants to thank Shota Rustaveli National Science Foundation for the financial support within the frame of project no. FR-19-676. The third author was supported by the Hungarian National Research Development and Innovation Office—NKFIH, KH130426. We are very grateful to several researchers around the world (co-authors, colleagues, etc.) for various kinds of contributions and support, which essentially have contributed to improve the quality of this book. As typical examples of researchers in this “supporting team”, we want to mention Roland Duduchava, Hans-Georg Feichtinger, Natasha Samko, Ferenc Schipp, Zurab Vashakidze, Georgi Tutberidze and Davit Baramidze. We also thank Dr. Hana Turcinová for permitting us to use her photo of the magic Nordic light we chosen as a symbol and which is taken exactly at the place (“Hotel Infinity”) where we finalized the book. Finally, and most important, our most cordial thanks go to our wonderful families for their patience, encouragement, support and love during all our work with this book.

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Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Vilenkin Groups and Functions . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 The Representation of the Vilenkin Groups on the Interval [0,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Convex Functions and Classical Inequalities . . .. . . . . . . . . . . . . . . . . . . . 1.5 Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Dirichlet Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7 Lebesgue Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8 Vilenkin-Fourier Coefficients . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.9 Partial Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.10 Final Comments and Open Questions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 5 11 15 25 34 43 52 56 67

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Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-Fourier Series . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 71 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 71 2.2 Conditional Expectation Operators . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 74 2.3 Martingales and Maximal Functions . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 79 2.4 Calderon-Zygmund Decomposition . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 89 2.5 Almost Everywhere Convergence of Vilenkin-Fourier Series .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 92 2.6 Almost Everywhere Divergence of Vilenkin-Fourier Series .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 106 2.7 Final Comments and Open Questions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 117

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Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Vilenkin-Fejér Kernels . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Approximation of Vilenkin-Fejér Means . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Almost Everywhere Convergence of Vilenkin- Fejér Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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3.5 3.6 4

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6

Approximate Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147 Final Comments and Open Questions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 151

Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Well-Known and New Examples of Nörlund and T Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Regularity of Nörlund and T Means . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Kernels of Nörlund Means . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Kernels of T Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Norm Convergence of Nörlund and T Means in Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 Almost Everywhere Convergence of Nörlund and T Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.8 Convergence of Nörlund and T Means in Vilenkin-Lebesgue Points . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.9 Riesz and Nörlund Logarithmic Kernels and Means . . . . . . . . . . . . . . . 4.10 Final Comments and Open Questions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Theory of Martingale Hardy Spaces . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Martingale Hardy Spaces and Modulus of Continuity . . . . . . . . . . . . . 5.3 Atomic Decomposition of the Martingale Hardy Spaces Hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Interpolation Between Hardy Spaces Hp . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Bounded Operators on Hp Spaces . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6 Examples of p-Atoms and Hp Martingales . . . .. . . . . . . . . . . . . . . . . . . . 5.7 Final Comments and Open Questions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Vilenkin-Fourier Coefficients and Partial Sums in Martingale Hardy Spaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Estimations of Vilenkin-Fourier Coefficients in Hp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Hardy and Paley Type Inequalities in Hp Spaces . . . . . . . . . . . . . . . . . . 6.4 Maximal Operators of Partial Sums on Hp Spaces . . . . . . . . . . . . . . . . 6.5 Convergence of Partial Sums in Hp Spaces . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Convergence of Subsequences of Partial Sums in Hp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.7 Strong Convergence of Partial Sums in Hp Spaces . . . . . . . . . . . . . . . . 6.8 Final Comments and Open Questions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

157 157 162 166 169 195 208 211 218 227 233 237 237 238 241 246 259 262 276 279 279 282 285 288 299 304 317 325

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8

9

Vilenkin-Fejér Means in Martingale Hardy Spaces . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces . . . . . . . 7.3 Convergence of Vilenkin-Fejér Means in Hp Spaces . . . . . . . . . . . . . . 7.4 Convergence of Subsequences of Vilenkin-Fejér Means in Hp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5 Strong Convergence of Vilenkin-Fejér Means in Hp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.6 Final Comments and Open Questions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Maximal Operators of Nörlund Means on Hp Spaces .. . . . . . . . . . . . 8.3 Maximal Operators of T Means on Hp Spaces . . . . . . . . . . . . . . . . . . . . 8.4 Strong Convergence of Nörlund Means in Hp Spaces . . . . . . . . . . . . . 8.5 Strong Convergence of T Means in Hp Spaces . . . . . . . . . . . . . . . . . . . . 8.6 Maximal Operators of Riesz and Nörlund Logarithmic Means on Hp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.7 Strong Convergence of Riesz and Nörlund Logarithmic Means in Hp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.8 Final Comments and Open Questions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Variable Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Doob’s Inequality in Variable Lebesgue Spaces . . . . . . . . . . . . . . . . . . . 9.4 The Maximal Operator Us . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.5 The Maximal Operator Vα,s . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6 Variable Martingale Hardy Spaces . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.7 Atomic Decomposition of Variable Hardy Spaces .. . . . . . . . . . . . . . . . 9.8 Martingale Inequalities in Variable Spaces . . . . .. . . . . . . . . . . . . . . . . . . . 9.9 Partial Sums of Vilenkin-Fourier Series in Variable Lebesgue Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.10 The Maximal Fejér Operator on Hp(·) . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.11 Final Comments and Open Questions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

10 Appendix: Dyadic Group and Walsh and Kaczmarz Systems . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Walsh Group and Walsh and Kaczmarz Systems . . . . . . . . . . . . . . . . . . 10.3 Estimates of the Walsh-Fejér Kernels . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4 Walsh-Fejér Means in Hp . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.5 Modulus of Continuity in Hp and Walsh-Fejér Means . . . . . . . . . . . . 10.6 Riesz and Nörlund Logarithmic Means in Hp .. . . . . . . . . . . . . . . . . . . . 10.7 Maximal Operators of Kaczmarz-Fejér Means on Hp . . . . . . . . . . . . .

xv

331 331 333 358 364 376 389 397 397 399 430 439 449 455 471 473 481 481 482 485 490 498 504 506 515 518 520 534 539 539 540 546 550 559 564 577

xvi

Contents

10.8 Modulus of Continuity in Hp and Kaczmarz-Fejér Means . . . . . . . . 595 10.9 Final Comments and Open Questions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 602 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 605 Notations . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 623 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 625

Chapter 1

Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

1.1 Introduction The fact that the Walsh system is the group of characters of a compact abelian group connects Walsh analysis with abstract harmonic analysis. It was discovered independently by Fine [96] and Vilenkin [389]. Later on, in 1947 Vilenkin [389– 391] actually introduced a large class of compact groups (now called Vilenkin groups) and the corresponding characters which includes the dyadic group and the Walsh system as a special case. For general references to harmonic analysis on groups see Pontryagin [279], Rudin [283], and Hewitt and Ross [163]. In particular, Vilenkin investigated the group Gm , which is a direct product of the additive groups Zmk =: {0, 1, . . . , mk − 1} of integers modulo mk , where m =: (m0 , m1 , . . .) are positive integers not less than 2, and introduced the Vilenkin systems {ψj }∞ j =0 . These systems include as a special case the Walsh system and many of the proofs presented for the Walsh system can be generalized readily to the Vilenkin case. It is well-known (see e.g. the books [3] and [295]) that if f ∈ L1 (Gm ) and the Vilenkin series T (x) =

∞ 

Cj ψj (x)

j =0

converges to f in L1 -norm, then  Cj = Gm

f ψ j dμ =: f(j ) , j = 0, 1, 2, . . . ,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 L.-E. Persson et al., Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series, https://doi.org/10.1007/978-3-031-14459-2_1

1

2

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

where Cj is called the j -th Vilenkin-Fourier coefficient and μ is the Haar measure on locally compact abelian groups Gm . This means that in this case the approximation series must be a Vilenkin-Fourier series: f ∼

∞ 

f(j ) ψj .

j =0

According to Riemann-Lebesgue lemma for Vilenkin system (for details see e.g. the books [3] and [295]) it is well-known that the Fourier coefficients of integrable functions vanish, that is f(k) → 0, as k → ∞, for each f ∈ L1 . Moreover, for f ∈ L1 (Gm ) and MN ≤ n ≤ MN+1 ,   f(n) ≤ cω1 (1/MN , f ) , where ω1 (δ, f ) is defined by ω1 (δ, f ) =: sup f (· − h) − f (·)1 . |h|≤δ

The classical theory of Hilbert spaces (for details see e.g the books [328, 378]) says that if we consider the partial sums Sn f =:

n−1 

f(k) ψk

k=0

with respect to any orthonormal systems and among them to Vilenkin systems, then Sn f 2 ≤ f 2 . It follows that for every f ∈ L2 , Sn f − f 2 → 0, as n → ∞. Moreover, the so called Bessel‘s equality (Parseval’s identity) asserts that for every L2 (Gm ), n    f(i)2 = f 2 . 2 i=1

It is well-known that (see e.g. [3] and [146]) Vilenkin systems do not form bases in the spaces L1 and L∞ . It follows that the boundedness does not hold for p = 1

1.1 Introduction

3

 and p = ∞. In these cases the Dirichlet kernel Dn defined by Dn =: n−1 k=0 ψk , plays a central role to study boundedness of partial sums from the space L1 (Gm ) to the space L1 (Gm ) and from the space L∞ (Gm ) to the space L∞ (Gm ). It is known that for every Vilenkin system, the Lebesgue constant Ln , defined by Ln =: Dn 1 satisfy the inequality Ln ≤ c log n. For some concrete systems it is possible to write two-sided estimations of the Lebesgue constants Lnk . In particular, for every bounded Vilenkin system Lukyanenko [218] proved two-sided estimates for the Lebesgue constants Lnk for some concrete indices nk ∈ N. Lukomskii [217] generalized this result and proved two-sided estimates for the Lebesgue constants Ln without the conditions  on the indexes. In particular, he showed that for every natural number n = ∞ j =0 nj Mj (M0 =: 1, Mk+1 =: mk Mk k ∈ N) and every bounded Vilenkin system we have the following two-sided estimates of Lebesgue constants: 1 1 3 1 v (n) + v ∗ (n) + ≤ Ln ≤ v (n) + 4v ∗ (n) − 1, 4λ λ 2λ 2

(1.1)

where v and v ∗ are defined by ∞    δj +1 − δj  + δ0 , v (n) =:

∗

v (n) =:

j =1

∞ 

δj∗ ,

(1.2)

j =1

with     δj = sign nj = sign nj

  and δj∗ =  nj − 1 δj .

(1.3)

In [44], the upper bound in (1.1) was improved and it was also proved a new similar lower bound by using a completely different new method. It is also known (see [44]) that any subsequence Snk is bounded from L1 (Gm ) to 1 L (Gm ) and from L∞ (Gm ) to L∞ (Gm ) if and only if nk has uniformly bounded variation, which means that   sup v (n) + v ∗ (n) < C < ∞. n∈N

As a corollary of this, the subsequence SMn of partial sums is bounded from the Lebesgue space L1 (Gm ) to the Lebesgue space L1 (Gm ) and from the Lebesgue space L∞ (Gm ) to the Lebesgue space L∞ (Gm ). Moreover, for any n ∈ N, 1 ≤ p ≤ ∞ and f ∈ Lp (Gm ), we have that SM f ≤ f p n p

4

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

and for any 1 ≤ p < ∞ and f ∈ Lp (Gm ), SMn f − f → 0, as n → ∞. p It follows that the Vilenkin polynomials are dense in the Lebesgue spaces Lp (Gm ), for all 1 ≤ p < ∞, which is very important to obtain almost everywhere convergence of sublinear operators of weak-(1,1) type. Onneweer [257] showed that if the modulus of continuity of f ∈ L1 satisfies the condition 

1 , as δ → 0, ω1 (δ, f ) = o (1.4) log (1/δ) then its Vilenkin-Fourier series converges in L1 -norm. He also proved that condition (1.4) can not be improved. Localization principle plays a central role to study pointwise convergence of partial sums with respect to Vilenkin systems. In particular, this result says that convergence of partial sums of Fourier series of a function f at the point x0 depend only on the neighborhood of x0 , since it is proved that for f ∈ L1 (Gm ), any x0 ∈ Gm and for some N ∈ N,  lim f (t) Dn (x0 − t) dt = 0. n→∞

Gm \IN (x0 )

From this it follows that if f, g ∈ L1 and f = g for some IN (x), then Sn f (x) and Sn g (x) converges or diverges simultaneously for every x ∈ IN (x). Moreover, Dini‘s test holds for any x ∈ Gm and f ∈ L1 (Gm ). In particular, if we define the function g by g (t) =:

f (t) − f (x) , x ∈ Gm , t ∈ Gm \{x}, f ∈ L1 (Gm ) |x − t|

and we have g ∈ L1 (Gm ), then Sn f (x) → f (x) , n → ∞. It follows that if f ∈ L1 (Gm ) and

|f (x − t) − f (x)| = O (log (1/|t|))−1−ε , ε > 0, t → 0, then Sn f (x) → f (x) , n → ∞.

1.2 Vilenkin Groups and Functions

5

1.2 Vilenkin Groups and Functions Denote by N+ the set of the positive integers, N =: N+ ∪ {0}, Z the set of the integers, R the real numbers, R+ the positive real numbers and C the complex numbers. Let m =: (m0 , m1 , . . .) be a sequence of positive integers not less than 2. Denote by Zmk =: {0, 1, . . . , mk − 1} the additive group of integers modulo mk . Define the group Gm as the complete direct product of the groups Zmk with the product of the discrete topologies of Zmk . The direct product μ of the measures μk (j ) =: 1/mk

(j ∈ Zmk )

is the Haar measure on Gm with μ (Gm ) = 1. If supn∈N mn < ∞, then we call Gm a bounded Vilenkin group. If the generating sequence m is not bounded, then Gm is said to be an unbounded Vilenkin group. In this book we discuss only bounded Vilenkin groups, i.e. the case when supn∈N mn < ∞. The elements of Gm are represented by sequences   x =: x0 , x1 , . . . , xj , . . .



 xj ∈ Zmj .

It is easy to give a base for the neighborhoods of Gm : I0 (x) : = Gm , In (x) : = {y ∈ Gm | y0 = x0 , . . . , yn−1 = xn−1 }

(x ∈ Gm , n ∈ N) .

We call subsets In (x) ⊂ Gm Vilenkin intervals. Let en =: (0, . . . , 0, xn = 1, 0, . . .) ∈ Gm

(n ∈ N) .

If we define In =: In (0) for n ∈ N and In =: Gm \ In , denotes the complement of In , then IN =

N−1  s=0

Is \Is+1 =

N−2 N−1  

INk,l k=0 l=k+1





N−1  k=1

 INk,N

,

(1.5)

6

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

where

INk,l

⎧ ⎪ ⎪ IN (0, . . . , 0, xk = 0, 0, . . . , 0, xl = 0, xl+1 , . . . , xN−1 , . . .), ⎨ for k < l < N, =: ⎪ (0, . . . , 0, xk = 0, xk+1 = 0, . . . , xN−1 = 0, xN , . . .), I N ⎪ ⎩ for l = N.

If we define the so-called generalized number system based on m in the following way: M0 =: 1, Mk+1 =: mk Mk (k ∈ N), then every n ∈ N can be uniquely expressed as n=

∞ 

nj Mj ,

j =0 

where nj ∈ Zmj (j ∈ N+ ) and only a finite number of nj ‘s differ from zero. The Vilenkin group is metrizable with the following metric: ρ (x, y) =: |x − y| =:

∞  |xk − yk |

Mk+1

k=0

For natural numbers n = n∗ =

∞  

∞

,

and k =

j =0 nj Mj

 mj nj Mj ,

(x, y ∈ Gm ) . ∞

j =0 kj Mj

we define

nj ∈ Zmj (j ∈ N),

j =0

k =: n+

∞ 

(ni ⊕ ki )Mi

i=0

and k =: n−

∞ 

(ni ki )Mi ,

i=0

where ai ⊕ bi =: (ai + bi )modmi , and is the inverse operation for ⊕.

ai , bi ∈ Zmi

(1.6)

1.2 Vilenkin Groups and Functions

For the natural number n =

7

∞

j =0 nj Mj

let us define the following numbers

n =: min{j ∈ N, nj = 0} and |n| =: max{j ∈ N, nj = 0}, ρ (n) =: |n| − n

(1.7)

and ∞    δj +1 − δj  + δ0 , v (n) =:

∗

v (n) =:

j =1

∞ 

δj∗ ,

j =1

where δj and δj∗ are defined by (1.3). By the definition, that is M|n| ≤ n < M|n|+1 and v(n) ≤ ρ(n) + 2. Next, we introduce on Gm an orthonormal system, which is called the Vilenkin system. At first, we define the complex-valued functions rk (x) : Gm → C, the so called generalized Rademacher functions, by

ı 2 = −1, x ∈ Gm , k ∈ N .

rk (x) =: exp (2πıxk /mk ) ,

(1.8)

Now, define the Vilenkin system ψ =: (ψn : n ∈ N) on Gm by: ψn (x) =:

∞ 

rknk (x) , (n ∈ N) .

(1.9)

k=0

Specifically, we call this system the Walsh-Paley system when m ≡ 2. Proposition 1.2.1 Let n ∈ N. Then |ψn (x)| = 1, ψn (x + y) = ψn (x) ψn (y) , ψn (−x) = ψn∗ (x) = ψ n (x) , ψn+  k (x) = ψk (x) ψn (x) , (k, n ∈ N, x, y ∈ Gm ) . Proof The proof is simple and it is based only on the definition of Vilenkin functions. Hence, we leave out the details.  A character on a commutative group I is a continuous complex-valued function which satisfies f (x + y) = f (x) f (y)

and

|f (x)| = 1,

for all x, y ∈ I. Let us denote by Ithe set of all character functions of I.

8

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Remark 1.2.2 I is a commutative group with ordinary multiplication of complex valued functions (fg) (x) = f (x) g (x). Proof The proof is simple and we leave out the details.



Remark 1.2.3 The Vilenkin functions are characters of Gm . Therefore every character has the form (1.9). Proof By using Proposition 1.2.1 we immediately can conclude that every Vilenkin function is a character of the group Gm . Now we shall prove that all characters have the form (1.2.1). By the definition |f (0)| = 1 = 0 and f (0) = f (0 + 0) = f (0) · f (0) = f 2 (0) . From these two equalities it follows that f (0) = 1. Since lim |ek − 0| = 0,

k→∞

by applying the continuity of characters, we find that lim f (ek ) = 1.

k→∞

(1.10)

On the other hand, ⎞ mk −times    f mk (ek ) = f ⎝ek + . . . + ek ⎠ = f (0) = 1, for all k ∈ N. ⎛

(1.11)

This equation has mk different solutions and for bounded Vilenkin groups we also have finite different mk s. It means that f (ek ) take finite different values, for all k ∈ N. According to (1.10) we get that f (ek ) = 1, for all k > N0 . In view of (1.11) there exists a sequence nk ∈ Zmk such that f (ek ) = exp (2πınk /mk ) , k ∈ N, where nk = 0 for k > N0 . We define n =:

∞  i=0

ni Mi =

N0  i=0

ni Mi .

1.2 Vilenkin Groups and Functions

9

By the definition f (ek ) = exp (2πınk /mk ) , k ∈ N. On the other hand, ψn (ek ) = exp (2πınk /mk ) , k ∈ N. Hence, f (ek ) = ψn (ek ) , Since x =

∞ 

(k ∈ N).

xi ei , for any x ∈ Gm , if we follow the analogical steps described

i=0

above we get that f (x) = ψn (x) , for all x ∈ Gm . 

The proof is complete. The direct product μ of the measures (j ∈ Zmk )

μk ({j }) =: 1/mk

is the Haar measure on Gm with μ (Gm ) = 1. Translation of a subset In (x) ∈ Gm by y is defined by τy (In (x)) = {In (x) + y}. Since μ is a product measure we get that μ (In (x)) =

n−1 

(μk {xk })

k=0

=

n−1  k=0

∞ 

(μk {0, . . . , mk − 1})

k=n ∞ 1  1 1= . mk Mn k=n

On the other hand,   μ τy (In (x)) = μ (In (x + y)) = 1/Mn for all y ∈ Gm . Hence,   μ τy (In (x)) = μ (In (x)) .

10

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

In particular, Gm is a locally compact abelian group and it can be equipped with the Haar measure (see Pontryagin [279]), which coincides with the product measure μ. We shall not construct the Haar measure here because for the groups we consider, the construction is trivial. Proposition 1.2.4 Let n, k ∈ N. Then 

 ψn dμ = Gm

1 n = 0, 0 n = 0.

Moreover, the Vilenkin systems are orthonormal, that is, 

 ψn ψk dμ = Gm

1 n = k, 0 n = k.

Proof It is evident that 

 1dμ (x) = 1.

ψ0 (x) dμ (x) = Gm

Gm

Let n = 0. Then there exists y ∈ Gm , such that ψn (y) = 1 and 

 ψn (x) dμ (x) = Gm

ψn (x + y) dμ (x + y) 

Gm

=

 ψn (x + y) dμ (x) = ψn (y)

ψn (x) dμ (x) .

Gm

Gm

Hence,  (1 − ψn (y))

ψn (x) dμ (x) = 0. Gm

Since 1 − ψn (y) = 0 we obtain that  ψn (x) dμ (x) = 0. Gm

k ∗ = 0 ⇐⇒ n = k. Let n = k. By On the other hand, it is easy to show that n+ using Proposition 1.2.1 we find that 

 ψn (x) ψk (x) dμ (x) = Gm

Gm

ψn+  k ∗ (x) dμ (x)

 =

 1dμ (x) = 1.

ψ0 (x) dμ (x) = Gm

Gm

1.3 The Representation of the Vilenkin Groups on the Interval [0,1)

11

Let n = k. Then ψn+  k ∗ (x) = ψ0 (x) and 

 ψn (x) ψk (x) dμ (x) = Gm

Gm

ψn+  k ∗ (x) dμ (x) = 0. 

The proof is complete.

1.3 The Representation of the Vilenkin Groups on the Interval [0,1) Let us define by Qm the set of rational numbers of the form pMn−1 , where 0 ≤ p ≤ Mn − 1 for some p ∈ N and n ∈ N. Any x ∈ [0, 1) can be written in the form x=

∞  xk , Mk+1

(1.12)

k=0

where each xk ∈ Zmk . For each x ∈ [0, 1) \ Qm there is only one expression of this form. When x ∈ Qm there are two expressions of this form, one terminates in 0’s and one does not terminate in 0’s. By expansion (1.12) of an element x ∈ Qm we shall mean the one which terminates in 0’s. Let G0 =: {x ∈ Gm : xn → 0, as n → ∞} and G∗0 =: {x ∈ Gm : x = y ∗ for some y ∈ G0 }. Obviously, the measure of G∗0 is 0. Define Fine’s map : [0, 1) → Gm by (x) =: (x0 , x1 , . . . ),

(1.13)

where x has expansion (1.12). Then is a bijective map from [0, 1) onto Gm \ G∗0 . The analogues of the Rademacher and Vilenkin functions are given by υn (x) = exp (2πıxk /mk )

(n ∈ N)

and φn =:

∞  k=0

υknk

(n ∈ N) ,

(1.14)

12

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

where x ∈ [0, 1) has the expansion (1.12) and n=

∞ 

nj Mj .

j =0

Comparing this with (1.8) and (1.9), we can see that υn = rn ◦

and φn = ψn ◦

on [0, 1)

and rn (x) = υn (|x|)

and ψn (x) = φn (|x|)

for x ∈ Gm \ G∗0 ,

where |x| was defined in (1.6). Fine’s map can be used to define a new addition and a new topology on [0, 1) which are closely related to those on Gm . Indeed, define the sum of two numbers x, y ∈ [0, 1) by ˙ = | (x) + (y)|. x +y In terms of the expansion (1.12) of x and y, we have ˙ = x +y

∞ 

−1 |xk + yk | Mk+1

k=0

˙ is evidently a commutative binary operation on [0, 1). Note that [0, 1) is Hence + ˙ We denote by − ˙ the inverse operation to +. ˙ not a group under +. These functions almost behave like characters with respect to dyadic addition, namely, ˙ = φn (x)φn (y) φn (x +y)

˙ ∈ (n ∈ N, x, y ∈ [0, 1), x +y / Qm ).

To prove (1.15) fix m ∈ N and x, y ∈ [0, 1). Notice that υm (x)υm (y) = (rm ◦ (x)) (rm ◦ (y)) = rm ( (x) + (y)) and that ˙ = rm ( (| (x) + (y)|)) . υm (x +y) It is clear that ˙ ∈ (| (x) + (y)|) = (x) + (y) for x +y / Q2 .

(1.15)

1.3 The Representation of the Vilenkin Groups on the Interval [0,1)

13

Consequently, (1.15) holds for the Rademacher case, i.e., for n = Mk . But the general case follows immediately since Vilenkin functions are finite products of Rademacher functions. Since for each fixed y ∈ [0, 1) the set of points x which ˙ ∈ Qm is a countable set, we observe that (2.6) holds for a.e. x, y ∈ satisfy x +y [0, 1). By a Vilenkin interval in [0, 1) we always mean an interval of the form

 I (p, n) =: pMn−1 , (p + 1)Mn−1 , where 0 ≤ p < Mn , n, p, ∈ N. Clearly, the topology is generated by the collection of Vilenkin intervals. Moreover, each Vilenkin interval is both open and closed in this topology. It follows that each Vilenkin function is continuous in this topology. Thus this topology differs from the usual topology in an essential way. For each x ∈ [0, 1) and n ∈ N we shell denote the Vilenkin interval of length Mn−1 which contains x by In (x). Thus In (x) =: I (p, n), where 0 ≤ p < Mn is uniquely determined by the relationship x ∈ I (p, n). This is the same notation used for Vilenkin intervals on the group but will not cause problems because the context will make it clear whether the Vilenkin interval is in the group or inside the unit interval. The measure μ on the group Gm induces the Lebesgue measure λ on the interval [0, 1), i.e., λ (I (p, n)) = 1/Mn

and λ ([0, 1)) = 1.

If we define translation of a subset I (p, n) ⊂ [0, 1) by τy (I (p, n)) = {I (p, n) + y}, then   λ τy (I (p, n)) = λ (I (p, n)) , for all y ∈ [0, 1). The following proposition is easy to prove (c.f. Proposition 1.2.4) Proposition 1.3.1 Let n, k ∈ N. Then 

 [0,1)

φn dλ =

1 n = 0, 0 n = 0.

14

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Moreover, the Vilenkin systems on [0.1) are orthonormal, that is, 

 [0,1)

φn φk dλ =

1 n = k, 0 n = k.

Paley’s Lemma on [0, 1) has the following form: Lemma 1.3.2 (Paley’s Lemma on [0.1)) Let n ∈ N and DMn =:

M n −1

φj .

j =0

Then  DMn (x) =

Mn x ∈ [0, 1/Mn ) , 0 x ∈ [1/Mn , 1) .

Proof By applying the orthonormality of the Vilenkin system (see Proposition 1.2.4), we get  0

M n −1    DM (t)2 dλ (t) = n

1

j =0

=

1

φj (t) φj (t) dλ (t)

M n −1  1  j =0

(1.16)

0

1dλ (t) = Mn .

0

It is easy to show that φk (x) = 1, k = 0, 1, . . . , Mn − 1 and DMn (t) = Mn , for x ∈ In . Hence, 

 DM (t)2 dλ (t) = n

1 0

 

1/Mn 0 1/Mn

= 0

  DM (t)2 dλ (t) + n  Mn2 dλ (t) +



= Mn +

1

1/Mn

1 1/Mn



1 1/Mn

1/Mn

  DM (t)2 dλ (t) n

  DM (t)2 dλ (t) n

  DM (t)2 dλ (t) . n

By combining (1.16) and (1.17) we have that 

1

  DM (t)2 dλ (t) = 0. n

(1.17)

1.4 Convex Functions and Classical Inequalities

15

  Since DMn (t) is a step function on intervals, we obtain that DMn (t) = 0, for [1/Mn , 1). The proof is complete.  For simplicity, in what follows we will write rn and ψn instead of υn and φn , respectively (see Chap. 9).

1.4 Convex Functions and Classical Inequalities Let I denote a finite or infinite interval on R+ . We say that a function f is convex on I if, for 0 < α < 1, and all x, y ∈ I, f ((1 − α)x + αy) ≤ (1 − α)f (x) + αf (y). If this inequality holds in the reversed direction, then we say that the function f is concave. Examples of convex functions are f (x) = ex , x ∈ R, f (x) = x a , x ≥ 0, a ≥ 1 or a < 0, and (1 + x p )1/p , x ≥ 0, p > 1. The following useful estimates are more or less easy consequences of the convexity (concavity) of the function f (x) = x p : Example 1.4.1 Let a1 , a2 , . . . , an be positive numbers. Then n 

(a)

p ai

≤

 n 

i=1

p−1

(b) n

p ai

≤ np−1

i=1

n 

 p ai

≤

i=1

n  i=1

n 

p

ai , p ≥ 1,

i=1

p ai

≤

n 

p

ai , 0 < p ≤ 1.

i=1

The next example is a consequence of the convexity of the function f (x) = ex . Example 1.4.2 (Young’s Inequality) For any a, b > 0, p, q ∈ R \ {0}, p1 + it yields that

1 q

= 1,

bq ap + , if p > 1, p q

(1.18)

bq ap + , if 0 < p < 1. p q

(1.19)

ab ≤ and ab ≥

16

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Proof By using convexity of the function f (x) = ex , we get that 1

ab = eln ab = e p

ln a p + q1 lnbq

≤

1 ln a p 1 ln bq 1 1 e + e = a p + bq . p q p q 

The proof of (1.4.2) is similar so the proof is complete.

Example 1.4.3 (Two Fundamental Inequalities) The following inequalities are "fundamental relations": If x > 0 and α ∈ R, then 

x α − αx + α − 1 ≥ 0 for α > 1 and α < 0 x α − αx + α − 1 ≤ 0 for 0 < α < 1.

(1.20)

Proof Inequalities in (1.20) follows directly from the fact that the function f (x) = x α is convex for α > 1 and α < 0 and concave for 0 < α < 1. The proof is complete.  Remark 1.4.4 In particular, several well-known inequalities follow directly from (1.20) e.g. the AG-inequality, Hölder’s inequality, Minkowski’s inequality, etc. We also note that Bennett’s inequalities (even in more precise form) in interpolation theory follows from (1.20). p Moreover, (1.18) follows directly from (1.20) applied with x = abq and α = p1 (the case 0 < α < 1) and (1.19) follows from (1.20) in the same way by instead applying (1.20) in the cases α > 1 and α < 0. We use different strategy and prove every classical inequalities by using Jensen’s inequality. Here we state Jensen’s inequality in the following fairly general form: Theorem 1.4.5 (Jensen’s Inequality) Let μ be a positive measure on a σ −algebra ℵ in a set  so that μ() = 1. If f is a real μ−integrable function, −∞ ≤ a < f (x) < b ≤ ∞ for all x ∈  and if  is convex on (a, b), then







f dμ ≤

 

(1.21)

(f )dμ. 

If  is concave, then (1.21) holds in the reversed direction. Proof Since  is convex, at each x0 ∈ R, there exist a, b ∈ R, such that (x0) = ax0 + b and (x) ≥ ax + b, ∀x ∈ R, (here, y = ax + b defines a supporting plane of the epigraph of  at x0 ). Let x0 =  f dμ, then we have



    f dμ = (x0) = ax0+b = a f dμ + b = (af + b) dμ ≤ (f )dμ.

 







If we apply concavity, analogously we can prove reversed direction of inequality (1.21). The proof is complete. 

1.4 Convex Functions and Classical Inequalities

17

Remark 1.4.6 In real analysis, we may require an estimate of





b



f (x) dμ(x) , a

where a, b ∈ R,  is convex and f : [a, b] → R is a non-negative μ-integrable function and μ is a positive measure. In this case, the Lebesgue measure of [a, b] need not to be unity. However, by substitution, the interval can be rescaled so that it has measure 1. Then Jensen’s inequality can be applied to get related estimate



1 b−a



b

a



1 f dμ ≤ b−a



b

 (f ) dμ. a

Remark 1.4.7 If  = R+ , n = 2, 3, . . . , μ =

n 

λk δk

k=1

 (δk is the unity mass at t = k), λk > 0 and nk=1 λk = 1, then Jensen’s inequality (1.21) coincides with discrete Jensen’s inequality with f (k) = xk . In particular, for a real convex function , numbers x1 , x2 , . . . , xn in its domain, and positive weights ai Jensen’s inequality can be stated as  n

n λk (xk ) k=1 λk xk n ≤ k=1  n k=1 λk k=1 λk

(1.22)

and the inequality is reversed if ψ is concave, which is

n  n λk (xk ) k=1 λk xk n n  . ≥ k=1 k=1 λk k=1 λk

(1.23)

Equality holds if and only if x1 = x2 = · · · = xn or  is linear on a domain containing x1 , x2 , · · · , xn . As a particular case, if the weights ai are all equal, then (1.22) and (1.23) become

n  n (xk ) k=1 xk  (1.24) ≤ k=1 n n and

n  respectively.

k=1 xk

n

n

 ≥

k=1 (xk )

n

,

(1.25)

18

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Remark 1.4.8 Conversely, if we put the mass 1 − λ at x and λ at y (x, y ∈ I ) and assume that (1.21) holds for a positive function , then, we have that ((1 − λ)x + λy) ≤ (1 − λ)(x) + λ(y), i.e. the function  is convex. These considerations show in fact that Jensen’s inequality is more or less equivalent to the notion of convexity. Example 1.4.9 For instance, the function log x is concave, so substituting (x) = log x in the previous formula (1.25) establishes the (logarithm of the) familiar arithmetic-mean/geometric-mean inequality:

n

k=1 xk n

log

" # n !n # log( log x x ) k k k=1 k=1 n = = log $ xk n n

n

 ≥

k=1

so that " # n # x k=1 k n ≥$ xk . n

n

k=1

Theorem 1.4.10 (Hölder’s inequality) Let p > 1 and 1/p 



 |fg|dμ ≤

+

1 q

|f | dμ

|g| dμ q



= 1. Then

1/q

p



1 p

(1.26)

.



For the case 0 < p < 1 (1.26) holds in the reverse direction. Proof The standard proof of (1.26) is obtained by just applying Young’s inequality (1.18) with a = f (x), b = g(x) and integrating. Another proof showing that (1.26) follows directly from Jensen’s inequality reads: Without loss of generality we may assume that 0 <  |g|dμ < ∞ and apply Jensen’s inequality to obtain that

1 |g|dμ 

p

 |fg|dμ

 ≤



|g|dμ

−1 



|f |p |g|dμ 

i.e. that 1−1/p 



 |fg|dμ ≤ 

1/p

|g|dμ 

|f | |g|dμ p



.

1.4 Convex Functions and Classical Inequalities

19

Put |f ||g|1/p = |f1 | and |g|1/q = |g1 | and we find that





1/p 

|f1 g1 |dμ ≤

1/q

|f1 |p dμ



|g1 |q dμ



.





We just change notation and (1.26) is proved.

Remark 1.4.11 We have equality in Hölder’s inequality when g(x) = (f (x))p−1 . In particular, this means that the following important relation 1/p

 |f |p dμ

 = sup



|f |ϕdμ,

(1.27)



holds for each p > 1, where the supremum is taken over all ϕ ≥ 0 such that q  ϕ dμ = 1. For the case 0 < p < 1 (1.27) holds with "sup" replaced by "inf". The investigations above show that (1.26) can be generalized to a version with finite many functions f1 , f2 , . . . , fn involved: Example 1.4.12 Let p1 , p2 , . . . , pn , n = 3, 4, . . . , be positive numbers such that 1 1 1 p1 + p2 + · · · + pn = 1. Then



 |f1 f2 · · · fn |dμ ≤ 

|f1 |p1 dμ

1/p1



 ···

|fn |pn dμ

1/pn .

(1.28)



By stating the following further generalization (with infinite many functions involved), we can prove an inequality which sometimes is called a “continuous form ” of Hölder’s inequality. Theorem 1.4.13 Let K(x, y) be positive and measurable on (1 × 2 , μ × ν) , where dν = 1. Then 2

 1

⎛ ⎜ exp ⎝



2

⎞ ⎟ log K(x, y)dν ⎠ dμ ≤ exp



2

⎛ ⎞  ⎜ ⎟ log ⎝ K(x, y)dμ⎠ dν.

(1.29)

1

Here, and in the sequel, we assume that (1 , μ) and (2 , μ) are two finite measure spaces with non-negative weights. Remark 1.4.14 The proof of (1.29) can be performed by using Jensen’s inequality and the Fubini theorem in a suitable way. However, we will present a new proof later on by just using the fact that (1.29) is a limit inequality of another useful continuous

20

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

integral inequality (see Remark 1.4.20). By applying (1.29) with  2 = Y1 ∪ Y2 ∪ . . . ∪ Yn , K(x, y) = |f pi (x)| on Yi and

dν = Yi

1 , i = 1, 2, . . . , n, pi

we get (1.28). A special case of (1.29) was crucial when an interpolation theory for infinite many spaces was created. Theorem 1.4.15 (Minkowski’s Inequality) If p > 1, then 1/p

 |f + g| dμ



1/p

≤

p



|f | dμ



1/p

+

p

|g| dμ p



(1.30)

.



Proof The standard proof of (1.30) is to use Hölder’s inequality but here we present another proof based on the (quasi-linearization) formula (1.27). It holds that q = p/(p − 1)



1/p |f + g| dμ

 =

p



|f + g|ϕdμ

sup ϕ

q dμ=1





≤ sup

ϕq =1 

 |f |ϕdμ +



q   ϕ dμ=1

1/p

=

|f | dμ

 +

p

|g|ϕdμ

sup



1/p

|g| dμ p

.





The proof is complete.

This proof is easy to generalize to obtain the following more general continuous version of (1.30): Theorem 1.4.16 (Minkowski’s Integral Inequality) Let the positive kernel K(x, y) be measurable on (1 × 2 , μ × ν). If p ≥ 1, then





p K(x, y)dν

1

1/p





1/p

≤

dμ

2

K p (x, y)dμ 2

1

For the case 0 < p < 1, (1.31) holds in the revered direction. Proof Let p > 1. We use again the idea from (1.27) and obtain that



p



I0 =:

K(x, y)dν 1

=

2





sup 

ϕ q dμ=1 1

1/p dμ

ϕ(x)

K(x, y)dνdμ, 2

dν.

(1.31)

1.4 Convex Functions and Classical Inequalities

21

where supremum is taken over all measurable ϕ such that 

b

ϕ q (x)dx = 1, q = p/(p − 1).

a

Hence, by using the Fubini theorem and an obvious estimate, we have that  I0 =



sup ϕ

K(x, y)ϕ(x)dμdν

q dμ=1



2



1





≤

sup 2

K(x, y)ϕ(x)dμ dν

q 1  ϕ dμ=1



 =

1/p

p

K (x, y)dμ 2

dν.

1

For p = 1 we have even equality in (1.31) because of the Fubini theorem, so the proof is complete. The proof of the case 0 < p < 1 is similar (we just need to use the representation formula (1.27) with “sup” replaced by “inf”).  Remark 1.4.17 By putting point masses δi in the points yi and K(x, yi ) = fi (x), i = 2, 3, . . . we obviously get a well-known generalization of (1.30) with n functions involved. For applications the following special case of Theorem 1.4.16 is useful e.g. when working with mixed-norm Lp spaces and we need some estimates replacing the Fubini theorem. More exactly, we let 1 = 2 = R with Lebesgue measure and put  K(x, y) =

k(x, y)(y)1/p (x), a ≤ y ≤ x, 0, x < y ≤ b,

where k(x, y), (y) and (x) are measurable so that Minkowski’s integral inequality can be used. Theorem 1.4.18 (Minkowski’s Integral Inequality of Fubini Type) If p ≥ 1, ∞ ≤ a < b ≤ ∞, then



b



p

x

k(x, y)(y)dy a

a

1/p (x)dx

 ≤ a

b



b

1/p (x)k p (x, y)dx

(y)dy.

y

In fact, our previous continuous form of Hölder’s inequality (see Theorem 1.4.13) may be regarded as a limit case of (1.31) but in order to understand this, we need to consider (the scale of) power means.

22

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

The scale of power means {Pα (f ; μ)} , −∞ < α < ∞, of a function f on a finite measure space (, μ) is defined as follows:

Pα (f ; μ) =:

⎧ ⎪ ⎪ ⎨

1/α |f |α dμ

1 μ()

⎪ ⎪ ⎩ exp

 1 μ()

, −∞ < α < ∞, α = 0,

(1.32)

log |f |dμ, α = 0, 

(for the case α > 0 we assume that f > 0 almost everywhere.) A special case for positive sequences a = {ai }ni=1 , n ∈ N+ , is obtained by letting μ=

n 

δi

and fi = ai , i = 1, 2, . . . , n ( = R+ ) :

i=1

⎧

1/α n ⎪ ⎪ 1  α ⎪ a , −∞ < α < ∞, α = 0, ⎨ n i Pα (a) =: nn=1 1/n ⎪ ! ⎪ ⎪ ai , α = 0. ⎩ i=1

As a generalization of the usual harmonic-geometric-arithmetic mean inequality we have the following: Theorem 1.4.19 (The “Power Mean Inequality”) The scale of powermeans {Pα (f ; μ)} , defined by (1.32), is a non-decreasing function of α (for fixed f and μ). Proof First, we let 0 < α < β < ∞. Then, by using Jensen’s inequality (1.21) with (u) = uβ/α , we find that ⎛ 1 Pαβ (f ; μ) =: ⎝ μ()

⎞β/α



|f | dμ⎠ α

1 ≤ μ()





β

|f |β dμ = Pβ (f ; μ), 

and we conclude that Pα (f ; μ) ≤ Pβ (f ; μ). Next, let 0 < α < ∞. Then, by again using Jensen’s inequality, now with the convex function (u) = exp u, we obtain that ⎛ 1 P0α (f ; μ) = exp ⎝ μ()



⎞ log |f | dμ⎠ ≤ α



1 μ()

 |f |α dμ = Pαα (f ; μ), 

1.4 Convex Functions and Classical Inequalities

23

so that P0 (f ; μ) ≤ Pα (f ; μ). If α < 0, then −1



1 ;μ Pα (f ; μ) = P−α |f |

−1 and, moreover, P0 (f ; μ) = P0 |f1 | ; μ and the proof of the remaining cases follows by just using what we already have proved.  It is also clear that lim Pα (f ; μ) = P0 (f ; μ).

α→0

Remark 1.4.20 In particular, by letting 2 dν = 1, replacing K(x, y) in (1.31) by K(x, y)1/p and letting p → ∞ (α = p1 → 0+ ), we obtain that (1.29) holds and that this version of Hölder’s inequality is a limit case of (1.31). Remark 1.4.21 As we have seen, standard Hölder’s inequality imply both the standard and the continuous versions of Minkowski’s inequality. Moreover, as we see above, we have also implication in the reversed direction on this more general continuous level. In the sequel also the following two Hardy type inequalities are needed:   Theorem 1.4.22 Hardy s Inequalities If 1 ≤ q < ∞, r > 0 and f is a measurable and non-negative function defined on (0, ∞), then



∞  t

q f (u) du

0

t −r

0

dt t

1/q ≤

q r



∞

(tf (t))q t −r

0

dt t

1/q (1.33)

and



q

∞  ∞

f (u) du 0

t

dt t t

1/q

r

q ≤ r



∞ 0

dt (tf (t)) t t

Proof Observe that the measure dμ =:

r −r/q r/q−1 t u du q

q r

1/q .

(1.34)

24

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

is a probability measure on [0, t] for a fixed t. Applying Jensen’s inequality with the convex function φ(x) = x q , we obtain



q

t

=

f (u) du 0

≤ =



q q

t

tr

r q q

0



t

tr

r

q q−1 r

q f (u) u1−r/q dμ

f (u)q uq−r dμ

0



t

t r−r/q

f (u)q uq−r+r/q−1 du.

0

Henceforth, by the Fubini theorem, q

∞  t



t −r−1 dt

f (u) du 0

≤ = =

0

q q−1  r

q q−1  r q q  r

∞

t

−1−r/q



0

0 ∞



t

q

f (u) u 0

∞

(uf (u))q u−r+r/q−1

q−r+r/q−1



∞

du dt

 t −1−r/q dt du

u

(uf (u))q u−r−1 du,

0

which proves (1.33). We prove that (1.33) implies (1.34). Applying (1.33) to the function g(u) =: u−2 f (u−1 ), we find that ∞  t



q g (u) du

0

t

−r−1

dt ≤

q q  r

0

∞

(tg (t))q t −r−1 dt.

0

The left hand side is equal to 

q

∞  ∞

f (v) dv 0

t −r−1 dt =

1/t

q

∞  ∞



f (v) dv 0

s r−1 ds,

s

while the right hand side is equal to q q  r

∞

(tg (t))q t −r−1 dt =

0

This finishes the proof.

q q  r

∞

(sf (s))q s r−1 ds.

0



1.5 Lebesgue Spaces

25

1.5 Lebesgue Spaces By a Vilenkin polynomial we mean a finite linear combination of Vilenkin functions. We denote the collection of Vilenkin polynomials by P. Let L0 (Gm ) represent the collection of functions which are almost everywhere limits with respect to a measure μ of sequences in P. For 0 < p < ∞ let Lp (Gm ) represent the collection of f ∈ L0 (Gm ) such that

 f p =:

1/p |f |p dμ Gm

is finite. Denote by L∞ (Gm ) the space of all f ∈ L0 (Gm ) for which f ∞ =: inf {C > 0, μ {x ∈ Gm , |f | > C} = 0} < +∞. The space C(Gm ) consists all continuous functions for which f C =: sup |f (x)| < c < ∞. x∈Gm

It is evident that if f ∈ C(Gm ), then and C (Gm ) ⊂ L∞ (Gm ).

f C = f ∞

Moreover, if 1 < p1 < p2 ≤ ∞, then Lp2 (Gm ) ⊂ Lp1 (Gm ). Theorem 1.5.1 If f ∈ Lp (Gm ) for all p > 0, then f ∈ L∞ (Gm ) and f ∞ = lim f p . p→∞

Proof By the definition we have that

 f p =

1/p |f (x)|p dμ(x) Gm

 p

≤ Gm

f ∞ dμ(x)

1/p = f ∞ .

On the other hand, for every ε > 0, there exists set E ∈ Gm , such that μ (E) = C > 0 and |f (x)| > f ∞ − ε.

26

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Hence

 f p ≥

1/p |f (x)|p dμ(x)



E

≥ E

  f ∞ − εp dμ(x)

1/p

  = f ∞ − ε μ1/p (E) Since lim μ1/p (E) = lim C 1/p = 1

p→∞

p→∞

and ε is any positive real number we obtain that lim f p ≥ f ∞ − ε

p→∞

and

lim f p = f ∞ .

p→∞



The proof is complete. By using Minkowski’s inequality (see Theorem 1.4.15), we easily obtain that f p = 0 ⇐⇒ f = 0 a.e., cf p = |c| f p

(c ∈ C) ,

  f + gp ≤ Cp f p + gp , where Cp = 1, for p ≥ 1 and Cp ≤ 21/p for p < 1. Because of the last property ·p is a norm for p ≥ 1 and quasi-norm for p < 1. Remark 1.5.2 Note that Minkowski’s integral inequality (see Theorem 1.4.16) can also be written as   ≤ f (·, t)p dt, for all p ≥ 1. f t) dt (·, Gm

Gm

p

The convolution of two functions f, g ∈ L1 (Gm ) is defined by  f (x − t) g (t) dt (x ∈ Gm ) .

(f ∗ g) (x) =: Gm

1.5 Lebesgue Spaces

27

It is easy to see that  f (t) g (x − t) dt

(f ∗ g) (x) =

(x ∈ Gm ) .

Gm

Theorem 1.5.3 Let f ∈ Lr (Gm ) , g ∈ L1 (Gm ) and 1 ≤ r < ∞. Then f ∗ g ∈ Lr (Gm ) and f ∗ gr ≤ f r g1 . Proof The inequality is trivial for r = ∞. For 1 ≤ r < ∞, we can conclude that



1/r |(f ∗ g) (x)|r dx

r





|f (x − t)| |g (t)| dt

≤

Gm

Gm

1/r dx

.

Gm

By using Minkowski’s integral inequality, we find that 1/r





(|f (x − t)| |g (t)|) dx

f ∗ gr ≤

r

Gm

Gm





1/r

|g (t)|

=

|f (x − t)|r dx

Gm

dt dt

Gm

= f r g1 , 

which shows the inequality.

Theorem 1.5.3 is clearly a special case of the next one when q = 1 and p = r. Theorem 1.5.4 Let f ∈ Lp (Gm ), g ∈ Lq (Gm ) and 1 ≤ p, q, r < ∞ such that 1 1 1 + =1+ . p q r Then f ∗ g ∈ Lr (Gm ) and f ∗ gr ≤ f p gq . Proof Since 1 ≤ p , q < ∞, then 1 1 1 + +  = 1,  q r p

p p + = 1,  q r

q q +  = 1, r p

28

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

where p and q  denote the conjugate indexes to p and q, respectively. By applying Hölder’s inequality to the indexes q  , r  and p , we can conclude that 

    |f (t)|p/q |f (t)|p/r |g (x − t)|q/r |g (x − t)|q/p dt

|(f ∗ g) (x)| ≤ Gm

  ≤ |f (t)|p/q  |f (t)|p/r |g (x − t)|q/r r |g (x − t)|q/p

p

q

p/q 

1/r

 |f (t)|p |g (x − t)|q dt

= f (t)p

q/p 

g (t)q

.

Gm

Hence, by using Fubini’s theorem and the relation between the parameters, we find that p/q 

f ∗ gr ≤ f (t)p

q/p 

 |f (t)|p Gm

p/q 

= f (t)p

q/p 

g (t)q

1/r



g (t)q

|g (x − t)|q dxdt Gm

q/r

g (t)q

p/r

f (t)p

= f p gq . 

The proof is complete. The space weak

− Lp

(Gm ) consists of all measurable functions f , for which

f weak−Lp =: sup y μ{(f > y)}1/p < +∞. y>0

The following properties of the weak-Lp (Gm ) spaces can be proved easily: f weak−Lp = 0 ⇐⇒ f = 0 a.e. cf weak−Lp = |c| f weak−Lp

(c ∈ C) ,

  f + gweak−Lp ≤ Cp f weak−Lp + gweak−Lp ,   where Cp = max 2, 21/p . Because of the last property ·weak−Lp is a quasinorm. Proposition 1.5.5 (Tsebisev’s Inequality) If 0 < p ≤ ∞, then  y p μ (|f | > y) ≤

|f (x)|p dx. Gm

1.5 Lebesgue Spaces

29

Proof It is easy to see that 

 |f (x)|p dx ≥ Gm

{x:|f (x)|>y}

|f (x)|p dx ≥ y p μ (|f | > y) , 

which proves the proposition. weak-Lp (G

Lp (G

We will show that the m ) spaces are larger than the m ) spaces. In particular, Tsebisev’s inequality can be rewritten in the following form: Proposition 1.5.6 If 0 < p ≤ ∞, then Lp (Gm ) ⊂ weak − Lp (Gm ) and f weak−Lp ≤ f p . Note that the inclusion Lp (Gm ) ⊂ weak − Lp (Gm ) is proper for any 0 < p < ∞. Indeed, let h(x) =: |x|−1/p . Then obviously h ∈ / Lp (Gm ), but h ∈ weak − Lp (Gm ) because ypμ

' ( x : |x|−1/p > y = 2y p y −p = 2.

Recall that the space weak-Lp (Gm ) is also complete for each p ≥ 1. We define the distribution function λf as follows: λf (y) =: μ {x : |f (x)| > y} . We will establish a useful formula which allows us to express the Lp (Gm ) norm of f in terms of the function λf (y), namely the following: Proposition 1.5.7 We have that 

∞ |f (x)| dμ(x) = p p

Gm

y p−1 λf (y) dy,

p ≥ 1.

(1.35)

0

Proof First, write the right hand side of (1.35) in the following form ∞ p

y 0



∞ p−1

λf (y) dy = p

y 0

p−1 Gm

 χ{x:|f (x)|>y} (x, y) dx dy.

30

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

After that we apply Fubini‘s theorem to this last integral and rewrite it using the fact that for a fixed x, we have  χ{x:|f (x)|>y} (x, y) =

1 y < |f (x)| , 0 y ≥ |f (x)| .

We obtain that



∞



y p−1

p 0

Gm

⎛



χ{x:|f (x)|>y} (x, y) dx dy =

Gm



⎜ ⎝p

|f(x)|

⎞

⎟ y p−1 dy ⎠ dx

0

|f (x)|p dμ(x).

= Gm



The proof is complete.

An operator T which maps a linear space of measurable functions on Gm in the collection of measurable functions on Gm is called sublinear if |T (f + g)| ≤ |T (f )| + |T (g)| a.e. on Gm and |T (αf )| = |α||T (f )| for all scalars α and all f in the domain of T . An operator T is a σ -sublinear operator if, for any α ∈ C,   ∞  ∞       fk  ≤ |T (fk )| T   k=1

and

|T (αf )| = |α||T (f )|.

k=1

An operator T from Lp (Gm ) into L0 (Gm ) is said to be of strong type (p, q) (or (p, q) type) for some 1 ≤ p ≤ ∞ and 1 < q < ∞ if there is a constant A > 0 such that Tf q ≤ A f p for all f ∈ Lp (Gm ). An operator T from Lp (Gm ) into L0 (Gm ) is said to be of weak type (p, q) for some 1 ≤ p ≤ ∞ and 1 < q < ∞ if there is a constant A > 0 such that Tf weak−Lq ≤ A f p for all f ∈ Lp (Gm ). Hence, any operator of type (Lp , Lq ) is necessarily of weak type (p, q).

1.5 Lebesgue Spaces

31

Theorem 1.5.8 (Marcinkiewicz Interpolation Theorem) Let 1 ≤ p1 < p2 and suppose that the sublinear operator T satisfies that p

y pi λTf (y) ≤ Ci i

 |f (y)|pi dμ(y), Gm

for all y > 0, f ∈ Lp (Gm ) and i = 1, 2, where the Ci ‘s are absolute constants independent of y and f. In the case when p2 = ∞, suppose that T is of strong type, i.e. that Tf ∞ ≤ C2 f ∞ . Then p Tf p

p −p 2 1

p1 p 2−p

≤ p2 C1 p

p−p1 2 −p1

p2 p

C2



 1 1 p f p , + p − p1 p2 − p

for all f ∈ Lp (Gm ) , p1 < p < p2 . In particular, if p1 < p < p2 and the operator T is of weak (p1 , p1 ) type and weak (p2 , p2 ) type, then it will be of (p, p) type. Proof Let ) A≡

p1 p −p1

C1 2 C2−1

−p2 p −p1

C2 2

p2 < ∞ p2 = ∞.

Fix y > 0, set  f y (x) =

f (x) |f (x)| ≤ Ay C2−1 |f (x)| > Ay

and fy (x) ≡ f (x) − f y (x) . Since T is a sublinear operator and f (x) ≡ fy (x) + f y (x) , we have that λTf (2y) ≤ λTfy (y) + λTf y (y)

32

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Thus, in the case when p2 < ∞ we have by hypotheses that the following holds true      y p2 p fy (x)p1 dμ(x) + C p2 y −p2 f (x) dμ(x) λTf (2y) ≤ C1 1 y −p1 2 Gm

= C1 1 y −p1 p

Gm



Gm

p +C2 2 y −p2

χ{x:|f (x)|>Ay} (x, y) |f (x)|p1 dμ(x)



Gm

χ{x:|f (x)|≤Ay} (x, y) |f (x)|p2 dμ(x).

Therefore, by using Proposition 1.5.7 and some obvious arguments we find that  |Tf (x)|p dμ(x) Gm



∞

=p

(2y)p−1 λTf (2y) d (2y)

0

≤ p2

p

p C1 1

+p2

p



y

+p2 = p2

p

p C1 1

∞

y 0

|f (x)|/A



|f (x)|

y p−p1 −1 d (y) dμ(x)

∞

p2

|f (x)|/A

Gm

|f (x)|p1 Gm



Gm

1 p − p1

y p−p2 −1 d (y) dμ(x)

1 |f (x)|p−p1 Ap1 −p dμ(x) p − p1

|f (x)|p2

= p2p C1 1 Ap1 −p +p2

χ{x:|f (x)|≤Ay} (x, y) |f (x)|p2 dμ(x)d (y)

0





p

p

Gm



Gm

+p2p C2 2 p



p−p2 −1

|f (x)|p1

p C2 2

χ{x:|f (x)|>Ay} (x, y) |f (x)|p1 dμ(x)d (y)

Gm





= p2p C1 1 p



p−p1 −1

0

p C2 2 p

∞



1 p C2 2 Ap2 −p p2 − p

1 |f (x)|p−p2 Ap2 −p dμ(x) p2 − p |f (x)|p dμ(x)

Gm



|f (x)|p dμ(x). Gm

By now substituting the value for A chosen at the beginning of this proof, we obtain the claimed inequality. In the case p2 = ∞ we have y f ≤ Ay = C −1 y. 2 ∞

1.5 Lebesgue Spaces

33

Hence, y Tf ≤ C2 f y ≤ y ∞ ∞ and we have λTf y = 0. Consequently, in the string of inequalities above we can simply omit the terms involving p2 when p2 = ∞. The proof is complete.  Theorem 1.5.9 Suppose that (X(Gm ), Lp (Gm )) are normed linear spaces in L0 and X0 is dense in X. Let T , Tn : X → Lp (Gm ) be sublinear operators, for some 1 ≤ p < ∞ with T bounded and Tn f → Tf a.e. on Gm as n → ∞, for each f ∈ X0 . Set T ∗ f =: sup |Tn f | ,

f ∈ X.

n∈N

If there is a constant C > 0, independent of f and n, such that p

y p μ ({|Tf | > y}) ≤ C f X

(1.36)

and ypμ

* ∗ + p T f > y ≤ C f X ,

(1.37)

for all y > 0 and f ∈ X, then Tf = lim Tn f n→∞

a.e. on Gm , for every f ∈ X. Proof Fix f ∈ X and set  =: lim sup |Tn f − Tf | . n→∞

It suffices to show that  = 0 a.e on Gm . Choose fm ∈ X0 (m ∈ N) such that f − fm X → 0, as m → ∞. Observe that by hypothesis  ≤ lim sup |Tn (f − fm )| + lim sup |Tn fm − Tfm | + |T (f − fm )| n→∞ ∗

n→∞

≤ T (f − fm ) + |T (f − fm )| , a.e on Gm .

34

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

a.e for all m ∈ N. In particular, * + μ ({ > 2y}) ≤  T ∗ (f − fm ) > y  + |{T (f − fm ) > y}| p

≤

p

f − fm X f − fm X + T  yp yp

holds for any y > 0 and m ∈ N. Since f − fm → 0 in X as m → ∞ it follows that μ ({ > 2y}) = 0 for all y > 0. We conclude that  = 0 a.e. on Gm . The proof is complete.



1.6 Dirichlet Kernels If f ∈ L1 (Gm ), we can define the Fourier coefficients, the partial sums of VilenkinFourier series, the Dirichlet kernels with respect to Vilenkin systems in the usual manner:   f ψ n dμ, f (n) =: (n ∈ N) , Gm

Sn f =:

n−1 

f(k) ψk ,

(n ∈ N+ ) ,

k=0

Dn =:

n−1 

(n ∈ N+ ) ,

ψk ,

k=0

respectively. It is easy to see that  Sn f (x) =

f (t) 

Gm

=

n−1 

ψk (x − t) dμ (t)

k=0

f (t) Dn (x − t) dμ (t) = (f ∗ Dn ) (x) . Gm

To prove upper and lower estimates for Dirichlet kernels we need an estimate of some sums of Rademacher functions: Lemma 1.6.1 Let k ∈ N and x ∈ Gm . Then m k −1 s=0

 rks (x) =

mk , if xk = 0,

0. 0, if xk =

1.6 Dirichlet Kernels

35

Proof If xk = 0 then rk (x) = 1 and rks (x) = 1 for all 1 ≤ s ≤ mk − 1. It follows that m k −1

rks (x) = mk , when xk = 0.

s=0

On the other hand, since rkmk (x) = e2πıxk = cos(2πxk ) + ı sin(2πxk ) = 1 if xk = 0 and rk (x) = 1 if xk = 0, we get that m k −1

rks (x) =

s=0

rkmk (x) − 1 rk (x) = 0, when xk = 0. rk (x) − 1 

The proof is complete. Lemma 1.6.2 Let k, s ∈ N and x ∈ Gm . Then s k −1

rku (x) =

cos (πsk xk /mk ) sin (π (sk − 1) xk /mk ) ı sin (πxk /mk )

+

sin (πsk xk /mk ) sin (π (sk − 1) xk /mk ) . sin (πxk /mk )

u=1

Proof Since s k −1 u=1

rku (x)

=

s k −1 u=1



2πuxk cos mk

 +

s k −1 u=1



2πuxk ı sin mk

 ,

if we apply the following well-known identities n  k=1

cos(kx) =

(n+1)x sin nx 2 cos 2 sin x2

(1.38)

36

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

and n 

sin(kx) =

k=1

(n+1)x sin nx 2 sin 2 , sin x2

(1.39) 

we immediately get the proof. Lemma 1.6.3 Let n ∈ N and xn = 1. Then  s −1 n     u rn (x) ≥ 1, for some 1 ≤ sn ≤ mn − 1    u=0

and  s −1 n     u rn (x) ≥ 1, for some 2 ≤ sn ≤ mn − 1.    u=1

Proof Let xn = 1. Since sin (π (mk − 1) /mk ) sin (π/mk ) = = 1, sin (π/mk ) sin (π/mk ) if we take the graph of sin x into account we obtain that s −1   n    r sn (x) − 1  sin (πs x /m ) sin (πsn /mn ) n n n   n u = rn (x) =  = ≥ 1.    rn (x) − 1  sin (πxn /mn ) sin (π/mn ) u=0

Analogously, we can prove that s −1    n sn −1     sin π (s − 1, m  r − 1 (x) n n     n u ≥ 1. rn (x) = rn (x)  =    rn (x) − 1  sin (π/mn ) u=1



The proof is complete.

The next well-known identities with respect to Dirichlet kernels (see Lemmas 1.6.4 and 1.6.5, Corollaries 1.6.6 and 1.6.7) will be used many times in the proofs of our main results: Lemma 1.6.4 Let n ∈ N. Then Dj +Mn = DMn + ψMn Dj = DMn + rn Dj ,

j ≤ (mn − 1) Mn

(1.40)

1.6 Dirichlet Kernels

37

and DMn −j (x) = DMn (x) − ψ Mn −1 (−x)Dj (−x) = DMn (x) − ψMn −1 (x)D j (x),

(1.41) j < Mn .

Proof Let 0 ≤ j < Mn . Then ψj +Mn = ψMn ψj . Hence, Dj +Mn = DMn +

j +M n −1 

ψk

k=Mn

= DMn +

j −1 

ψk+Mn

k=0

= DMn + ψMn

j −1 

ψk

k=0

= DMn + rn Dj and (1.40) is proved. Now we prove second identity. By using the well known equalities ψn+  m = ψn ψm ,

ψn−  m = ψn /ψm ,

ψn (x) = ψ n (−x) and ψn ψn = 1

as well as the simple fact that v = Mk − 1 − v, f or v = 0, 1, . . . , Mk − 1, (Mk − 1)− we can conclude that DMk −j (x) = DMk (x) −

M k −1 

ψv (x)

v=Mk −j

= DMk (x) − ψ Mk −1 (−x)

M k −1 

ψMk −1 (−x)ψ v (−x)

v=Mk −j

= DMk (x) − ψ Mk −1 (−x)

M k −1  v=Mk −j

ψMk −1 (−x)/ψv (−x)

38

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

= DMk (x) − ψ Mk −1 (−x)

M k −1 

ψ(Mk −1)− v (−x)

v=Mk −j

= DMk (x) − ψ Mk −1 (−x)

M k −1 

ψv (−x)

v=Mk −j

= DMk (x) − ψ Mk −1 (−x)Dj (−x) = DMn (x) − ψMn −1 (x)D j (x) 

and also (1.41) is proved so the proof is complete. Lemma 1.6.5 Let n ∈ N and 1 ≤ sn ≤ mn − 1. Then Dsn Mn = DMn

s n −1

ψkMn = DMn

k=0

s n −1

rnk

(1.42)

k=0

and ⎛ ∞  Dn = ψn ⎝ DMj j =0

where n =

∞

i=0 ni Mi .

We note that

⎞

mj −1



rjk ⎠ ,

(1.43)

k=mj −nj

mj −1

k k=mj −nj rj

≡ 0, for all nj = 0.

Proof It is simple to show that DsMn =

s−1 (k+1)M  n−1 −1 k=0

ψj = DMn−1

j =kMn−1

s−1 

k rn−1

k=0

and (1.6.5) is proved. For the proof of (1.43), we let n=

r 

nsk Msk ,

1 ≤ nsk ≤ msk − 1, ns0 > ns1 > . . . > nsr

k=0

and n(i) =

r  k=i+1

nsk Msk ,

i = 0, . . . ., r.

1.6 Dirichlet Kernels

39

It is easy to show that n(0) = n and n(r) = 0. We have that ns1 Mns −1

1

Dn =

n−1 

ψj +

ψj

j =ns1 Mns

j =0

1

n−nns Mns −1 1 

= Dns1 Mns + ψns1 Mns 1

1

1

ψj

j =0

ns

= Dns1 Mns + r1 1 Dn(1) . 1

By using this method n-times, we get that Dn =

−1 r j  j =0 l=0

= ψn

ns

rsl l Dnsj Mns

j

r  r  j =0 l=j

= ψn

r  r  j =0 l=j

ns

rsl l Dnsj Mns

j

ns rsl l DMns j

nsj −1



rskj .

k=0

Since rknk

n k −1

m k −1

rjl =

l=0

rjl ,

l=mk −nk

by applying (1.42) we obtain that r  ns Dn = ψn rsj j DMns



j

j =0

rjk

k=0

r  = ψn DMns j =0

nsj −1

j

msj −1



k=msj −nsj

rskj .

It is evident that if we take classical representation of n = 0 ≤ nk ≤ mk − 1, we get (1.43). The proof is complete.

r

k=0 nk Mk ,

where 

40

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Corollary 1.6.6 Let n ∈ N. Then DMn+1 =

⎛

⎞

⎝

rks ⎠ .

m k −1

n  k=0

(1.44)

s=0

Proof If we use (1.42) in Lemma 1.6.5 when s = mn − 1 we get that DMn+1 =

m n −1 (k+1)M n −1

ψj

j =kMn

k=0

= DMn

m n −1

rnk .

k=1

If we unfold DMk k = n, n − 1, . . . , 0 in a similar way, we obtain (1.44) and the proof is complete.  Corollary 1.6.7 (Paley’s Lemma) Let n ∈ N. Then  DMn (x) =

Mn , x ∈ In , 0, x ∈ / In .

Proof It is obvious that if xl = 0 for some l = 0, 1, . . . , n − 1, then m l −1 

rlk = 0

k=0

and DMn (x) = 0. On the other hand, if xl = 0 for every l = 0, 1, . . . , n − 1, then m l −1 

rlk = ml , l = 0, 1, . . . , n − 1

k=0

and DMn = Mn . This completes the proof. We also need the following estimate: Corollary 1.6.8 Let n ∈ N. Then DMn 1 = 1.



1.6 Dirichlet Kernels

41

Proof By using Corollary 1.6.7 we get that 



DMn 1 =

Mn dμ = 1.

DMn (x) dμ (x) = In

In



The proof is complete. Lemma 1.6.9 Let x ∈ Is \Is+1 for some s = 0, . . . , N − 1. Then |Dn (x)| ≤ Ms+1 ≤ cMs and  |Dn (x − t)| dμ (t) ≤ IN

cMs , MN

where c is an absolute constant. Proof By combining (1.43) in Lemma 1.6.5 and Corollary 1.6.7, we have that |Dn (x)| ≤

s s   nj DMj (x) = nj Mj j =0

≤

j =0

s  (mj − 1)Mj = Ms+1 − M0 ≤ Ms+1 ≤ cMs . j =0

Since t ∈ IN and x ∈ Is \Is+1 , s = 0, . . . , N − 1, we obtain that x − t ∈ Is \Is+1 . By using the estimate above we get that |Dn (x − t)| ≤ cMs and  |Dn (x − t)| dμ (t) ≤ IN

cMs . MN 

The proof is complete. Lemma 1.6.10 Let n ∈ N and x ∈ Gm . Then |Dn (x)| < where C is an absolute constant.

C , (x = 0) , |x|

42

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Proof Let x ∈ Is \Is+1 for some s = 0, . . . , N − 1. Then x = (0, . . . , 0, xs = 0, xs+1, xs+2 , . . .) and ∞

 mi − 1 1 1 R ≤ |x| ≤ = ≤ , where R =: sup mn . Ms Mi Ms−1 Ms n∈N i=s

Hence, if we apply the first inequality of Lemma 1.6.9, we get that |Dn (x)| ≤ Ms+1 ≤ RMs ≤

C R2 ≤ . |x| |x| 

The proof is complete.

The following estimate plays an important role to prove some negative results in the next sections: Lemma 1.6.11 Let n ∈ N, |n| = n and x ∈ In \In+1 . Then   |Dn (x)| = Dn−M|n| (x) ≥ Mn . Proof Let x ∈ In \In+1 . Since n = nn Mn +

|n|−1 

nj Mj + n|n| M|n|

j =n

and n − M|n| = nn Mn +

|n|−1 

  nj Mj + n|n| − 1 M|n| ,

j =n

if we apply Lemma 1.6.3, Corollary 1.6.7 and (1.43) in Lemma 1.6.5 we can conclude that     mn −1      s  Dn−M  ≥ ψn DM rn  |n| n    s=mn −nn     mj −1 |n|     s  − ψn DMj rj    s=mj −nj j =n+1

1.7 Lebesgue Constants

43

    mn −1    s  = DMn rn    s=mn −nn     nn −1    m −α n n s   = DMn rn rn    s=0   n  −1  n  s   = DMn  rn  ≥ DMn ≥ Mn .  s=0  Analogously we can show that  nn     s  |Dn | = DMn  rn  ≥ Mn   s=0



and the proof is complete.

1.7 Lebesgue Constants To study boundedness of subsequences of partial sums of integrable functions in L1 norm, it is important to find two-sided estimations for Lebesgue constants Ln , which are defined as an L1 (Gm ) norm of Dirichlet kernels with respect to the Vilenkin system: Ln =: Dn 1 . Here, we prove two-sided estimates for Lebesgue constants with respect to bounded Vilenkin systems. In particular, the following result holds true: Lemma 1.7.1 Let n=

∞ 

ni Mi .

i=1

Then 1 1 v (n) + 2 v ∗ (n) ≤ Ln ≤ v (n) + v ∗ (n) , 4R R

44

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

where R =: sup mn . n∈N

Proof First we choose indices 0 ≤ 1 ≤ α1 < 2 ≤ α2 < . . . < s ≤ αs < s+1 = ∞ such that αj + 1 < j +1 , for j = 1, 2, . . . , s, nk = 0,

for 0 < k < 1 ,

nk ∈ {1, 2, . . . , mk − 1} for j ≤ k ≤ αj and nk = 0,

for αj < k < j +1 .

According to (1.43) in Lemma 1.6.5 we have that ⎛ ⎞ m ∞ k −1  Dn = ψn ⎝ DMk rku ⎠ k=0

(1.45)

u=1

⎛ ⎞ mk  −nk −1 ∞  DMk rku ⎠ − ψn ⎝ k=0

u=1

⎛ ⎞ αj m s  k −1  = ψn ⎝ DMk rku ⎠ j =1 k=j

u=1

j =1 k=j

u=1

⎛ ⎞ αj

n s  k −1   DMk rku ⎠ − ψn ⎝ =: I − I I. Since Mk − 1 =

k−1    mj − 1 Mj , j =0

(1.46)

1.7 Lebesgue Constants

45

if we apply again (1.43) in Lemma 1.6.5, we get that

DMk −1

⎛ ⎞ mj −1 k−1   = ψMk −1 ⎝ DMj rju ⎠ . j =0

u=1

Hence, ⎛ ⎛ ⎞⎞ αj j −1 m m s k −1 k −1    ⎝ I = ψn ⎝ DMk rku − DMk rku ⎠⎠ j =1

k=0

u=1

k=0

⎛  ⎞ s DMj −1 DMαj +1 −1  ⎠ − = ψn ⎝ ψMαj +1 −1 ψMj −1

(1.47)

u=1

j =1

⎛  ⎞ s DMj − ψMj −1 DMαj +1 − ψMαj +1 −1  ⎠ − = ψn ⎝ ψMαj +1 −1 ψMj −1 j =1

⎛  ⎞ s DMj DMαj +1  ⎠ − = ψn ⎝ ψMαj +1 −1 ψMj −1 j =1

and s  I 1 ≤ DMαj +1 + DMj = 2s ≤ v (n) . 1

j =1

1

Moreover, I I 1 ≤

αj s      nj − 1 δj DM j 1 j =1 j =j

αj s      nj − 1 δj ≤ v ∗ (n) . = j =1 j =j

The proof of the upper estimate follows by combining the last two estimates. Let x ∈ Ik+1 (xk ek ) , where 1 ≤ xk ≤ nk − 1. Then, by the definition of Vilenkin functions, if we apply (1.46) and equalities x0 = x1 = . . . = xk−1 = 0,

46

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

we find that ψMl −1 (x) =

l−1 

rtmt −1 (x)

(1.48)

t =0

=

l−1 

e

(2πıxt (mt −1))/mt

=

t =0

l−1 

e0 = 1,

t =0

for any 0 ≤ l ≤ k. Let j ≤ k ≤ αj and x ∈ Ik+1 (xk ek ) , where 1 ≤ xk ≤ nk − 1. Then, in view of Corollary 1.6.7 and (1.47), we get that I = −ψn (x) ⎛ + ψn (x) ⎝ ⎛

DMj (x) ψMj −1 (x) j −1 



DMαl +1 (x) ψMαl +1 −1 (x)

l=1

= ψn (x) ⎝−Mj +

j −1  

−

DMl (x)

⎞

ψMl −1 (x) 

⎠

⎞

Mαl +1 − Ml ⎠ .

l=1

By using Lemma 1.6.2 we have that ⎛ I I = ψn (x) ⎝DMk (x) ⎛ + ψn (x) ⎝

mk  −nk −1

⎞ rku (x)⎠

u=1 k−1 

l=j

DMl (x)

n l −1 

rlu (x) +

u=1

j −1  αs  s=0 l=s

DMl (x)

n l −1  u=1

= ψn (x) Mk

cos (π ( nk ) xk /mk ) sin (π ( nk − 1) xk /mk ) ı sin (πxk /mk )

+ ψn (x) Mk

sin (π ( nk ) xk /mk ) sin (π ( nk − 1) xk /mk ) sin (πxk /mk )

+ ψn (x)

k−1  l=j

Ml ( nl − 1) + ψn (x)

j −1  αs  s=0 l=s

Ml ( nl − 1) .

⎞ rlu (x)⎠

1.7 Lebesgue Constants

47

It is obvious that   II − I    |I I − I | =  ψn 





1/2 2 II − I 2 II − I = Re . + Im ψn ψn On the other hand,

 cos (π ( nk ) xk /mk ) sin (π ( nk − 1) xk /mk ) II − I Im = Mk ψn sin (πxk /mk )

(1.49)

(1.50)

and

II − I Re ψn

 = Mk +

sin (π ( nk ) xk /mk ) sin (π ( nk − 1) xk /mk ) sin (πxk /mk )

k−1 

Ml ( nl − 1) +

l=j

j −1  αs 

Ml ( nl − 1)

s=0 l=s

+ Mj −

j −1  

 Mαl +1 − Ml .

l=1

Let x ∈ Ik+1 (ek ) and R =: supn∈N mn . Then we find that sin (π ( nk ) xk /mk ) sin (π ( nk − 1) xk /mk ) ≥ 0, for xk = 1. sin (πxk /mk ) Moreover, since k−1 

Ml ( nl − 1) ≥ 0,

l=j j −1  αs 

Ml ( nl − 1) ≥ 0,

s=0 l=s

Mj −

j −1    Mαl +1 − Ml ≥ 0, l=1

for x ∈ Ik+1 (ek ) , we obtain that

II − I Re ψn

 ≥

sin (π ( nk ) xk /mk ) sin (π ( nk − 1) xk /mk ) ≥ 0. sin (πxk /mk )

(1.51)

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1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

If we apply (1.49), (1.50), (1.51) and Lemma 1.6.3 for x ∈ Ik+1 (ek ), we immediately get that





1/2 2 II − I 2 II − I |I I − I | = Re + Im ψn ψn 

 Mk cos (π ( nk ) xk /mk ) sin (π ( nk − 1) xk /mk ) 2 ≥ sin (πxk /mk )

 1/2 Mk sin (π ( nk ) xk /mk ) sin (π ( nk − 1) xk /mk ) 2 + sin (πxk /mk ) ≥

Mk sin (π ( nk − 1) xk /mk ) Mk | nk − 1| . ≥ Mk ≥ sin (πxk /mk ) R

By using (1.45) it follows that |Dn | ≥

Mk | nk − 1| for x ∈ Ik+1 (ek ) . R

Let   x ∈ Iαj +2 xαj +1 eαj +1 ,

where 1 ≤ xαj +1 ≤ mαj +1 − 1.

By using (1.43) if we invoke equalities (1.45), (1.47) and (1.48), we get that    ⎛ j α  k  DMαk +1 DMk  j |Dn | =  −⎝ − DMl ψMk −1  k=1 ψMαk +1 −1 k=1 l=k ⎛ ⎞ j αk     ⎝ Mαk +1 − Mk − | nl − 1| Ml ⎠ = k=1

l=k

k=1

l=k

k=1

l=k

ml  −nl −1

⎛ ⎞ j αk     ⎝ Mαk +1 − Mk − ≥ (ml − 2) Ml ⎠ ⎛ ⎞ j αk αk      ⎝ Mαk +1 − Mk − = Ml+1 + 2 Ml ⎠

=

j  αk  k=1 l=k

Ml ≥ Mαj .

l=k

u=1

⎞   u ⎠ rl  

1.7 Lebesgue Constants

49

Hence, Dn 1 ≥

+

 αl s   l=0 k=l +1 Ik+1 (ek ) s 

Mk | nk − 1| dμ R

mαj +1 −1 



j =0 xαj +1 =1



Mαj dμ Iαj +2 xαj +1 eαj +1

 αl s  s    mαj +1 − 1 Mαj Mk | nk − 1| 1 ≥ + R Mk+1 Mαj +2 j =0

l=0 k=l

≥

αl s  s  | nk − 1|  1 + R2 2R l=0 k=l

≥

j =0

1 ∗ 1 v (n) . v (n) + 2 R 4R 

The proof is complete. Corollary 1.7.2 Let n ∈ N. Then Dn 1 ≤ 4R log n, where R =: sup mn . n∈N

Proof The proof readily follows by just using Lemma 1.7.1 and inequalities v (n) = 2|n| =

2 ln n 2 ln 2|n| ≤ ≤ 4 ln n and v ∗ (n) = R|n| = 2R ln n. ln 2 ln 2 

The proof is complete. Corollary 1.7.3 Let qn = M2n + M2n−2 + . . . + M2 + M0 . Then n ≤ Dqn 1 ≤ 4Rn, where R =: sup mn . 2R n∈N

Proof The upper estimate readily follows from Corollary 1.7.2. The proof of the lower inequality follows from Lemma 1.7.1 and from the identity v (qn ) = 2n. Thus we leave out the details.



50

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Corollary 1.7.4 Let k ∈ N+ . Then, R =: supn∈N mn , DM ≤ 2 + R n 1 and DM

n +1

≤ 4 + 2R. 1

Proof The proof follows by just using Lemma 1.7.1 and the following identities v (Mn ) = 2, v (Mn + 1) = 4 and inequalities v ∗ (Mn ) < mn < R, v ∗ (Mn + 1) < m0 + mn < 2R. 

The proof is complete. Lemma 1.7.5 Let n ∈ N. Then Mn −1 1  2 v (k) ≥ 2 . nMn R k=1

Proof Let Mk−1 ≤ n < Mk . Then n = nk−1 Mk−1 + n(1) ,

where n(1) < Mk−1 .

Moreover, we have that M k −1 

mk−1 −1(r+1)Mk−1 −1

v (n) =

n=Mk−1





r=1

n=rMk−1

v (n)

mk−1 −1Mk−1 −1

=





r=1

n=0

v (n + rMk−1 )

mk−1 −1Mk−2 −1

=





r=1

n=0

mk−1 −1Mk−1 −1

v (n + rMk−1 ) +





r=1

n=0



v (n + rMk−1 )

r=1 n=Mk−2

mk−1 −1Mk−2 −1

=



mk−1 −1Mk−1 −1

v (n + rMk−1 ) +





r=1 n=Mk−2

v (n + rMk−1 )

1.7 Lebesgue Constants

51

mk−1 −1Mk−2 −1

=





r=1

n=0

mk−1 −1Mk−1 −1



(v (n) + 2) + ⎛

= (mk−1 − 1) ⎝ ⎛ = (mk−1 − 1) ⎝



v (n)

r=1 n=Mk−2 Mk−2 −1



Mk−1 −1

(v (n) + 2) +

n=0 Mk−1 −1





⎞ v (n)⎠

n=Mk−2

⎞

v (n) + 2Mk−2 ⎠ .

n=0

This gives that M k −1 

Mk−1 −1

v (n) =

n=0



M k −1 

v (n) +

v (n)

n=Mk−1

n=0 Mk−1 −1

=



⎛

v (n) + (mk−1 − 1) ⎝

n=0

Mk−1 −1



v (n) + 2Mk−2 ⎠

n=0 Mk−1 −1

= mk−1



v (n) + 2Mk−2 .

n=0

Let T (Mk ) =:

M k −1 

v (n) .

n=0

Then T (Mk ) = mk−1 T (Mk−1 ) + 2Mk−2 . It is easy to see that this is valid for all k ∈ N. If we set A (k) =: T (Mk ) /Mk , then A (k) ≥ A (k − 1) + 2/R 2 , A (k − 1) ≥ A (k − 2) + 2/R 2 , ········· A (1) ≥ A (0) + 2/R 2 .

⎞

52

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Summing up these inequalities, we find that A (k) ≥ 2k/R 2 and T (Mk ) ≥ 2kMk /R 2 , 

which completes the proof.

1.8 Vilenkin-Fourier Coefficients The characteristic function χH of a set H is defined by  χH (x) =:

1, x ∈ H, 0, x ∈ / H.

First by using the previous findings we can formulate the following important proposition: Proposition 1.8.1 Since the Vilenkin function ψm is constant on In (x) for every x ∈ Gm and 0 ≤ m < Mn , it is clear that each Vilenkin function is a complexvalued step function, that is, it is a finite linear combination of the characteristic functions. On the other hand, notice that, by Corollary 1.6.7 (Paley’s Lemma), it holds that χIn (t ) (x) =

Mn −1 1  ψj (x − t) , x ∈ In (t), Mn j =0

for each x, t ∈ Gm and n ∈ N. Thus each step function is a Vilenkin polynomial. Consequently, we obtain that the collection of step functions coincides with the collection of Vilenkin polynomials P. Since the Lebesgue measure is regular it follows that given f ∈ L1 , there exist Vilenkin polynomials P1 , P2 . . . ,

such that Pn → f

a.e., as n → ∞.

Moreover, any f ∈ Lp (Gm ) can be written in the form f = g − h where the functions g and h are almost everywhere limits of increasing sequences of nonnegative Vilenkin polynomials. In particular, P is dense in the space Lp , for all p ≥ 1.

1.8 Vilenkin-Fourier Coefficients

53

Later on, we will prove that such Vilenkin polynomials Pn which converge almost everywhere to f ∈ Lp (Gm ) can for example be  n

 k  f (k) ψk (x) . 1− n k=0

It is evident that if f ∈ Lp (Gm ), then |f(i) | ≤ f p for all p ≥ 1. By using the density of Vilenkin polynomials in L1 (Gm ), we can prove a stronger result, which is called the Riemann-Lebesgue Lemma: Lemma 1.8.2 (Riemann-Lebesgue Lemma) If f ∈ L1 (Gm ), then f(k) → 0,

as k → ∞.

Proof By using Proposition 1.8.1, we know that the Vilenkin polynomials are dense in L1 and given ε > 0, there is P ∈ P such that f − P 1 < ε. Since P (n) = 0, for large n, we have that 

 Gm

f ψ n dμ =

Gm

(f − P ) ψ n dμ

for n > degP . Therefore   lim sup f(n) ≤ f − P 1 < ε n→∞

and the proof follows.



Lemma 1.8.3 Let f ∈ L1 and MN ≤ n ≤ MN+1 . Then there exists an absolute constant c, such that

   f(n) ≤ cω1 f, 1 . MN

54

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Proof By the definition 

f(n) =



Gm

= Gm

(1.52)

f (x) ψ n (x) dμ (x) f (x + eN ) ψ n (x + eN ) dμ (x + eN ) 

= ψ n (eN )

Gm

f (x + eN ) ψ n (x) dμ (x)

and ψn (eN ) f(n) =

 Gm

f (x + eN ) ψ n (x) dμ (x) .

We also know that n=

N 

nj Mj ,

nj ∈ Zmj

and nN = 0.

j =1

Since xk = 0, for k = 0, . . . , N − 1 and xN = 1, we obtain that ψn (eN ) =

N 

n rk k

(x) = exp 2πı

N 

k=0

 nk xk /mk

= exp 2πınN /mN .

k=0

It is easy to prove that exp 2πınN /m = 1. Moreover, by using (1.52), we get that     |exp 2πınN /mN − 1| f(n) = 2 sin2 πnN /mN f(n)     =  (f (x − eN ) − f (x)) ψ n (x) dμ (x) Gm  |f (x − eN ) − f (x)| dμ (x) ≤ Gm

 1 ≤ ω1 f, , MN which implies that   f(n) ≤ The proof is complete.



ω1 f, M1N 2 sin2 πnN /mN

≤ cω1

1 f, MN

 . 

1.8 Vilenkin-Fourier Coefficients

55

Lemma 1.8.4 (Bessel‘s Inequality) Let f ∈ L2 (Gm ) and n ∈ N. Then n    f(i)2 ≤ f 2 . 2 i=1

Proof Let f ∈ L2 (Gm ). Since 



f ψi dμ = f(i),

f ψi dμ = Gm

Gm

if we apply orthonormality of Vilenkin functions (see Proposition 1.2.4), we immediately get that 2 n   f (i) ψi f − i=1





f−

= Gm

n  i=1

 = +

f 22 n 

−



f(i) ψi 

f Gm

n 



n 

 f(i) ψi dμ

i=1

f(i) ψi dμ −



 f Gm

n 

 f(i) ψi dμ

i=1

 ψi ψi dμ Gm

i=1 n 

f(i) f(i) −

i=1

= f 22 −

f−

i=1

f(i) f(i)

= f 22 −

(1.53)

2

n 

n  i=1

f(i) f(i) +

n 

f(i) f(i)

i=1

|f(i) |2 ≥ 0.

i=1



The proof is complete. Finally, we present a useful property of Fourier coefficients:

Proposition 1.8.5 Under the map  , convolution of functions corresponds to pointwise multiplication of sequences, i.e., f ∗ g (k) = f(k)  g (k) , k ∈ N.

56

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Proof By combining Fubini‘s theorem and translation invariance of the Haar measure and Proposition 1.2.1, we get that f ∗ g (k) =



 f (t) g ((x − t)) dμ (t) ψk (x) dμ (x)

Gm





Gm

f (t) g ((x − t)) dμ (t) ψk (t) ψk (x − t) dμ (x)

= Gm





Gm

f (t) g ((x − t)) ψk (x − t) ψk (t) dμ (x) dμ (t)

= Gm

 =

Gm

 g ((x − t)) ψk (x − t) dμ (x − t)

f (t) ψk (t) dμ (t) · Gm

Gm

= f(k)  g (k) , k ∈ N. 

The proof is complete.

1.9 Partial Sums By using the density of Vilenkin polynomials in Lp (Gm ), we can also derive the following information for the partial sums SN : Lemma 1.9.1 Let f ∈ Lp (Gm ) or f ∈ C(Gm ) if p = ∞. Then the following statements are equivalent for any 1 ≤ p ≤ ∞ : (i) SN f − f p → 0 as N → ∞; (ii) supN SN f p < Cp f p . Proof If (i) holds, then SN f − f p < Cp and SN f p = SN f − f + f p ≤ SN f − f p + f p < Cp + f p , so (ii) holds. On the other hand, assume that (ii) holds. Then, by using the density of the Vilenkin polynomials (see Proposition 1.8.1) we get that, for any ε > 0, f − T p < ε/2Cp , for sufficiently large n > deg T . We get that for such N, SN f − f p = SN f − SN T + SN T − f p ≤ SN (f − T )p + SN T − f p

1.9 Partial Sums

57

= SN (f − T )p + T − f p < Cp T − f p + T − f p < ε, so (i) holds and the proof is complete.



Theorem 1.9.2 The Fourier series does not converge on L1 and L∞ , i.e. there exists f ∈ L1 (Gm ) such that Sn f − f 1  0, as n → ∞, and there exists f ∈ L∞ (Gm ) such that Sn f − f ∞  0, as n → ∞. Proof According to Lemma 1.9.1 it is sufficient to prove the first statement to verify that sup

sup Sn f 1 = ∞

n∈N+ f 1 ≤1

(1.54)

By using Lemma 1.7.1 we see that DN 1 → ∞, as N → ∞. Let qk−1 < M2k . Then Sqk−1 (DM2k )1 = Dqk−1 1 .

(1.55)

If we apply Corollary 1.7.3, since qn < M2n we get that Sqn−1 (DM2n )1 = Dqn−1 1 ≥ cn → ∞, as n → ∞. Hence, according to the fact that DMn 1 = 1 (see Corollary 1.6.8) we conclude that (1.54) holds. The proof of the first statement is complete. The easiest proof of the divergence of the partial sums with respect to Vilenkin systems in the space L∞ uses the non-boundedness of Dirichlet’s kernel in L1 and the Banach–Steinhaus uniform boundedness principle. We omit the details.  Theorem 1.9.3 Let f ∈ L1 (Gm ) and (αk ) be an increasing sequence of natural numbers. Then there exists an absolute constant c, such that Sα f ≤ c f 1 , k 1

(1.56)

58

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

and Sα f − f → 0, as k → ∞, k 1

(1.57)

  sup v (αk ) + v ∗ (αk ) < C < ∞,

(1.58)

if and only if

k∈N

where v and v ∗ are defined in (1.2). Proof By using Theorem 1.5.3 we can conclude that Sα f ≤ f 1 Dα 1 , k k 1 so (1.56) holds. If (1.58) holds, then Lemma 1.7.1 implies that Dαk 1 < C < ∞ and Sα f ≤ C f 1 , k 1 so that by Lemma 1.9.1, we get that (1.57) holds. On the other hand, if supk∈N (v (αk ) + v ∗ (αk )) Lemma 1.7.1 and (1.55) we get that

=

∞, by combining

sup Sαk f 1 = sup f ∗ Dαk 1 = Dαk 1 → ∞, as k → ∞,

f 1 =1

f 1 =1

which implies that (1.57) does not hold, so the proof is complete. Corollary 1.9.4 Let n ∈ N, 1 ≤ p < ∞ and f ∈ Lp (Gm ). Then SM f ≤ f p n p and SM f − f → 0, as n → ∞. n p Corollary 1.9.5 Let N ∈ N and f ∈ Lp (Gm ) for some 1 ≤ p < ∞. Then lim sup f (x − t) − f (x)p = 0.

N→∞ t ∈IN



1.9 Partial Sums

59

Proof By using Corollary 1.9.4 for any ε > 0, there exists k ∈ N such that SMk ∈ P and f − SM f < ε/2. k p Let N > k. Then, by using Proposition 1.2.1 (see also (1.59)), we find that SMk f (x − t) = SMk f (x), for t ∈ IN . Hence, sup f (x − t) − SMk f (x − t) + SMk f (x − t) − f (x)p

t ∈IN

= sup f (x − t) − SMk f (x − t) + SMk f (x) − f (x)p t ∈IN

≤ sup f (x − t) − SMk f (x − t)p + sup SMk f (x) − f (x)p t ∈IN

t ∈IN

≤ ε/2 + ε/2 = ε. 

The proof is complete. Our next important theorem reads: Theorem 1.9.6 Let f ∈ L2 (Gm ). Then we have that (i) (Boundedness of the partial sums) Sn f 2 ≤ f 2 , (ii) (Fourier expansion) Sn f − f 2 → 0, as n → ∞, (iii) (Bessel‘s equality) f 22 =

∞    f(i)2 , i=1

(iv) if f(i) = 0 for all i ∈ N, then f = 0 a.e. on Gm . This means that the Vilenkin systems are complete in L2 (Gm ).

60

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Proof (i) By using the orthonormality of the Vilenkin functions and Bessel‘s inequality (see Lemma 1.8.4) we can conclude that 

 Sn f 22

=

n 

Gm

=

 f(i) ψi

i=1

n  n 

n 

  f (k) ψk dμ

k=1

f(i) f(k)

 ψi ψk dμ Gm

i=1 k=1

=

n 

|f(i) |2

i=1

≤ f 22 . (ii) If we apply Lemma 1.9.1 in the case when p = 2 we immediately get convergence of partial sums of f ∈ L2 (Gm ) in L2 (Gm )-norm. (iii) According to the identity (1.53) and (ii) we get that f 22

−

n  i=1

2 n  2   |f (i) | = f − f (i) ψi i=1

= f −

Sn f 22

2

→ 0, as n → ∞.

(iv) If f(i) = 0, for all i ∈ N, then by using (iii) we get that f 22 =

∞    f(i)2 = 0 i=1

which implies that f (x) = 0 a.e. on Gm . The proof is complete.  Lemma 1.9.7 Let f ∈ L1 , y ∈ Gm and N ∈ N. Then  f (t) Dn (x − t) dt = 0

lim

n→∞ Gm \IN (y)

for all x ∈ IN (y).

1.9 Partial Sums

61

Proof Since x − t ∈ / IN (0) , by using (1.43) in Lemma 1.6.5 for n = we get that

∞

i=0 ni Mi ,

 f (t) Dn (x − t) dt Gm \IN (y)

=

N−1 

⎛



=



f (t) ψn (x − t) ⎝DMj (x − t)

⎞ rjk (x − t)⎠ dt

k=mj −nj

j =0 G \I (x) m N N−1 

mj −1

gˆ j (n) ,

j =0

where ⎛ gj (t) =: f (t) ⎝DMj (x − t)

mj −1



⎞ rjk (x − t)⎠ .

k=mj −nj

According to the fact that gj ∈ L1 , if we apply the Riemann-Lebesgue Lemma (Lemma 1.8.2) we can conclude that gˆj (n) → 0,

n → ∞.

Hence,  lim

n→∞ Gm \IN (y)

f (t) Dn (x − t) dt = 0.



The proof is complete.

Theorem 1.9.8 (Localization Principle) Let y ∈ Gm , N ∈ N, f, g ∈ L1 (Gm ) and f = g on IN (y). Then Sn f (x) and Sn g (x) simultaneously converge or diverge for every x ∈ IN (y). Proof By using Lemma 1.9.7 we get that  Sn f (x) − Sn g (x) =

(f (t) − g (t)) Dn (x − t) dt → 0. Gm \IN (y)

The proof is complete.



62

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Theorem 1.9.9 (Dini’s Test) Let x ∈ Gm , f ∈ L1 (Gm ) and define the function g by g (t) =:

f (t) − f (x) , |x − t|

t ∈ Gm \{x}.

If g ∈ L1 (Gm ), then Sn f (x) → f (x) , n → ∞. Proof By using Lemma 1.6.10 we find that           |Sn f (x) − f (x)| ≤  |g (t)| dt. (f (t) − f (x)) Dn (x − t) dt  + C   Gm \IN (x)  IN (x) Moreover, by Theorem 1.9.8,  (f (t) − f (x)) Dn (x − t) dt → 0, n → ∞. Gm \IN (x)

Since g ∈ L1 (Gm ) we get that, for every ε > 0,  |g (t)| dt ≤ ε IN (x)



when N is large enough and the proof follows. Dini‘s test immediately implies the following: Corollary 1.9.10 Let f ∈ L1 (Gm ) and

|f (x − t) − f (x)| = O (log (1/|t|))−1−ε ,

ε > 0, t → 0.

Then Sn f (x) → f (x) , n → ∞. The modulus of continuity of functions in the Lebesgue spaces f ∈ Lp (Gm ) is defined by

ωp

1 ,f Mn

 =: sup f (· − h) − f (·)p h∈In

1.9 Partial Sums

63

The best approximation En (f, Lp ) of function f ∈ Lp (Gm ) 1 ≤ p < ∞ is defined by   inf f − ψp = f − En f, Lp p

ψ∈Pn

if it exists, where Pn is the set of all Vilenkin polynomials of order less than n ∈ N. Set   En f, Lp =: inf f − ψp , ψ∈Pn

Lemma 1.9.11 For any function f ∈ L2 (Gm ), we have that min f − P 2 = f − Sn f 2 ,

P ∈Pn

i.e.,   En f, Lp = Sn f

in the space L2 (Gm ).

Proof Analogously to (1.53) we can conclude that 2 n n n n     2   f (i) αi − αi ψi = f 2 − αi f (i) + |αi |2 f − i=1

2

= f 22 −

i=1

i=1

n 

n 

i=1

|f(i) |2 +

i=1

|f(i) − αi |2 .

i=1



This identity implies the proof. Proposition 1.9.12 Let n ∈ N. Then



   1 1 1 p ωp f, ≤ EMn f, L ≤ f − SMn f p ≤ ωp f, . 2 Mn Mn Proof First we note that we have that   EMn f, Lp =

inf f − P p ≤ f − SMn f p

P ∈PMn

 = 

Gm

≤ Gm

(f (x − t) − f (x)) DMn (t) dμ (t)

p

(f (x − t) − f (x))p DMn (t) dμ (t)

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1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

 = Mn In (0)

(f (x − t) − f (x))p dμ (t)

 1 ≤ ωp f, . Mn To prove the first inequality we observe that if P ∈ PMn and y ∈ In (0) , then P (x − y) =

M n −1

αk ψk (x − y)

(1.59)

k=0

=

M n −1

αk ψk (x) ψk (y)

k=0

=

M n −1

αk ψk (x) = P (x) .

k=0

Hence, τy f (x) − f (x) = f (x − y) − P (x − y) + P (x) − f (x) and

 1 = sup τy f (x) − f (x) p ≤ 2 f − P p ωp f, Mn y∈In (0) for every P ∈ PMn . It follows that

   1 1 ωp f, ≤ EMn f, Lp , 2 Mn and also the first inequality is proved so the proof is complete.



Theorem 1.9.13 Let f ∈ L1 (Gm ) and Mk < n ≤ Mk+1 . Then there is an absolute constant C such that 

1 Sn f − f 1 ≤ C log nω1 ,f . Mk

1.9 Partial Sums

65

Proof Let Mk < n ≤ Mk+1 . Then, by using Corollary 1.7.2, we can conclude that   Sn f − f 1 ≤ Sn SMk f − f 1 + SMk f − f 1 ≤ C (log n + 1) SMk f − f 1

 1 ≤ C log nω1 ,f . Mk 

The proof is complete. Theorem 1.9.14 (a) Let f ∈ L1 (Gm ) and

ω1

1 ,f Mn

 =o

 1 , as n → ∞. n

Then Sk f − f 1 → 0, as k → ∞. (b) There exists a function f ∈ L1 (Gm ), for which

ω1

1 ,f Mn



 1 , as n → ∞ =O n

and Sk f − f 1  0, as k → ∞. Proof Let f ∈ L1 (Gm ) and

ω1

1 ,f Mn

 =o

 1 , as n → ∞. n

By using Theorem 1.9.13 we get that Sn f − f 1 → ∞,

as n → ∞.

To prove part (b) we consider the function f =

∞  1 (DM2k +1 − DM2k ). 2k k=0

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1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

Then f ∈ L1 (Gm ) and ⎧ 1 * + ⎨ 2k , j ∈ M2k , . . . , M2k +1 − 1 , k ∈ N+ , ∞ * + f(j ) = M2k , . . . , M2k +1 − 1 . / ⎩ 0, j ∈ k=0

Moreover, SM2n f =

n−1  1 (DM2k +1 − DM2k ) 2k k=0

and f − SM2n f =

∞  1 (DM2k +1 − DM2k ). 2k k=n

By using Proposition 1.9.12 we find that

ω1

1 ,f M2n

 ∼ f − SM2n f 1 =

 ∞  1 1 = O , as n → ∞. k 2 2n

(1.60)

k=n

Let M2k ≤ j < M2k +1 , k ∈ N+ . Then Sj f = SM2k f +

Dj − DM2k 2k

= SM2k f +

Dj −M2k 2k

.

(1.61)

By combining (1.60) and (1.61) with Corollary 1.7.3, we find that ψM2k Dq2k−1 −1 lim sup Sq2k−1 f − f 1 = lim sup + SM2k f − f k 2 k→∞ k→∞ 1

 1 ≥ lim sup k Dq2k−1 −1 1 − SM2k f − f 1 2 k→∞ ∞  1 ≥ C − lim sup = C > 0. 2i k→∞ i=k

The proof is complete.



1.10 Final Comments and Open Questions

67

1.10 Final Comments and Open Questions (1) The definitions and proofs in Sect. 1.2 can be found in [3]. Similar results concerning the Walsh system are proved in [295]. (2) Representation of the Vilenkin groups on the interval [0, 1) in Sect. 1.3 can be found in [3]. Similar results concerning the Walsh group on the interval [0, 1) are proved in [295]. (3) The proofs of the classical inequalities in Sect. 1.4 by using a convexity approach, which we think is not presented in this complete form in some book before, are mainly taken from Persson [267] (c.f. also [200], [205], [251] and [268]). (4) For the definitions and proofs of Sect. 1.5 see [378] and [429]. (5) The proof Lemma 1.6.1 can be found in Agaev et al. [3]. (6) The proofs of Lemmas 1.6.2 and 1.6.3 can be found in Blahota et al. [44]. (7) The first equality of Lemma 1.6.4 was proved in Vilenkin [389] and the second identity in Gát and Goginava [114]. (8) The proofs of Lemma 1.6.5, Corollaries 1.6.6, 1.6.7, and 1.6.8 for the Walsh system were presented in the book [295] and for bounded Vilenkin systems in the book [3]. (9) The proof of Lemmas 1.6.9 and 1.6.11 can be found in Tephnadze [351]. (10) The first proof of Lemma 1.7.1 with some different constants was given in Lukomskii [217]. For a new and shorter proof described in this book which improved the upper bound and provided a similar lower bound see [44]. (11) The proofs of Corollaries 1.7.2, 1.7.3 and 1.7.4 for the Walsh system can be found in the book [295] and for bounded Vilenkin systems in the book [3]. (12) The proof of Lemma 1.7.5 for bounded Vilenkin systems can be found in Memic et al. [223] and for the Walsh system in [96]. (13) Proofs of classical results on Fourier coefficients with respect to Vilenkin systems in Sect. 1.8 can be found in the books [3] and [328]. (14) Proofs of classical results on partial sums with respect to Vilenkin systems in Sect. 1.9 can be found in the books [3], [378] and [328]. Similar results concerning Walsh system are proved in the book [295]. (15) The proof of almost everywhere divergence of partial sums with respect to Vilenkin systems which is given in Sect. 2.6 are presented in [277]. Similar problem for the Walsh system was treated in the book [295]. (16) Since the Vilenkin system is the group of all characters, there is a link between Fourier analysis and abstract harmonic analysis. A generalization of characters was introduced by Gát [106] by means of the concept of ψα systems. These systems are applicable in some mathematical theories, for instance in the study of almost even arithmetical functions and limit periodic arithmetic functions. The properties of these characters was studied by Toledo [376]. (17) The fact that every locally compact group (abelian or not) has a Haar measure was proved in Pontryagin [279], for example. For a slick construction of such

68

1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces

a measure see Bredon [53]. In the abelian case, the construction can be carried out efficiently using Kakutani’s fixed point theorem (see Rudin [284]). (18) Uniform and point-wise convergence and some approximation properties of partial sums with respect to Vilenkin (Walsh) and trigonometric systems in L1 norms was investigated by Antonov [12], Avdispahi´c and Memi´c [18], Baramidze [20], Goginava [124, 128], Shneider [297], Simon [300, 302], Sjölin [321], Persson [262, 263], Onneweer and Waterman [256, 258]. Moreover, Fine [96] derived sufficient conditions for the uniform convergence, which are in complete analogy with the Dini-Lipschitz conditions. Guli´cev [150] estimated the rate of uniform convergence of a Walsh-Fourier series by using Lebesgue constants and modulus of continuity. Uniform convergence of subsequences of partial sums was investigated also in Goginava and Tkebuchava [137], Fridli [100] and Gát [110]. Approximation properties of the two-dimensional partial sums with respect to Vilenkin and trigonometric systems can be found [295] and [464]. (19) It has been pointed out that the behaviour of the Dirichlet kernel of the WalshKaczmarz system is worse in a certain sense than that of the kernel of the ˘ Walsh-Paley system considered more often. In 1948 Sneider [296] introduced the Walsh-Kaczmarz system and showed that the inequality lim sup n→∞

Dnκ (x) ≥C>0 log n

holds almost everywhere. This “spreadness” of the Dirichlet kernel of this system makes it easier to construct examples of divergent Fourier series. A number of other pathological properties of this system is also connected with this “spreadness” property of the kernel. Thus, for example, for Fourier series with respect to the Walsh-Kaczmarz system, it is impossible to establish any local test for convergence at a point or on an interval, since the principle of localization does not hold for this system. In 1974 Schipp [288, 289] and Young [452] proved that the Walsh-Kaczmarz system is a convergence system. (20) For the trigonometric system it is important to note that results of Fejér and Szego, later given in [336], in particular, contains an explicit formula for Lebesgue constants, namely, Ln =

 ∞ 1 1 1 1 16  1 + + + . . . + . π2 4k 2 − 1 3 5 2k(2n + 1) − 1 k=1

(21) An investigation of the hamming cube and the analogous consideration concerning the Rademacher and Walsh functions which are connected to dyadic analysis and Walsh system can be found in [89, 170, 171].

1.10 Final Comments and Open Questions

69

(22) The most important property of Lebesgue constants with respect to the WalshPaley system were obtained by Fine in [96]. In [295], the two-sided estimate V (n) ≤ Ln ≤ V (n) 8 was proved, where n =

∞

j =1 nj 2

V (n) =:

j

and V (n) is defined by

∞    nj +1 − nj  + n0 . j =1

In [217], Lukomskii presented the estimate Ln ≥ V (n)/4. Moreover, Malykhin et al. [220] (see also Astashkin and Semenov [17]) improved this estimate above and proved the following: V (n) + 1 ≤ Ln ≤ V (n). 3 Here the factors 1/3 and 1 are sharp. Remark 1.10.1 We note that v ∗ (n) = 0 for the Walsh system.  Open Problem Let n = ∞ i=1 ni Mi . Investigate if or not it is possible to derive two-sided estimates of the Lebesgue constants Ln for unbounded Vilenkin systems. Remark 1.10.2 By making a simple observation related to Lemma 1.7.1 we obtain that the proof of the upper estimate of the Lebesgue constants is true also for unbounded Vilenkin systems. To obtain the lower estimate of the Lebesgue constants for unbounded Vilenkin systems with absolute constant we need more precise estimates than those in the proof of Lemma 1.7.1. ∞ Open Problem Let n = i=1 ni Mi . Derive the Lebesgue constants Ln with respect to the Walsh-Kaczmarz system and prove two-sided estimates in this case. Moreover, use this estimate to find necessary and sufficient conditions for the indexes nk , such that κ S f − f → 0, as k → ∞. nk 1 for all f ∈ L1 , where Snκ f is n-th partial sum with respect to Kaczmarz system.

Chapter 2

Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-Fourier Series

2.1 Introduction Topics of almost everywhere convergence and convergence in Lp -norm of classical operators related to Vilenkin-Fourier series widely use methods of martingale theory. By applying methods of martingale theory, Calderón–Zygmund decomposition of integrable function can be also given, which is a fundamental result in Fourier analysis, harmonic analysis, and singular integral. Given an integrable function f : Gm → C, this gives a precise way of partitioning Gm into two sets, one where f is essentially small and the other is a countable collection of Vilenkin intervals where f is essentially large, but where some control of the function is retained. This leads to the associated Calderón–Zygmund decomposition of f, where f is written as the sum of “good” and “bad” functions, using the above sets. By using Calderón–Zygmund decomposition, we easily obtain a sufficient condition which provides weak-(1, 1) type inequality for the σ -sublinear operator V which is bounded from Lp1 to Lp1 for some 1 < p1 ≤ ∞. In particular, if  |Vf | dμ ≤ C f 1 I

for f ∈ L1 and Vilenkin interval I which satisfy suppf ⊂ I and Gm f dμ = 0, then the operator V is of weak-type (1, 1). This sufficient condition works for summability methods with respect to a Vilenkin system with integrable kernels, but it is useless to study various type of convergences of partial sums. Despite this complexity (see e.g. the book [295]), it was proved that there exists an absolute constant Cp depending only on p, such that Sn f p ≤ Cp f p , when 1 < p < ∞. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 L.-E. Persson et al., Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series, https://doi.org/10.1007/978-3-031-14459-2_2

71

72

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

The boundedness does not hold for p = 1 or p = ∞, but Watari [393] (see also Gosselin [147] and Young [453]) proved that there exists an absolute constant C such that, for n = 1, 2, . . . , the weak type estimate yμ {|Sn f (x)| > y} ≤ C f 1 , f ∈ L1 (Gm ), y > 0, holds, where μ is the Haar measure on the locally compact abelian group Gm . For an integrable function f ∈ L1 (Gm ) and for any x ∈ Gm , we define the maximal function M(f ) by     .  f dμ (t) (t)   n∈N μ (In (x)) In (x)

M(f ) (x) =: sup

1

(2.1)

Then, by applying the technique of martingale theory, it can be proved that it is bounded from L1 (Gm ) to the space weak-L1 (Gm ) , that is yμ {|M(f )(x)| > y} ≤ c f 1 , f ∈ L1 (Gm ), y > 0. S#∗ defined by Moreover, if we consider restricted maximal operator .   . S#∗ f =: sup SMn f  , (Mk =: m0 . . . mk−1 , k = 0, 1 . . .),

(2.2)

n∈N

according the fact that supn∈N |SMn f (x)| = M(f )(x), we also get that * ∗ + yμ . S# f > y ≤ c f 1 , f ∈ L1 (Gm ), y > 0. Hence, if f ∈ L1 (Gm ), then SMn f → f a.e. on Gm . The almost everywhere convergence of subsequences of Vilenkin-Fourier series was considered in [45], where some methods of martingale theory were used. The almost-everywhere convergence of Fourier series for L2 functions was postulated by Luzin [219] in 1915 and the problem was known as Luzin’s conjecture. Carleson’s theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of L2 functions, proved by Carleson [66] in 1966. The name is also often used to refer to the extension of the result by Hunt [166] which was given in 1968 to Lp functions for p ∈ (1, ∞) (also known as the Carleson-Hunt theorem). Carleson’s original proof is exceptionally hard to read, and although several authors have simplified the arguments there are still no easy proof of his theorem. Expositions of the original paper Carleson include these by Kahane [184], Mozzochi [229], Jorsboe and Mejlbro [182] and Arias de Reyna [16]. Moreover, Fefferman [91] published a new proof of Hunt’s extension, which was done by bounding a maximal operator S ∗ of the partial sums defined by S ∗ f =: sup |Sn f | . n∈N

2.1 Introduction

73

This, in its turn, inspired a much simplified proof of the L2 result by Lacey and Thiele [204], explained in more details in Lacey [201]. In the books Fremlin [99] and Grafakos [149] there are also given proofs of Carleson’s theorem. An interesting extension of the Carleson-Hunt result, which is much more closer to L1 (Gm ) space then Lp (Gm ) for any p > 1, was done by Carleson’s student Sjölin [321] and later on, by Antonov [12]. Already in 1923, Kolmogorov [192] showed that the analogue of Carleson’s result for L1 is false by finding such a function whose Fourier series diverges almost everywhere (improved slightly in 1926 to diverging everywhere), and this result indeed was guiding for efforts for many authors after that Carleson proved his results in 1966. In 2000, Kolmogorov’s result was improved by Konyagin [193], by finding functions with everywhere-divergent Fourier series in a space smaller than L1 (Gm ), but a suitable candidate for such a space that is consistent with the results of Antonov and Konyagin is still an open problem. The famous Carleson theorem was very important and surprising when it was proved in 1966. Since then this interest has remained and a lot of related research has been done. In fact, in recent years this interest has even been increased because of the close connections to e.g. scattering theory [231], ergodic theory [80, 81], the theory of directional singular integrals in the plane [28, 77, 79, 203] and the theory of operators with quadratic modulations [209]. We refer to [201] for a more detailed description of this fact. These connections have been discovered from various new arguments and results related to Carleson’s theorem, which have been found and discussed in the literature. We mean that these arguments share some similarities, but each of them has also a distinct new idea behind, which can be further developed and applied. It is also interesting to note that, for almost every specific application of Carleson’s theorem in the aforementioned fields, mainly only one of these new arguments was used. The analogue of Carleson’s theorem for the Walsh system was proved by Billard [32] for p = 2 and by Sjölin [320] for 1 < p < ∞, while for bounded Vilenkin systems it was proved by Gosselin [147]. Schipp [289, 290, 295] investigated the so called tree martingales, i.e., martingales with respect to a stochastic basis indexed by a tree, and generalized the results about maximal function, quadratic variation and martingale transforms to these martingales (see also [294, 400]). Using these results, he gave a proof of Carleson’s theorem for Walsh-Fourier series. A similar proof for bounded Vilenkin systems can be found in Schipp and Weisz [294, 400]. In each proof, they show that the maximal operator of the partial sums is bounded on Lp (Gm ), i.e., there exists an absolute constant Cp such that ∗ S f ≤ Cp f p , when f ∈ Lp , 1 < p < ∞. p A recent proof of almost everywhere convergence of subsequences of Walsh-Fourier series was given by Demeter [78] in 2015. In this Chapter we give a new and shorter proof of almost everywhere convergence of Vilenkin-Fourier series, which use the theory of martingales, described in Persson et al. [277].

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2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

Stein [325] constructed an integrable function whose Vilenkin-Fourier (WalshFourier) series diverges almost everywhere. In [295] it was proved that there exists an integrable function whose Fourier-Walsh series diverges everywhere. Kheladze [190, 191] proved that for any set of measure zero there exists a function in f ∈ Lp (Gm ) (1 < p < ∞) whose Vilenkin-Fourier series diverges on the set, while the result for continuous or bounded function was proved by Harris [160] or Bitsadze [33]. From the above-mentioned result of Hunt for Vilenkin-Fourier series for a function of the class Lp , 1 < p < ∞, it follows that the Vilenkin-Fourier series can be divergent only on a set of measure zero. Moreover, Simon [301] constructed an integrable function such that its Vilenkin-Fourier series diverges everywhere. Similar results for the bounded Vilenkin systems was proved by Persson et al. [277]. Bochkarev [49] considered rearrangements of Vilenkin-Fourier series of bounded type. It is not known whether Carleson’s theorem holds for unbounded Vilenkin systems. However, some theorems were proved for unbounded Vilenkin systems by Gát [109, 111, 112], Simon [300, 301] and Tarkaev [340]. Antonov [13] proved that for f ∈ L1 (log+ L)(log+ log+ log+ L)(Gm ) its Walsh-Fourier series converges almost everywhere. Similar result for the bounded Vilenkin systems was proved by Oniani [255]. However, there exists a function from L1 (log+ log+ L)(Gm ) whose Vilenkin-Fourier series diverges everywhere, where in this result Gm is a general (not necessary “bounded”) Vilenkin group (see Tarkaev [340]).

2.2 Conditional Expectation Operators Let (, G, P ) be a complete probability space and G1 be a complete σ -algebra, such that G1 ⊂ G and ξ, η are G -measurable and G1 -measurable functions, respectively, such that   |ξ | dP < ∞ and E(|η|) =: |η| dP < ∞, E(|ξ |) =: 



where E is called expectation operator. Definition 2.2.1 We say that η is the conditional expectation operator of ξ relative to G1 if E(ξ χA ) = E(ηχA )

for all A ∈ G1 ,

2.2 Conditional Expectation Operators

75

that is, 

 ξ dP = A

χdP ,

for all A ∈ G1 .

A

In this case E(ξ ) = E(η). We use the notation η =: E (ξ | G1 ). Lemma 2.2.2 Let H ≡ L2 (, G, P )

and H1 ≡ L2 (, G1 , P ) .

Then E (ξ | G1 ) is the orthogonal projection operator of the space H to the space H1 . Proof Let us denote by ξG1 the orthogonal projection operator of H to H1 . Since ξG1 − ξ ⊥ H1 , we have that    E η ξG1 − ξ = 0 for every η ∈ H1 . If we take η = χA , we get that   E(ξ χA ) = E χA ξG1 . Since ξG1 ∈ H1 , we obtain that ξG1 is G1 -measurable and E (ξ | G1 ) = ξG1 . 

The proof is complete. Lemma 2.2.3 Let H ≡ L2 (, G, P )

and H1 ≡ L2 (, G1 , P ) .

Then the best approximation in L2 norm of ξ ∈ H in H1 is the conditional expectation operator E (ξ | G1 ) relative to G1 . Proof We have that



2 = E (ξ − E (ξ | G1 ))2 . inf E (ξ − η)2 = E ξ − ξG1

η∈H1



The proof is complete. Lemma 2.2.4 Let D = {D1 , D2 , . . .}

76

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

be a set of finite or countable many subsets of , such that ∞ 

Di = , P (Di ) > 0, i ≥ 1.

i=1

If G = σ (D) and Eξ exists, then   E ξ χD i E (ξ | G) = P (Di )

a.e on the set Di , i ≥ 1.

Proof It is well known that if η is G-measurable, where G = σ (D) , then it takes constant values on the sets Di , i ≥ 1, i.e., it can be represented in the following way: ξ (x) =

∞ 

ξk χDk (x) .

k=1

Denote by Ci the constant value of E (ξ | G) on the set Di . Then   / 0 E ξ χDi = E E (ξ | G) χDi = Ci P (Di ) and Ci =

1 Eξ χDi . P (Di ) 

The proof is complete.

Now we prove uniqueness of conditional expectation operator relative to a σ algebra G : Theorem 2.2.5 Let η1 and η2 be two conditional expectation operators of ξ relative to a σ -algebra G. Then η1 = η2 a.e. Proof By the definition, we have that E (η1 χA ) = E (η2 χA ) for any A ∈ G. It follows that E ((η1 − η2 ) χA ) = 0 for any A ∈ G.

2.2 Conditional Expectation Operators

77

By defining B =: {x : η1 (x) − η2 (x) > 0} , we can conclude that B ∈ G,   0 = E ((η1 − η2 ) χB ) = E (η1 − η2 ) χ{η1 −η2 >0} and (η1 − η2 ) χ{η1 −η2 >0} = 0

a.e.

It follows that η1 − η2 ≤ 0

a.e.

η1 − η2 ≥ 0

a.e.

Analogously we can prove that

By combining the last two inequalities we can conclude that η1 − η2 = 0

a.e. 

The proof is complete. Theorem 2.2.6 (i) Let E |ξ | < ∞ and c be a constant. Then E (cξ | G) = cE (ξ | G) a.e. For example, E (0 | G) = 0 a.e. (ii) Let E |ξ | < ∞. Then E (E (ξ | G)) = E (ξ ) . (iii) Let E |ξ1 | < ∞ and E |ξ2 | < ∞. Then E (ξ1 ± ξ2 | G) = E (ξ1 | G) ± E (ξ2 | G) (iv) Let E |ξ | < ∞ and ξ be G-measurable, then E (ξ | G) = ξ a.e.

a.e.

78

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

(v) Let E |ξ | < ∞ and G1 ⊂ G. Then E (E (ξ | G1 ) | G) = E (E (ξ | G) | G1 ) = E (ξ | G1 ) a.e. Proof (i) it is easy to see that E (cξ χA ) = cE (ξ χA ) , for any A ∈ G, which implies the first equality. If we take c = 0, then we get that E (0 | G) = E (0 · ξ | G) = 0 · E (ξ | G) = 0, a.e. (ii) If we take A = , then we have that E (E (ξ | G)) = E (ξ ) . (iii) Since E (ξ1 ± ξ2 ) χA = E (ξ1 χA ) ± E (ξ2 χA )

for any A ∈ G,

we find that E (ξ1 ± ξ2 | G) = E (ξ1 | G) ± E (ξ2 | G) a.e. (iv) is obvious. (v) Since E (ξ | G1 ) is G-measurable, by applying (iv) we obtain that E (E (ξ | G1 ) | G) = E (ξ | G1 ) a.e. Finally, we prove that E (E (ξ | G) | G1 ) = E (ξ | G1 ) a.e. Let E (ξ | G1 ) = ξ1 and E (ξ | G) = ξ2 . If A ∈ G1 , then A ∈ G and E (ξ2 χA ) = E (ξ χA ) = E (ξ1 χA ) and also last equality holds. The proof is complete. 

2.3 Martingales and Maximal Functions

79

2.3 Martingales and Maximal Functions In this Chapter, we investigate some special martingales, which are very useful in Fourier analysis. Define the σ -algebras generated by Vilenkin intervals: Fn =: {In (x) , x ∈ Gm } . The conditional expectation operators of f relative to Fn is denoted by En f (n ∈ N). A map ν : Gm → N ∪ ∞ is called a stopping time relative to (Fn , n ∈ N) if {x ∈ Gm : ν(x) = n} =: {ν = n} ∈ Fn . It is well-known that {ν ≤ n} ∈ Fn (n ∈ N) and {ν ≥ n} ∈ Fn−1 (n ∈ N). The conditional expectation operator En f coincides with the SMn -th partial sum of f. Theorem 2.3.1 For f ∈ L1 (Gm ), we have En f = SMn f (n ∈ N). Proof By using Paley’s Lemma (see Corollary 1.6.7), we find that SMn f (x) =

M n −1

f(k) wk (x) = |In (x)|−1

k=0

 f (t)dμ(t), In (x)

where |In (x)| = Mn−1 denotes the length of In (x) . By using Theorem 2.2.4, we get that En f (x) = SMn f (x) . The proof is complete.  Now we introduce the notion of martingales with respect to the σ -algebras Fn (n ∈ N) as follows. Definition 2.3.2 A sequence f = (fn , n ∈ N) of integrable functions fn is said to be a martingale relative to the σ -algebras (Fn , n ∈ N) if (1) fn is Fn measurable for all n ∈ N, (2) En fk = fn for all n ≤ k. Martingales with respect to (Fn , n ∈ N) are called Vilenkin martingales.

80

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

In particular, fn is a Vilenkin polynomial, such that deg(fn ) < Mn , i.e. fn =

M n −1

αi ψi ∈ PMn ,

fk = fn +

i=0

M k −1 

αi ψi .

(2.3)

i=Mn

It is easy to prove that the sequence (Fn , n ∈ N) is regular, i.e., fn ≤ Rfn−1 (n ∈ N+ ), R =: sup mn ,

(2.4)

n∈N

for all non-negative Vilenkin martingales f = (fn , n ∈ N). The martingale f = (fn , n ∈ N) is said to be Lp -bounded (0 < p ≤ ∞) if fn ∈ Lp and f Lp =: sup fn p < ∞. n∈N

The set of Lp -bounded martingales is denoted by Lp (Gm ). If 1 ≤ p < ∞ and f ∈ Lp (Gm ) , then it is easy to prove that the sequence F = (En f, n ∈ N) is a martingale. This type of martingales are called regular. Moreover, F is Lp -bounded and lim En f − f p = 0.

n→∞

(2.5)

Consequently, F Lp = f p . The converse of the latest statement holds also if 1 < p < ∞: for an arbitrary Lp -bounded martingale f = (fn , n ∈ N) there exists a function f ∈ Lp (Gm ) for which fn = En f. If p = 1, then there exists a function f ∈ L1 (Gm ) of the preceding type if and only if f is uniformly integrable, namely, if  lim sup

y→∞n∈N {|f |>y} n

|fn (x)| dμ (x) = 0.

Thus the map f → f =: (En f, n ∈ N) is isometric from Lp onto the space of Lp -bounded martingales when 1 < p < ∞. Consequently, these two spaces can be identified with each other. Similarly, the space L1 (Gm ) can be identified with the space of uniformly integrable martingales.

2.3 Martingales and Maximal Functions

81

Definition 2.3.3 For a martingale f = (fn , n ∈ N) and a stopping time τ with respect to (Fn , n ∈ N), the stopping martingale (fnτ , n ∈ N) is defined by fnτ =:

n 

χ{τ ≥k} dk f,

k=0

where dk f =: fk − fk−1 (k ∈ N)

(2.6)

are called martingale differences and f−1 =: 0. It is easy to check that fnτ = fk on the set {τ = k} whenever n ≥ k and fnτ (n ∈ N) is really a martingale. For a martingale f = (fn , n ∈ N), we define fn∗ by fn∗ =: max |fk | . k≤n

The maximal function f ∗ of a martingale f is defined by f ∗ =: M(f ) =: sup |fn | . n∈N

For an integrable function f ∈ L1 (Gm ), the maximal function f ∗ can also be given by    1  f dμ (t) (t) ,  In (x) n∈N |In (x)|

f ∗ (x) =: sup |En f (x)| = sup |SMn f (x)| = sup n∈N

n∈N

which is the same as given in (2.1). The following inequality holds true: Lemma 2.3.4 The martingale f = (fn , n ∈ N) satisfies the weak-type inequality   ∗  |fn |dμ. ρ μ fn > ρ ≤ {fn∗ >ρ } Proof Consider the stopping time νρ =: inf {n ∈ N, |fn | > ρ} .

82

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

Then we have that n      ρ μ νρ ≤ n = ρ μ νρ = k k=0

≤

 n 

 n 

|fk | dμ =

k=0

≤

(νρ =k )  n 

|Ek fn | dμ

k=0

(νρ =k )

 |fn | dμ =

k=0

(νρ =k )

|fn | dμ. (νρ ≤n)

Since   * + νρ ≤ n = fn∗ > ρ , 

the lemma is proved.

Next, we prove the following important inequality for the maximal function f ∗ of a regular martingale generated by f ∈ L1 (Gm ): Lemma 2.3.5 Let f ∈ L1 (Gm ). Then   sup ρ μ f ∗ > ρ ≤ f 1 . ρ>0

Proof The proof is the same as that of Lemma 2.3.4. Indeed, ∞      ρ μ νρ < ∞ = ρ μ νρ = k k=0

≤

∞ 

 |fk | dμ =

k=0

≤

(νρ =k )  ∞  k=0

(νρ =k )

 ∞ 

|Ek f | dμ

k=0

(νρ =k )

 |f | dμ =

|f | dμ (νρ ρ . 

2.3 Martingales and Maximal Functions

83

Theorem 2.3.6 Let 1 < p < ∞. For every Lp -bounded martingale f = (fn , n ∈ N) ∈ Lp (Gm ), we have that f ∗ belongs to Lp (Gm ), more precisely ∗ f ≤ p

p f p . p−1

Proof First we write the inequality in Lemma 2.3.4 in the following form:     ρE χ{x, fn∗ (x)>ρ} ≤ E fn χ{x, fn∗ (x)>ρ} (n ∈ N, ρ > 0) . Integrating both sides with respect to the measure pρ p−2 dρ (p > 1) and applying Fubini‘s theorem, we get that  ∗p  E fn =

∞ pρ

p−1

  E χ{x:fn∗ >ρ} dρ ≤

0

∞

  pρ p−2 E fn χ{x:fn∗ (x)>ρ} dρ

0



p E fn (fn∗ )p−1 . = p−1

On the other hand, Hölder’s inequality implies that

E fn (fn∗ )p−1 ≤ fn p (fn∗ )p−1

p/(p−1)

p−1 = fn p fn∗ p .

p−1 After division by fn∗ p < ∞ the combination of the two proceeding inequalities gives that ∗ f ≤ n p

p fn p . p−1

By (2.5), there exists f ∈ Lp (Gm ) such that fn = En f. Since fn∗ is non-decreasing and fn p → f p , by taking the limit, the proof becomes complete.



84

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

Theorem 2.3.7 Let 1 < p ≤ ∞ and f ∈ Lp (Gm ). Then f p ≤ f ∗ p ≤ Cp f p . Note that Cp =

p p−1

for 1 < p < ∞ and C∞ = 1.

Proof The inequality is trivial for p = ∞. If 1 < p < ∞, then the left hand side follows from f p = lim fn p ≤ f ∗ p . n→∞



The right hand side was proved in Theorem 2.3.6.

Now we can state a version of the Lebesgue’s differentiation theorem, which says that the derivative of the integral of a function f is almost everywhere equal to f . Corollary 2.3.8 If f ∈ L1 (Gm ), then  f (t) dt = f (x)

lim Mn

n→∞

a.e.

In (x)

Proof First of all, we note that f is integrable on every set In (x) and thus the left hand side is well defined for any f ∈ L1 (Gm ). Let Tf (x) =: f (x) and Tn f (x) =:

1 |In (x)|



 f (t) dt = Mn In (x)

f (t) dt. In (x)

These operators are linear and sup y μ (|Tf | > y) = sup y μ (|f | > ρ) y>0

y>0

≤ sup



ρ>0 {|f |>y}

|f | dμ ≤ f 1

implies (1.36). Inequality (1.37) follows from Lemma 2.3.5. Denote by X0 the set of continuous functions. If f ∈ X0 , then the result holds obviously. Since X0 is dense in L1 (Gm ), Theorem 1.5.9 implies the proof of the corollary for all f ∈ L1 (Gm ).  We also prove the following result concerning a subsequence of partial sums with respect to the Vilenkin system.

2.3 Martingales and Maximal Functions

85

Corollary 2.3.9 If f ∈ L1 (Gm ), then ∗ . S# f weak−L1 ≤ C f 1 . where . S#∗ f was defined in (2.2). Moreover, lim SMn f (x) = f (x) a.e.

n→∞

Proof Since  SMn f (x) = Mn

(2.7)

f (t) dt, In (x)

by combining Corollary 1.6.7 and Lemma 2.3.5, we immediately get the weak(1,1) type inequality. The almost everywhere convergence follows from Corollary 2.3.8.  To characterize points of convergence of partial sums, the following definition is important: Definition 2.3.10 A point x ∈ Gm is called a Lebesgue point of f ∈ L1 (Gm ) , if  f (t) dt = f (x)

lim Mn

n→∞

a.e.

In (x)

According to Corollary 2.3.9, almost every point is a Lebesgue point of f ∈ L1 (Gm ). On the other hand, the second statement in Corollary 2.3.9 can also be given in the following way: Corollary 2.3.11 For all Lebesgue points x of f ∈ L1 (Gm ), lim SMn f (x) = f (x).

n→∞

Proof According to the identity (2.7), the proof immediately follows from Corollary 2.3.9.  We will also use the following convexity-concavity theorem: Theorem 2.3.12 Let T be a countable index set and (At , t ∈ T) be an arbitrary (not necessarily monotone) sequence of σ -algebras. Suppose that for all h ∈ Lp (Gm ) and all 1 < p < ∞ Doob’s inequality sup |Et h| ≤ Cp hp t ∈T

p

86

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

holds, where Et denotes the conditional expectation operator relative to At . If (ft , t ∈ T) is a sequence of non-negative measurable functions, then for all 1 ≤ p < ∞, E

 

p Et ft

 

≤ Cp E

t ∈T

p ft

t ∈T

and for all 0 < q ≤ 1,  E



q ft

≤ Cq E

t ∈T

 

q Et ft

.

t ∈T

Proof These inequalities are obvious for p = 1 and q = 1. Assume that 1 < p < ∞. By Riesz’s representation theorem           Et ft = sup  Et ft g dμ ,  gp ≤1  Gm t ∈T

t ∈T

p

where 1 1 + = 1. p p Hölder’s and Doob’s inequalities imply that           Et ft gdμ ≤ ft |Et g|dμ    Gm t ∈T t ∈T Gm    

sup |Et g| dμ ≤ ft Gm

t ∈T

 ≤ ft t ∈T

p

t ∈T

sup |Et g| t ∈T

 ≤ Cp ft . t ∈T

The first inequality is proved.

p

p

2.3 Martingales and Maximal Functions

87

To verify the second one, let 1 < p < ∞ and r be the conjugate index to 2p, i.e., 1 1 + =1 r 2p and set 1/(2p)

gt =: ft

,

g = (gt , t ∈ T).

Let us use the notation gLp (r )

 1/r  r < ∞. =: |g | t t ∈T p

Then ⎛ ⎝



 Gm



⎞1/2

1/p

dμ⎠

ft

= gL2 (2p )

t ∈T

       = sup  gt ht dμ .   h 2 ≤1 Gm L (r )

t ∈T

Moreover, it follows from Hölder’s inequality that        gt ht dμ    Gm t ∈T  Et |gt ht | dμ ≤ t ∈T Gm

≤



t ∈T Gm

=





1/r 2p 1/(2p)  dμ Et gt Et |ht |r  1/r dμ (Et ft )1/(2p) Et |ht |r

t ∈T Gm



 ≤ Gm

⎛  ≤⎝



1/(2p)  Et ft

t ∈T

  Gm

t ∈T



1/r Et |ht |

r

dμ

t ∈T

1/p Et ft

⎞1/2 ⎛  dμ⎠ ⎝

 Gm

 t ∈T

2/r Et |ht |r

⎞1/2 dμ⎠

.

88

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

Since 1 < 2/r = 2 − 1/p < 2, it follows from the first inequality that  

 Gm

2/r Et |ht |

r



 dμ ≤ Cp



Gm

t ∈T

=

2/r |ht |

r

dμ

t ∈T

Cp h2L2 ( ) r

≤ Cp .

Thus  

 Gm

1/p ft



 dμ ≤ Cp Gm

t ∈T



1/p Et ft

dμ,

t ∈T

which is exactly the second inequality of the theorem with q = 1/p. The proof is complete.  For a martingale f = (fn )n≥0 , the square function and the conditional square function of f are defined by S(f ) =

∞ 



1/2 |dn f |2

and

s(f ) =

n=0

∞ 

1/2 En−1 |dn f |2

,

n=0

respectively, where dn f are the martingale differences defined by (2.6), E−1 =: E0 and f−1 =: 0. The following theorem was proved in Burkholder [57], Garsia [105], Long [213] and Weisz [400]. Using some methods of Chap. 9, we give another proof of Theorem 2.3.13. Theorem 2.3.13 If 0 < p < ∞, then ∗ f ∼ S(f )p ∼ s(f )p . p

(2.8)

If in addition 1 < p ≤ ∞, then ∗ f ∼ f p . p

(2.9)

Proof The inequality is trivial for p = 2. Using the method of Corollary 9.8.3, we get the inequality for all 0 < p < 2. It is easy to show that |dn f |2 ≤ REn−1 |dn f |2

(n ∈ N),

2.4 Calderon-Zygmund Decomposition

89

in other words, (Fn ) is regular. Hence S(f )p ≤ Cp s(f )p

(0 < p < ∞).

By applying Theorem 2.3.12 with the choice ft =: |dn f |2 , we have that  1/2  2 s(f )p = |d | E f n−1 n n∈N 1/2  2 = En−1 |dn f | n∈N

p

p/2

1/2  |dn f |2 ≤ Cp = Cp S(f )p n∈N

p/2

for 2 < p < ∞. Finally the equivalence S(f )p ∼ f p

for 1 < p < ∞

can be proved using Khintchine’s inequality (see Burkholder [57] or Weisz [400]). The equivalence (2.9) was proved in Theorem 2.3.7 so the proof is complete. 

2.4 Calderon-Zygmund Decomposition The idea of Calderon-Zygmund decomposition is to split an arbitrary integrable function to small and large parts and then to investigate each part by different techniques: Lemma 2.4.1 (Calderon-Zygmund Decomposition) () Let f ∈ L1 and y > f 1 . Then there exist Vilenkin intervals I0 , I1 , . . . , and functions g, b such that (i) f = g + b, (ii) g∞ ≤ y, (iii) supp b ⊂ , where  =

∞ k=0

Ik ,

(iv) μ () ≤ C f 1 /y, |b| dμ ≤ 2yμ (Ik ) , (k ∈ N) , (v) bdμ = 0 and Ik

(vi) g1 ≤ f 1 .

Ik

90

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

Proof Let ρ =: inf {n ∈ N : En |f | > y} and τ (x) =: inf {n ∈ N : x ∈ Fn+1 } , where Fn+1 ∈ Fn is the smallest set containing {ρ = n + 1} . In other words, if {ρ = n} ∈ Fn is decomposed into the disjoint union of Vilenkin intervals In,l ∈ Fn and In,l ∈ Fn−1 denotes the Vilenkin interval which contains In,l , then Fn =



In,l .

l

Then {ρ (x) = n} implies that {τ (x) ≤ n − 1} . This means that ρ < τ on the set {ρ = ∞}. It is easy to see that μ ({τ = ∞}) ≤

∞ 

μ (Fn ) ≤

n=1

∞   n=1





μ In,l ≤ R

l

* +

∞}) = Rμ( f ∗ > y ), = Rμ ({ρ =

∞   n=1

  μ In,l

l

where R = supn∈N mn . Let g =: f τ ,

b =: f − f τ .

Then (i), (ii) and (vi) evidently hold. If  =: {τ (x) = ∞} , then =

∞ 

{τ = k}

k=0

and hence  can be decomposed into the union of disjoint Vilenkin intervals and (iii) holds. It follows from Lemma 2.3.5 that μ ({τ = ∞}) ≤ Cμ

*

f∗ > y

+

≤ C f 1 /y,

2.4 Calderon-Zygmund Decomposition

91

which shows also (iv) holds. Suppose that Ik ∈ Fn . Then Ik ⊂ {τ = n} and 

 bdμ = Ik

  χ{τ =n} f − f τ dμ =

Ik



  χ{τ =n} En f − f τ dμ = 0.

Ik

Moreover,       |b| dμ ≤ |f | dμ + f τ  dμ ≤ χ{τ =n} En |f | dμ + yμ (Ik ) ≤ 2yμ (Ik ) , Ik

Ik

Ik

Ik



i.e. also (v) holds so the proof is complete.

The next theorem is important to study almost everywhere convergence in the next Chapter: Theorem 2.4.2 Suppose that the σ -sublinear operator V is bounded from Lp1 to Lp1 for some 1 < p1 ≤ ∞ and  |Vf | dμ ≤ C f 1 I

for f ∈ L1 and Vilenkin interval I which satisfy  suppf ⊂ I,

f dμ = 0. Gm

Then the operator V is of weak-type (1, 1), i.e sup yμ ({Vf > y}) ≤ f 1 .

y>0

Proof Let us apply the Calderon-Zygmund decomposition. Set  =:

∞ 

Ik .

k=0

Since |Vf | ≤ |V g| + |V b| ,

V gp1 ≤ A gp1

and g∞ < 1, y

(2.10)

92

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

we have that p

μ ({|V g| > (1 + A) y}) ≤

p

V gp11 gp11 f 1 g1 ≤ . ≤ ≤ p p p 1 1 1 A y y y y

Moreover,

1 {|V b| > y} μ ({|V b| > y}) ≤ μ () + μ   1 C f 1 |V b| dμ. + ≤ y y 

If bk = χIk b, then (2.10) is satisfied and therefore  |V bk | dμ ≤ C bk 1 . Ik

Hence, by (v) of Lemma 2.4.1 we find that  |V bk | dμ ≤ 

∞   k=0

Ik

|V bk | dμ ≤

∞ 

2yμ (Ik ) ≤ C f 1 .

k=0

Consequently, μ ({|Vf | > (1 + A) y}) ≤

C f 1 . y

Since this inequality holds also for y ≤ f 1 we have proved the theorem.



2.5 Almost Everywhere Convergence of Vilenkin-Fourier Series In this section we prove one of the most important results of this chapter and also the whole book. In particular we prove an analogy of the Carleson-Hunt theorem with respect to Vilenkin systems. In particular, we use the theory of martingales and give a new and shorter proof of the almost everywhere convergence of Vilenkin-Fourier series of f ∈ Lp (Gm ) for 1 < p < ∞ in case the Vilenkin system is bounded. Moreover, we also prove sharpness by stating an analogy of the Kolmogorov theorem for p = 1 and construct a function f ∈ L1 (Gm ) such that the partial sums with respect to Vilenkin systems diverge everywhere.

2.5 Almost Everywhere Convergence of Vilenkin-Fourier Series

93

First we introduce some notations. For j, k ∈ N we define the following subsets of N : 1 IjkMk =: [j Mk , j Mk + Mk ) N and I =: {IjkMk : j, k ∈ N}. We also define the partial sums sI k

jMk

taken in these intervals as follows: 

sI k f =: jMk

f(i)ψi .

k i∈IjM

k

For simplicity, we suppose that f(0) = 0. Lemma 2.5.1 For an arbitrary n ∈ IjkMk , sI k f = ψn Ek (f ψ n ).

(2.11)

jMk

Proof It is easy to see that sI k f = jMk

  k j ∈IjM

 Gm

f ψ j dμ ψj

k

=

M k −1   i=0

Gm

 (f ψ j Mk )ψ i dμ ψj Mk ψi

= ψj Mk SMk (f ψ j Mk ) = ψj Mk Ek (f ψ j Mk ). 

Equality (2.11) can be proved in the same way. For n=

∞ 

nj Mj

(0 ≤ nj < mj ),

j =0

we introduce n(k) =:

∞  j =k

nj Mj ,

1 / k In(k) = n(k), n(k) + Mk N

(n ∈ N).

(2.12)

94

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

k For I = In(k) , let



k

T I f =: T In(k) f =:

(2.13)

sJ f.

[n(k+1),n(k))⊃J ∈I |J |=Mk k k Since In(k) = In(k) implies ˜

n(k + 1) = n(k ˜ + 1), the operators T I (I ∈ I) are well defined. Note that there are nk summands in (2.13). k We can rewrite Sn f as a sum of T In(k) f ’s. In particular, the following identity holds Lemma 2.5.2 For all n ∈ N, we have that Sn f =

∞ 

k

T In(k) f

(2.14)

k=0

= ψn

∞ n k −1 



n −l n −l , r k k Ek dk+1 (f ψ n )rk k

k=0 l=0 k where In(k) is defined in (2.12). Moreover,

⎛

n k −1

⎝

⎞

n −l n −l ⎠ r k k Ek dk+1 (f ψ n )rk k

l=0

(2.15) n∈N

is a martingale difference sequence. k+1 k k Proof Since n is contained in In(k) and In(k) ⊂ In(k+1) , define

gn,k =:

 j ∈[n(k+1),n(k+1)+Mk+1 )

f(j )ψj −



f(j )wj .

j ∈[n(k),n(k)+Mk )

By (2.11), gn,k = ψn Ek+1 (f ψ n ) − ψn Ek (f ψ n ) = ψn dk+1 (f ψ n ). Since [n(k + 1), n(k)) ⊂ [n(k + 1), n(k + 1) + Mk+1 ) \ [n(k), n(k) + Mk ),

(2.16)

2.5 Almost Everywhere Convergence of Vilenkin-Fourier Series

95

we have that 



f(j )ψj =

j ∈[n(k+1),n(k))

g2 n,k (j )ψj

j ∈[n(k+1),n(k))

=

n k −1 n(k+1)+(l+1)M  k −1

g2 n,k (j )ψj .

j =n(k+1)+lMk

l=0

It is easy to see that  / n − (nk − l)Mk ∈ n(k + 1) + lMk , n(k + 1) + (l + 1)Mk

(0 ≤ l < mk ),

which implies by (2.11) that 

f(j )ψj =

j ∈[n(k+1),n(k))

n k −1

ψn−(nk −l)Mk Ek (gn,k ψ n−(mk −l)Mk )

l=0 n k −1

= ψn

  ψ (nk −l)Mk Ek (gn,k ψ n )ψ(nk −l)Mk

l=0 n k −1

= ψn



n −l n −l . r k k Ek (gn,k ψ n )rk k

l=0

Therefore, by taking into account (2.16), we can conclude that 

k

T In(k) f =

f(j )ψj

j ∈[n(k+1),n(k))

= ψn

n k −1



r nk k −l Ek dk+1 (f ψ n )rknk −l .

l=0

Since [0, n) =

∞  /  n(k + 1), n(k) , k=0

we get that Sn f =

∞ 



k=0 j ∈[n(k+1),n(k))

and (2.14) is proved.

f(j )ψj =

∞  k=0

k

T In(k) f,

(2.17)

96

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

Since rk is Fk+1 measurable and Ek (rki ) = 0

for i = 1, . . . , mk − 1,

we can see that ⎛

n k −1

Ek ⎝

⎞

r nk k −l Ek dk+1 (f ψ n )rknk −l ⎠ = 0,

l=0

Hence, (2.15) is really a martingale difference sequence so the proof is complete.  Next, we state the following important estimate: Lemma 2.5.3 For all k, n ∈ N, we have that

k |T In(k) f | ≤ REk |sI k+1 f − sI k f | , n(k)

n(k+1)

where R =: max(mn , n ∈ N). Proof Equalities (2.17) and (2.11) imply that   k |T In(k) f | ≤ mk Ek |dk+1(f ψ n )|   ≤ REk |ψn Ek+1 (f ψ n ) − ψn Ek (f ψ n )|

= REk |sI k+1 f − sI k f | , n(k+1)

(2.18)

n(k)



and the proof is complete.



k Lemma 2.5.4 For all n ∈ N, ψ n T In(k) f is a martingale difference sequence k∈N with respect to (Fk+1 )k∈N . k

Proof First, we note that ψ n T In(k) f is Fk+1 measurable because of (2.17) and the fact that rk is Fk+1 measurable. Since Ek (rki ) = 0

for i = 1, . . . , mn − 1,

we can see that ⎞ ⎛ n k −1



k n −l n −l In(k) ⎠ = 0, Ek ψ n T f = Ek ⎝ r k k Ek dk+1 (f ψ n )rk k l=0

2.5 Almost Everywhere Convergence of Vilenkin-Fourier Series

97

and, hence, ⎛ ⎝

n k −1

r nk k −l Ek

⎞

nk −l ⎠ dk+1 (f ψ n )rk

l=0

k∈N

is a martingale difference sequence. The proof is complete.



Before stating our main theorems, we need some further notations. In what follows, I , J , K denote elements of I. Let FK =: Fn

and EK =: En

if |K| = Mn .

Assume that  = (K , K ∈ I) is a sequence of functions such that K is FK measurable. Set  T;I,J f =: K T K f I ⊂KJ

and ∗ T;I f =: sup |T;I,J f |,

∗ T∗ f =: sup |T;I f |. I ∈I

I ⊂J

If K (t) = 1 for all K ∈ I and t ∈ Gm , then we omit the notation  and we write simply TI,J f , TI∗ f and T ∗ f . For I ∈ I with |I | = Mn , let I + ∈ I be such that I ⊂ I+

|I + | = Mn+1 .

and

Moreover, let I − ∈ I denote one of the sets I− ⊂ I

|I − | = Mn−1 .

with

Note that FI − = Fn−1 and EI − = En−1 are well defined. We introduce the maximal functions sI and s ∗ as follows: sI∗ f =: sup EK − |sK f | K⊂I

and s ∗ f =: sup sI∗ f. I ∈I

98

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

Since |sI + f | is FI + measurable, by the regularity condition (2.4), |sI + f | ≤ REI |sI + f | ≤ RsI∗+ f.

(2.19)

Lemma 2.5.5 For any real number x > 0 and K ∈ I, let K =: χ{xx,sI∗ f ≤x,I K} . Then ∗ T∗ f ≤ 2 sup αK T;K f + 4R 2 xχ{s ∗ f >x} .

(2.20)

K∈I

Proof Let us fix I  J in I and t in Gm . Set τK =: χ{sK∗ f >x}

(K ∈ I).

Therefore K = τK + K . Consequently, if the set {K ∈ I : I ⊂ K  J, τK + (t) = 1} is empty, then T;I,J f (t) = 0 or else let K1 be its minimum element. Moreover, denote by K0 one of the minimum elements of the set {K ∈ I : K ⊂ K1+ , τK (t) = 1}. This means that if L  K0 , then τL (t) = 0. Thus αK0 (t) = 1 and T;I,J f (t) = T;K1 ,J f (t) = K1 T K1 f (t) + T;K + ,J f (t) 1

= K1 T K1 f (t) + αK0 (t) T;K0 ,J f (t) − T;K0 ,K + f (t) . 1

By Lemma 2.5.3 and (2.19), we find that

K1 |T K1 f | ≤ RK1 EK1 |sK + f − sK1 f | 1

∗ ≤ 2R 2 K1 EK1 K1 sK +f 1

≤ 4R xχ{s ∗ f >x} . 2

2.5 Almost Everywhere Convergence of Vilenkin-Fourier Series

99

On the other hand,     ∗ f (t). T;K0 ,J f (t) − T;K0 ,K + f (t) ≤ 2T;K 0 1

Taking the supremum over all I  J , we get (2.20) and the proof is complete.



We define the quasi-norm  · P p,q (0 < p, q < ∞) by f P p,q

⎛  p/q ⎞1/p  ⎠ , =: sup x ⎝E αI x>0

I ∈I

where αI is defined in Lemma 2.5.5. Observe that αI can be rewritten as αI =: χ{EI − |sI f |>x,EJ − |sJ f |≤x,J I } .

(2.21)

Denote by P p,q the set of functions f ∈ L1 which satisfy f P p,q < ∞. For q = ∞, we define f P p,∞

p 1/p



=: sup x E sup αI I ∈I

x>0

(0 < p < ∞).

It is easy to see that f P p,∞ ≤ f P p,q

(0 < q < ∞)

and f P p,∞ = sup xλ(s ∗ f > x)1/p = s ∗ f p,∞ . x>0

Lemma 2.5.6 Let max(1, p) < q < ∞, f ∈ P p,q and x, z > 0. Then

λ

∗ sup αI T;I f I ∈I

 > zx

≤ Cp,q z−q x −p f P p,q ,

where αI is defined in Lemma 2.5.5. Proof Equality (2.17) implies that ξ T K (f ) = T K (f ξ )

p

100

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

for F  any  K measurable function ξ . By Lemma 2.5.4, for a suitable n ∈ I , ψ n T K f I ⊂K is a martingale difference sequence relative to (FK + )I ⊂K . We have that               ∗ T;I f = sup  K ψ n T K f  = sup  ψ n T K (f K ) . I ⊂J I ⊂KJ  I ⊂J I ⊂KJ  Using Burkholder-Gundy’s inequality (see Theorem 2.3.13) together with (2.18), we obtain that  ∗  EI |T;I f |p0 ≤ Cp0 EI



 ≤ Cp0 EI

p0 /2



|ψ n T (f K )| K

2

I ⊂K



p0 /2 EK |dK + (f K ψ n )|

2

,

I ⊂K

where p0 > 1. Applying again Theorem 2.3.13, one can establish that 

∗ EI |T;I f | p0



 p 0     dK + (f K ψ n ) ≤ Cp0 EI    I ⊂K

 p 0     = Cp0 EI  K dK + (f ψ n ) .   I ⊂K

For fixed I and t ∈ Gm denote by K0 (t) ∈ I (K1 (t) ∈ I) the smallest (largest) interval K ⊃ I for which K0 (t ) = 1 (K1 (t ) = 1). Then ψn (t)



K (t)dK + (f ψ n )(t)

I ⊂K

= ψn (t)

 K0 (t )⊂K⊂K1 (t )

K (t)dK + (f ψ n )(t)

  = K1 (t ) (t) ψn (t)EK1 (t )+ (f ψ n )(t) − ψn (t)EK0 (t )(f ψ n )(t)   = K1 (t ) (t) sK1 (t )+ f (t) − sK0 (t )f (t) .

2.5 Almost Everywhere Convergence of Vilenkin-Fourier Series

101

By (2.19) and the definition of K , we get that   K1 (t )(t) sK1 (t )+ f (t) − sK0 (t )f (t)

∗ ∗ ≤ RK1 (t )(t) sK + f (t) + sK (t ) f (t) 0 1 (t )

(2.22)

∗ ≤ 2RK1 (t ) (t)sK + f (t) ≤ 4Rx. 1 (t )

Hence,  ∗  EI |T;I f |p0 ≤ Cp0 x p0 . By Tsebisev’s inequality (see Proposition 1.5.5) and the concavity theorem (see Theorem 2.3.12), for p0 ≥ q > 1, one can see that



q ∗ ∗ λ sup αI T;I f > zx ≤ (zx)−q E sup αI T;I f I ∈I

I ∈I

 ≤ (zx)

−q

E

 I ∈I

≤ Cp0 ,q (zx)−q E  ≤ Cp0 ,q z−q E

q/p0 ∗ αI T;I f p0

 

q/p0 ∗ αI EI (T;I f p0 )

I ∈I



q/p0

αI

.

I ∈I

We put p0 =: q 2 /p ≥ q > 1 and observe that

λ

∗ sup αI T;I f I ∈I

 > zx

≤ Cp,q z

−q

E

 

p/q αI

I ∈I

≤ Cp,q z−q x −p f P p,q . p

The proof is complete. Lemma 2.5.7 Let max(1, p) < q < ∞ and f ∈ P p,q . Then

sup y p λ T ∗ f > (2 + 8R 2 )y ≤ Cp,q f P p,q . y>0



102

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

k Proof First we define a decomposition generated by the sequences  k = (K ,K ∈ I), where k K =: χ{2k 0} T K f =



k K K T f

k∈Z

and T ∗f ≤



T∗k f.

k∈Z

Next we apply Lemma 2.5.5 to  k and x = 2k and find that k ∗ T∗k f ≤ 2 sup αK T k ;K f + 2k+2 R 2 χ{s ∗ f >2k } , K∈I

where k αK =: χ{s ∗ f >2k ,s ∗ f ≤2k ,I K} K

I

(K ∈ I).

By now choosing j ∈ Z such that 2j < y ≤ 2j +1 , we get that χ{s ∗ f ≤y} T ∗ f ≤ 2 ≤2



k ∗ sup αK T k ;K f +

k≤j K∈I





2k+2 R 2 χ{s ∗ f >2k }

k≤j

k ∗ sup αK T k ;K f + 8R 2 y.

k≤j K∈I

Hence, according to Lemma 2.5.6, for any k ∈ Z and zk > 0, we have that

 −q p k ∗ λ sup αK T k ;K f > zk 2k ≤ Cp,q zk 2−pk f P p,q . K∈I

(2.23)

2.5 Almost Everywhere Convergence of Vilenkin-Fourier Series

103

Consequently,

y p λ T ∗ f > (2 + 8R 2 )y

≤ y p λ(s ∗ f > y) + y p λ T ∗ f > (2 + 8R 2 )y, s ∗ f ≤ y ⎛ ⎞  p k ∗ ≤ f P p,∞ + y p λ ⎝ sup αK T k ;K f > y ⎠ k≤j K∈I

p

≤ f P p,q

⎛ ⎞  k ∗ + ypλ ⎝ sup αK T k ;K f > 2j ⎠ . k≤j K∈I

In order to be able use (2.23), we observe that Cβ



2β(k−j ) = 1 if

β > 0 and Cβ = 1 − 2−β .

k≤j

Set 2j Cβ 2β(k−j ) = Cβ 2(β−1)(k−j )2k =: zk 2k . Then, for β = (q − p)/(2q), we get that −q

zk 2−pk ≤ Cp,q 2−pj 2p(j −k)+q(β−1)(j −k) ≤ Cp,q y −p 2(q−p)(k−j )/2. Thus, by (2.23), we have that ⎞ ⎛  

 k ∗ k ∗ λ⎝ sup αK T k ;K f > 2j ⎠ ≤ λ sup αK T k ;K f > zk 2k k≤j K∈I

K∈I

k≤j

≤ Cp,q



−q

zk 2−pk f P p,q p

k≤j

≤ Cp,q y −p f P p,q p



2(q−p)(k−j )/2

k≤j

≤ Cp,q y The proof is complete.

−p

p f P p,q ,



Let  denote the closure of the triangle in R2 with vertices (0, 0), (1/2, 1/2) and (1, 0) except the points (x, 1 − x), 1/2 < x ≤ 1.

104

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

Lemma 2.5.8 Suppose that 1 < p, q < ∞ satisfy (1/p, 1/q) ∈ . Then, for all f ∈ Lp , f P p,q ≤ Cp,q f p . Proof For an arbitrary x > 0, we use the definition of αI given in (2.21). Then αI is FI − measurable and, obviously, αI αJ = 0

if I  J

or J  I.

For all I ∈ I, we introduce the projections FI =: αI sI and observe that sI ◦ sJ = 0 for every incomparable I and J . Therefore, for every g ∈ L1 and I, J ∈ I, we get that FI (FJ g) = αI sI (sJ (αJ g)) = sI (αI αJ sJ g) = δI,J FI g, where δI,J is the Kronecker symbol. Thus the projections FI are orthogonal and Bessel’s inequality implies for any g ∈ L2 that 

(FI g, I ∈ I)2L2 (l ) = 2

FI g22 ≤ g22 .

I ∈I

We also introduce the operators GI g =: EI − (ηI FI g)

(g ∈ L1 , I ∈ I),

where (ηI , I ∈ I) is a fixed sequence of functions satisfying ηI ∞ ≤ 1 for each I ∈ I. Then    2 2 (GI g, I ∈ I)L2 (l ) ≤ E EI − |FI g| 2

I ∈I

 =E



 |FI g|

2

≤ g22 .

I ∈I

Furthermore, by Doob’s inequality (see Theorem 2.3.12), we have that (GI g, I ∈ I)Ls (l∞ )

≤ sup EI − |g| ≤ Cs gs I ∈I

s

for any 1 < s ≤ ∞ and g ∈ Ls . It follows by interpolation that (GI g, t ∈ I)Lp (lq ) ≤ Cp,q gp

(g ∈ Lp ),

2.5 Almost Everywhere Convergence of Vilenkin-Fourier Series

105

where 1/p = (1 − t)/2 + t/s and 1/q = (1 − t)/2 for any 0 ≤ t ≤ 1. Hence, by setting g =: f and ηt =: sign sI f , we have that ⎛  p/q ⎞1/p  ⎝E ⎠ (αI EI − |sI f |)q ≤ Cp,q f p . I ∈I

Using the fact that αI EI − |sI f | > xαI , we can see that ⎛  p/q ⎞1/p  ⎠ x ⎝E αI ≤ Cp,q f p , I ∈I



which finishes the proof. Our first main result in this Section reads: Theorem 2.5.9 Let f ∈ Lp (Gm ), where 1 < p < ∞. Then ∗ S f ≤ Cp f p . p

Proof It is easy to see that Lemma 2.5.2 implies that S ∗ f ≤ T ∗ f . Hence, it follows from Lemmas 2.5.7 and 2.5.8 that   sup y p λ S ∗ f > y ≤ Cp f p y>0

for 1 < p < ∞. Now the proof of the theorem follows by using the Marcinkiewicz interpolation theorem (see Theorem 1.5.8).  The next norm convergence result in Lp spaces for 1 < p < ∞ follow from the density of the Vilenkin polynomials in Lp (Gm ) (see Proposition 1.8.1) and from Theorem 2.5.9 and Lemma 1.9.1. Theorem 2.5.10 Let f ∈ Lp (Gm ), where 1 < p < ∞. Then Sn f − f p → 0, as n → ∞. Our announced Carleson-Hunt type theorem reads: Theorem 2.5.11 Let f ∈ Lp (Gm ), where p > 1. Then Sn f → f a.e., as n → ∞.

106

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

Also here the proof follows directly from Theorems 1.5.9 and 2.5.9 by using in addition the density of Vilenkin polynomials in Lp (Gm ) (see Proposition 1.8.1).

2.6 Almost Everywhere Divergence of Vilenkin-Fourier Series A set E ⊂ Gm is called a set of divergence for Lp (Gm ) if there exists a function f ∈ Lp (Gm ) whose Vilenkin-Fourier series diverges on E. Lemma 2.6.1 If E is a set of divergence for L1 (Gm ), then there is a function f ∈ L1 (Gm ) such that S ∗ f = ∞ on E. 1 Proof We claim that given  any g ∈ L (Gm ), there is an unbounded monotone increasing sequence λ = λj , j ∈ N of positive real numbers and a function f ∈ L1 (Gm ) such that

fˆ(j ) = λj g(j ˆ )

(j ∈ N).

(2.24)

To prove this claim we use Corollary 1.9.4 for p = 1 to choose a strictly increasing sequence of positive integers n1 , n2 , . . . such that SMnk g − g1 < Mk−1

(k ∈ N+ ).

(2.25)

Consider the function f defined by ∞

 g − SMnk g . f =: g + k=1

By (2.25), the series converges in the norm of L1 (Gm ). In particular, f belongs to L1 (Gm ) and fˆ(j ) = g(j ˆ )+

∞   k=1 Gm

g − SMnk g ψ j dμ

for j ∈ N. Therefore, the claim follows from the orthogonality if we set λj =: 1 +



1

(j ∈ N).

k∈N+ :Mnk ≤j

Next we suppose that g ∈ L1 (Gm ) is a function whose Vilenkin-Fourier series diverges on E. Use the claim we proved above to choose a monotone increasing,

2.6 Almost Everywhere Divergence of Vilenkin-Fourier Series

107

unbounded sequence λ which satisfies (2.24). Moreover, by Abel’s transformation, we find that Sn g − Sm g =

n−1   1  Sj +1 f − Sj f λj

j =m

 n−1

 Sn f Sm f 1 1 = − + − Sj f λn−1 λm λj −1 λj j =m+1

for any integers n, m ∈ N with n > m. Since λ is increasing, it follows that |Sn g − Sm g| ≤

2 ∗ S f λm

(n, m ∈ N, n > m).

Therefore the fact that λ is unbounded, implies that Sn g converges at x when S ∗ f (x) is finite. In particular, S ∗ f (x) = ∞

for all x ∈ E. 

The proof is complete.

Lemma 2.6.2 A set E ⊆ Gm is a set of divergence for L1 (Gm ) if and only if there exist Vilenkin polynomials P1 , P2 , . . . such that ∞ 

Pj 1 < ∞

(2.26)

j =1

and sup S ∗ Pj (x) = ∞

j ∈N+

(x ∈ E).

(2.27)

Proof Suppose first that E is a set of divergence for L1 (Gm ). Let g ∈ L1 (Gm ) be a function whose Vilenkin-Fourier series diverges on E. By repeating the arguments in the proof of Lemma 2.6.1, we can choose an unbounded, increasing positive sequence (λj , j ∈ N) and a function f ∈ L1 (Gm ) such that  n−1

 Sn f Sm f 1 1 Sn g − Sm g = − + − Sj f λn−1 λm λj −1 λj j =m+1

108

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

for all integers n, m ∈ N, m < n. Let (ωj , j ∈ N) be an unbounded sequence of positive, increasing numbers which satisfies  ∞

 1 1 ωj < ∞. − λj λj +1 j =1

For example, let ωj =:

1 √1 λj

+ √1

.

λj+1

Indeed, then    ∞

∞   1 1 1 1 1 3 −3 = √ . ωj ≤ − λj λj +1 λ1 λj λj +1 j =1 j =1 Fix x ∈ E. If |Sj f (x)| = O(ωj ), as j → ∞, then |Sn g(x) − Sm g(x)| → 0, as n, m → ∞ and we get that Sn g(x) is a convergent series for any x ∈ E, which is a contradiction. Consequently, the inequality |Sn f (x)| > ωn

(2.28)

holds for infinitely many integers n ∈ N. Now use Corollary 1.9.4 for p = 1 to choose strictly increasing sequences of positive integers (nj , j ∈ N) and (αj , j ∈ N) which satisfy nj < αj + 1, f − SMnj f 1 < Mj−1

(2.29)

and S ∗ (SMnj f )∞
1 set nj =: 1 + max{nj −1 , αj }. Then (nj , j ∈ N) is a strictly increasing sequence of integers and it is easy to see that Mnj+1 ⊕ k1 > Mnj ⊕ k0

(2.31)

for any choice of integers k0 and k1 which satisfy 0 ≤ k0 ≤ Mαj , 0 ≤ k1 ≤ Mαj+1 and j ∈ N. Let f =:

∞  j =1

ψMnj Pj

and observe by (2.26) that f ∈ L1 (Gm ). It is clear that the series defining f converges in L1 (Gm ) norm. Consequently, this series is the Vilenkin-Fourier series of f. Moreover, (2.31) can be used to see that SMnj +k f − SMnj f = ψMnj Sk Pj for 0 ≤ k < Mnj+1 −Mnj , j ∈ N+ . In particular, (2.27) implies the Vilenkin-Fourier series of f diverges at each x ∈ E. The proof is complete. 

110

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

Corollary 2.6.3 If El , E2 , . . . are sets of divergence for L1 (Gm ), then E =: ∪∞ n=1 En is also a set of divergence for L1 (Gm ). (n)

(n)

Proof Apply Lemma 2.6.2 to choose Vilenkin polynomials P1 , P2 , . . . such that ∞ 

(n)

Pj 1 < ∞

j =1

and sup (S ∗ Pj )(x) = ∞ (n)

(x ∈ En , n ∈ N+ ).

j ∈N+

(2.32)

Thus, there exist integers α1 < α2 < . . . such that ∞ 

(n)

Pj 1
Cn. k=0

1

Consider the function ) gn (x) =

D

(x)

αn |Dαn (x)| , if Dαn (x) = 0, 0, if Dαn (x) = 0.

It is constant on any set of the form In (x) , x ∈ Gm . Hence gn is a Vilenkin polynomial of order at most Mn . Moreover, since ψk (x − t) = ψk (x)ψ k (t) = ψ k (t),

0 ≤ k < Mn , x ∈ In (0).

we get that Dαn (x − t) = D αn (t), x ∈ In (0). Hence, by the choice of αn ∈ [Mn−1 , Mn ) we have that 

 Sαn gn (x) =

 Gm

gn (t)Dαn (x − t) = Dαn 1 > Cn,

For k=

n−1  s=0

ks Ms (ks ∈ Zms ),

(x ∈ In (0)).

114

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . . (n)

we define the points xk

∈ Gm (0 ≤ k < Mn ) by (n)

=: (k0 , k1 , . . . , kn−1 , 0, 0, . . .)

xk and set

Qn =:

M n −1



mn+k −1 s=1

1 − τx (n) gn

mn+k − 1

k

k=0

s rn+k

 ,

where (n)

τx (n) gn (x) = gn (x − xk ). k

By using Lemma 1.6.1, we find that mn+k −1

s rn+k (x)

s=1

mn+k − 1

) =

if xn+k = 0, if xn+k = 0

(2.35)

if xn+k = 0, , if xn+k = 0. mn+k −1

(2.36)

1,

−1 mn+k −1 ,

and mn+k −1 s=1

1−

s (x) rn+k

mn+k − 1

) =

0, 1+

1

It is easy to prove that for any x ∈ Gm there exists xj(n) =: (j0 , j1 , . . . , jn−1 , 0, 0, . . .) (n)

such that In (x) = In (xj ), that is x0 = j0 , x1 = j1 , xn−1 = jn−1 . (n)

Consider the j -th term of the expression of Qn and let x ∈ In (xj ). Since τx (n) gn (xj(n) ) = gn (0) = 1, j

according to (2.36), we can conclude that mn+j −1 1 − τx (n) gn (x) j

s=1

s rn+j (x)

mn+j − 1

mn+j −1 =1−

s=1

s rn+j (x)

mn+j − 1

=0

2.6 Almost Everywhere Divergence of Vilenkin-Fourier Series

115

(n)

if xn+j = 0. Since x ∈ In (xj ) for some 0 ≤ k ≤ Mn − 1, we find that Qn (x) =

M n −1



mn+k −1 s=1

1 − τx (n) gn (x)

 = 0 if xn+k = 0.

mn+k − 1

k

k=0

s rn+k (x)

On the other hand, (2.35) and |τx (n) gn (x)| ≤ 1 k

imply that if xn+k = 0, then we get that   mn+k −1 s  mn+k −1 s    rn+k  rn+k    s=1 s=1  ≤ 1 + τx (n) gn  1 − τx (n) gn  k  k mn+k − 1  mn+k − 1  ≤ 1+

1 mn+k = . mn+k − 1 mn+k − 1

It follows that |Qn (x)| ≤

M n −1 k=0

mn+k mn+k − 1

if xn+k = 0 for all 0 ≤ k ≤ Mn − 1. Hence, we can conclude that Qn ∈ L1 (Gm ). Indeed,  |Qn |dμ Gm

≤

m 0 −1

···

x0 =0

=

m 0 −1 x0 =0

=

=





···



!Mn −1 k=0

mn+Mn −1 −1

!Mn −1 k=0



···

xn+Mn −1 =1

M −1 n  mn+k

k=0

M −1 n  mn+k

In+Mn (x)

xn+Mn −1 =1

xn−1 =0 xn =0

1

Mn

···

mn−1 −1 mn −1

1

Mn



xn−1 =0 xn =0



M −1 n 

mn+Mn −1 −1 

mn−1 −1 mn −1

k=0

1 Mn+Mn

mn+k mn+k − 1

k=0

M −1 n  k=0

 m −1 0 

 mn+k dμ mn+k − 1

mn+k mn+k − 1



mn+Mn −1 −1

mn−1 −1 mn −1

···

x0 =0





xn−1 =0 xn =0



···

xn+Mn −1 =1

 M n −1 mn+k Mn (mn+k − 1) = 1. mn+k − 1 k=0

1

116

2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-. . .

Clearly, Qn is a Vilenkin polynomial. Moreover, since the terms of the expanded product have pairwise disjoint spectra, by expanding the product used to define Qn , it is easy to see that for k = 0, 1, . . . , Mn − 1 , SMn+k +αn Qn − SMn+k Qn =

±1 rn+k Sαn (τx (n) gn ), k mn+k − 1

where + sign is used if Mn − 1 is even number and − sign is used if Mn − 1 is odd. Therefore, since   SM

n+k

  Qn (x) ≤ DMn+k 1 Qn 1 ≤ 1,

the choice of the integers αn , for sufficiently large n implies that   SM

n+k +αn

     Qn (x) > Cn −  SMn+k Qn (x) >

C n, 2

(2.37)

(n)

(x ∈ In (xk )).

Let n1 < n2 < . . . be positive integers chosen so that ∞  1 √ y) ≤ c f 1 , f ∈ L1 (Gm ), y > 0. This result can be found in Zygmund [464] for trigonometric series, in Schipp [287] for Walsh series and in Pál and Simon [260] for bounded Vilenkin series. The boundedness does not hold from Lebesgue space L1 (Gm ) to the space L1 (Gm ). On the other hand, if we consider restricted maximal operator . σ#∗ of Fejér means defined by   . σ#∗ f =: sup σMn f  ,

(3.1)

n∈N

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 L.-E. Persson et al., Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series, https://doi.org/10.1007/978-3-031-14459-2_3

119

120

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

then there exists a function f ∈ L1 (Gm ) such that . σ#∗ f 1 = ∞. In the one-dimensional case Yano [450] proved that σn f − f p → 0,

n → ∞, (f ∈ Lp (Gm ), 1 ≤ p < ∞).

as

However  (see [181, 295]), the rate of convergence can not be better then  O n−1 (n → ∞) for non-constant functions, i.e., if f ∈ Lp , 1 ≤ p < ∞ and 

σMn f − f = o 1 , as n → ∞, p Mn then f is a constant function. It is also known that (see e.g the books [3] and [295]) for any 1 ≤ p < ∞ and n ∈ N, we have the following estimate

σn f − f p ≤ Cp ωp

1 ,f MN

 + Cp

N−1  s=0

Ms ωp MN



1 ,f Ms

 ,



where ωp M1n , f is the modulus of continuity of function f ∈ Lp . By applying this estimate, we immediately obtain that if f ∈ lip (α, p) , i.e.,

ωp

1 ,f Mn



=O

1 Mnα

 , n → ∞,

then ⎧ ⎪ O 1 , if α > 1, ⎪ ⎪ ⎨ MN σn f − f p = O MNN , if α = 1, ⎪ ⎪ ⎪ ⎩ O 1α , if α < 1. M N

The books [104] and [230] investigated very general approximation kernels with special properties, called an approximate identity, which consists of a class of summability methods and provide norm and a.e convergence of these summability methods with respect to the trigonometric system. Investigations of these summations can be used to obtain norm convergence of Fejér means with respect to the Vilenkin system also, but these methods are not useful to study a.e convergence in this case, because of some special properties of the kernels of Fejér means.

3.2 Vilenkin-Fejér Kernels

121

3.2 Vilenkin-Fejér Kernels It is obvious that 1 (Dk ∗ f ) (x) n k=0  = (f ∗ Kn ) (x) = f (t) Kn (x − t) dμ (t) , n−1

σn f (x) =

Gm

where Kn are the so called Fejér kernels: 1 Dk . n n−1

Kn =:

k=0

Using Abel transformation we get another representation of Fejér means σn f (x) =

 n−2

 k  1− f (k) ψk (x) . n k=0

We frequently use the following well-known result: Lemma 3.2.1 Let A > t, t, A ∈ N. Then ⎧ Mt ⎪ ⎨ 1−rt (x) , x ∈ It \It +1 , KMA (x) = MA2+1 , x ∈ IA , ⎪ ⎩ 0, otherwise.

x − xt et ∈ IA ,

 Proof Let k = ∞ j =1 kj Mj and x ∈ It \It +1 . Then, by using (1.43) in Lemma 1.6.5 and Paley’s Lemma (see Corollary 1.6.7), we get that ⎛ t −1  Dk (x) = ψk (x) ⎝ kj Mj + Mt j =0

m t −1  i=mt −kt

⎞ rti (x)⎠ .

122

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

Here, we note that

mj −1

k k=mj −nj rj MA 

MA KMA (x) =

k=1 MA 

=

≡ 0, for all nj = 0. It follows that

Dk (x) ⎛ t −1  ⎝ ψk (x) kj Mj + Mt j =0

k=1

⎞

m t −1 

rti

(x)⎠

i=mt −kt

=: I1 + I2 . First, we prove that I1 = 0. By applying Lemma 1.6.1 we find that m t −1

rlkl (x) = 0, for x ∈ It \It +1

kt =0

and I1 =

m 0 −1 k0 =1



mt−1

...

mA−1 −1



mt+1



...

kt−1 =0 kt+1 =0

⎛

A−1 

⎝

kA−1 =0

⎞ rl l (x)⎠ k

l=0, l =t

t −1 

m t −1

kj Mj

j =0

rtkt (x) = 0.

kt =0

Since m 0 −1 k0 =1



mt−1

...



mA−1 −1

mt+1

kt−1 =0 kt+1 =0

...



kA−1 =0

⎛ ⎝

⎞

A−1 

rlkl (x)⎠ =

l=0, l =t

⎛

m l −1 

A−1 

⎝

l=0, l =t

⎞ rlkl (x)⎠ ,

kl =0

it follows that if x − xt et ∈ / IA , then we also have I2 = 0. That is, KMA (x) = 0 in this case. On the other hand, if x − xt et ∈ IA , then MA KMA (x) = Mt

A−1 

ml

l=0,l =t

= Mt

kt =0

rtkt

m t −1

(x)

rti (x)

i=mt −kt

mt −1 m t −1 MA  rtkt (x) rti (x) mt kt =0

=

m t −1 

i=mt −kt

mt −1 kt Mt MA  rt (x) − 1 . mt rt (x) − 1 kt =0

3.2 Vilenkin-Fejér Kernels

123

Since m t −1 kt =0

rtkt (x) = 0, rt (x) − 1

we get that MA KMA (x) = Mt MA

1 . (1 − rt (x))

We also recall that for x ∈ IA we have that MA KMA (x) = MA KMA (0) =

MA −1 MA − 1 1  . k= MA 2 k=0



The proof is complete. Corollary 3.2.2 Let t, n ∈ N and x ∈ Gm . Then KMn (x) =

j −1 n−1 m  Mj 1 1 −1 (Mn + 1)DMn (x) + DMn (x − lej ). 2 Mn 1 − e−2πıl/mj

j =0 l=0

Proof By using Corollary 1.6.7 (Paley’s Lemma) and Lemma 3.2.1, we easily get the proof. We leave out the details.  The proof of the next lemma can easily be done by using Lemma 3.2.1: Lemma 3.2.3 Let n ∈ N and x ∈ INk,l , where k < l. Then KMn (x) = 0 if n > l,

(3.2)

  KMn (x) ≤ cMk

(3.3)

m n s −1    KM (x) ≤ c M χIn (x−rs es ) . s n

(3.4)

and

s=0

rs =1

Moreover,  Gm

where C is an absolute constant.

  KM  dμ ≤ C < ∞, n

(3.5)

124

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

We also need the following useful result: Lemma 3.2.4 Let n, t, s ∈ N and 1 ≤ s ≤ mn − 1. Then sMn KsMn =

 l−1 s−1   l=0

 rni Mn DMn +

 s−1 

i=0

 rnl Mn KMn

(3.6)

l=0

and   sMn KsM (x) ≥ Mn Mn−1 ≥ M 2 n n−1 2

for x ∈ In+1 (en−1 + en ) .

(3.7)

Moreover, if x ∈ It \It +1 , x − xt et ∈ / In and n > t, then KsMn (x) = 0.

(3.8)

Proof We have that sMn KsMn =

s−1 (l+1)M  n −1 l=0

=

Dk

k=lMn

s−1 M n −1 

Dk+lMn .

l=0 k=0

Let 0 ≤ k < Mn . Then, by using (1.42) in Lemma 1.6.4 we find that Dk+lMn =

lM n −1 

ψm +

m=0

lM n +k−1

ψm

m=lMn

= DlMn +

k−1 

ψm+lMn

m=0

= DlMn + rnl

=

 l−1  i=0



k−1 

ψm

m=0

rni DMn + rnl Dk .

(3.9)

3.2 Vilenkin-Fejér Kernels

125

According to (3.9) we readily get that sMn KsMn =

s−1 M n −1  l=0 k=0

=

s−1 M n −1 

Dk+lMn  l−1 

l=0 k=0

=

 l−1 s−1   l=0

=

l=0

 DMn + rnl Dk

i=0



Mn DMn +

rni

i=0

 l−1 s−1  

 rni

rnl

l=0

 rni

 s−1 

Mn DMn +

i=0

 s−1 

 M −1 n  

rnl

Dk

k=0

Mn KMn

l=0

and (3.6) is proved. Let x ∈ In+1 (en−1 + en ) . By using Lemma 3.2.1 we have that   KM (x) = n

Mn−1 Mn−1 Mn−1 . ≥ = |1 − rn−1 (x)| 2 sin π/mn−1 2

By combining Lemmas 1.6.3 and 3.2.1 and (3.6), we get that   s−1         Mn Mn−1   l sMn KsM (x) =  r K M (x) (x)  ≥ Mn KMn (x) ≥ n Mn n n   2 l=0

and   KsM (x) ≥ Mn−1 , n 2 which implies (3.7). If n > t, x ∈ It \It +1 and x − xt et ∈ / In , then, by combining Corollary 1.6.7 and Lemma 3.2.1, we obtain that DMn (x) = KMn (x) = 0. By (3.6) and identity (3.10), we obtain (3.8) so the proof is complete.

(3.10) 

126

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

The next equality for Fejér kernels is very important for our further investigations:  Lemma 3.2.5 Let n = ri=1 sni Mni , where n1 > n2 > · · · > nr ≥ 0 and 1 ≤  sni < mni for all 1 ≤ i ≤ r as well as n(k) = n − ki=1 sni Mni , where 0 < k ≤ r. Then ⎛ ⎞ r k−1   snj ⎝ nKn = rnj ⎠ snk Mnk Ksnk Mnk (3.11) k=1

+

r−1 

j =1

⎛ ⎝

k=1

k−1 

j =1

⎞ sn rnj j ⎠ n(k) Dsnk Mnk .

Proof Let k, n ∈ N, 0 ≤ k < Mn . If we use the identity (1.40) in Lemma 1.6.4, we get that sn1 Mn1

nKn =

 k=1

n 

Dk +

Dk

k=sn1 Mn1 +1 (1)

= sn1 Mn1 Ksn1 Mn1 +

n 

Dk+sn1 Mn1

k=1 n

 sn Dsn1 Mn1 + rn11 Dk + (1)

= sn1 Mn1 Ksn1 Mn1

k=1 sn

= sn1 Mn1 Ksn1 Mn1 + n(1) Dsn1 Mn1 + rn11 n(1) Kn(1) . If we calculate n(1) Kn(1) in a similar way, we find that sn

n(1) Kn(1) = sn2 Mn2 Ksn2 Mn2 + n(2) Dsn2 Mn2 + rn22 n(2) Kn(2) , so sn

nKn = sn1 Mn1 Ksn1 Mn1 + rn11 sn2 Mn2 Ksn2 Mn2 sn

sn

sn

+ rn11 rn22 n(2) Kn(2) + n(1) Dsn1 Mn1 + rn11 n(2) Dsn2 Mn2 .

3.2 Vilenkin-Fejér Kernels

127

By using this method successively with n(2) Kn(2) , . . . , n(r−1) Kn(r−1) , we obtain that ⎛ ⎛ ⎞ ⎞ r k−1 r   snj  snj ⎝ nKn = rnj ⎠ snk Mnk Ksnk Mnk + ⎝ rnj ⎠ n(r) Kn(r) k=1

+

r−1  k=1

⎛ ⎝

j =1 k−1  j =1

j =1

⎞ sn

rnj j ⎠ n(k) Dsnk Mnk .

Since n(r) = 0, we conclude that (3.11) holds so the proof is complete.



r Corollary 3.2.6 Let n = · · · > nr ≥ 0 and i=1 sni Mni , where n1 > n2 >  1 ≤ sni < mni for all 1 ≤ i ≤ r as well as n(k) = n − ki=1 sni Mni , where 0 < k ≤ r. Then ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ |n| |n| r r−1     snj snj ⎝ ⎝ nKn = ψn ⎝ rnj ⎠ snk Mnk Ksnk Mnk + rnj ⎠ n(k) Dsnk Mnk ⎠ . k=1

j =k

k=1

j =k

Proof The proof follows from Lemma 3.2.5 and the definition of n-th Vilenkin function ψn =

|n| 

sn

rnj j .

j =1



We leave out the details. We will also frequently use the next estimation of the Fejér kernels: Corollary 3.2.7 Let n ∈ N. Then n |Kn | ≤ C

|n|  l=n

|n|      Ml KMl  ≤ C Ml KMl  ,

(3.12)

l=0

where C is an absolute constant. Proof By using Corollary 1.6.7 and Lemma 3.2.1, we get that   Mn DMn ≤ CMn KMn 

(3.13)

for every l ∈ N. By combining (1.42) in Lemmas 1.6.5, 3.2.4 and (3.13), we can conclude that     sMn KsMn  ≤ CMn KMn 

128

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

and     sMn DsMn  ≤ CMn DMn  . Hence, it follows from Lemma 3.2.5 that n |Kn (x)| ≤ C

|n| 

  Ml KMl (x) .

l=0



The proof is complete.

In the next Theorem we rewrite some above mentioned results in a different useful form: Theorem 3.2.8 Let n ∈ N and x ∈ [0, 1). Then KMn (x) = +

1 −1 (M + 1)DMn (x) 2 n

(3.14)

j −1 n−1 m  Mj

1 ˙ j−1 DMn (x +lM +1 ). Mn 1 − e−2πıl/mj

j =0 l=0

Moreover, for MN−1 ≤ n < MN we have that |Kn (x)| ≤ CMN−1

N−1  j =0

Mj

j −1 N−1  m

i=j

˙ j−1 DMi (x +lM +1 ),

(3.15)

l=0

˙ denotes the addition on the corresponding Vilenkin group. where + Proof By using Corollary 3.2.2 we easily get the proof of equality (3.14) while the inequality (3.15) is a consequence of Corollary 3.2.7 and equality (3.14), so we leave out the details.  The next result is very important for our further investigation in this Chapter to prove norm convergence in Lebesgue spaces of Fejér means. Corollary 3.2.9 For any n, N ∈ N+ , we have that  Kn (x)dμ(x) = 1, Gm

(3.16)

 |Kn (x)| dμ(x) ≤ C < ∞,

sup

(3.17)

n∈N Gm



Gm \IN

|Kn (x)| dμ(x) → 0, as n → ∞,

where C is an absolute constant.

(3.18)

3.2 Vilenkin-Fejér Kernels

129

Proof According to Corollary 1.6.8 we immediately get the proof of (3.16). By combining (3.5) in Lemmas 3.2.3 and 3.2.7 we can conclude that 

|n|

1 |Kn (x)| dμ (x) ≤ Ml n Gm l=0

 Gm

  KM (x) dμ (x) l

|n|

≤

1 Ml < C < ∞, n l=0

so also (3.17) is proved. Let x ∈ INk,l , k = 0, . . . , N − 2, l = k + 1, . . . , N − 1. By using (3.3) in Lemmas 3.2.3 and 3.2.7 we get that |Kn (x)| ≤

l   C Ms KMs (x) n

(3.19)

s=0

≤

l C Ms Mk n s=0

≤

CMl Mk . n

k,q

Let x ∈ INk,N where x ∈ Iq+1 for some N ≤ q < |n|, i.e.,   x = 0, . . . , 0, xk = 0, 0, . . . , 0, xq = 0, xq+1 , . . . , x|n|−1 , . . . . Then CMk Mq c |Kn (x)| ≤ . Mi Mk ≤ n n q−1

(3.20)

i=0

k,|n|

Let x ∈ I|n|

⊂ INk,N , i.e.,

  x = 0, . . . , 0, xk = 0, xk+1 = 0, . . . , xN = 0, . . . , x|n|−1 = 0, x|n| , . . . . Then |Kn (x)| ≤

|n|−1 c CMk M|n| . Mi Mk ≤ n n i=0

(3.21)

130

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

If we combine (3.20) and (3.21) we can conclude that  INk,N

|Kn | dμ =

k,q

|n|−1  q=N

≤

|Kn | dμ +

Iq+1

q=N

≤



|n|−1 

|Kn | dμ

k,|n|

(3.22)

I|n|

CMk CMk + n n

C(|n| − N)Mk . M|n|

Hence, if we apply (3.20) we find that  Gm \IN

|Kn | dμ =

N−2  N−1 

j−1 N−1  m



k,l k=0 l=k+1 j =l+1 xj =0 IN

≤ C

N−2  N−1 

|Kn | dμ +

N−1  k=0

INk,N

|Kn | dμ

 ml+1 . . . mN−1 Ml Mk Mk +C (|n| − N) MN n M|n| N−1

k=0 l=k+1

k=0

=: I + I I. It is evident that I =

N−2  N−1  k=0 l=k+1

≤c

N−2  k=0

≤c

N−2  k=0

Mk M|n|

(N − k)Mk M|n| |n| − k 2|n|−k

c(|n| − N) ≤ → 0, as n → ∞. 2|n|−N Analogously, we see that II ≤

c(|n| − N) → 0, as n → ∞, 2|n|−N

so also (3.18) holds and the proof is complete.



The next lemma will be important for the proof of almost everywhere convergence of Fejér means.

3.2 Vilenkin-Fejér Kernels

131

Lemma 3.2.10 Let n ∈ N. Then  sup |Kn | dμ ≤ C < ∞,

Gm \IN n>MN

where C is an absolute constant. Proof Let n > MN and x ∈ INk,l , k = 0, . . . , N − 2, l = k + 1, . . . , N − 1. By using (3.19) in the proof of Lemma 3.2.9 we get that sup |Kn (x)| ≤ n>MN

CMl Mk . MN

(3.23)

Let n > MN and x ∈ INk,N . Then, by using Lemma 1.6.9, we find that |Kn (x)| ≤ CMk so that sup |Kn (x)| ≤ CMk .

(3.24)

n>MN

Hence,  sup |Kn | dμ

(3.25)

Gm \IN n>MN

=

N−2  N−1 

j−1 N−1  m

 sup |Kn | dμ +

N−1 

k,l k=0 l=k+1 j =l+1 xj =0 IN n>MN

≤C

N−2  N−1  k=0 l=k+1

≤C

N−2  k=0

k=0

sup |Kn | dμ

INk,N n>MN

 Mk ml+1 . . . mN−1 Ml Mk +C MN MN MN N−1 k=0

(N − k)Mk + C ≤ C < ∞. MN

The proof is complete.



Now we prove some lower estimates for the Fejér kernels, which will be used to prove some negative results in the next Chapters:   Lemma 3.2.11 Let n ∈ N, n = |n| and x ∈ In+1 en−1 + en . Then 2 Mn    2 |nKn (x)| =  n − M|n| Kn−M|n| (x) ≥ Mn−1 ≥ , R2

where R = supn mn .

132

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces n−1,n

Proof Let x ∈ In+1

. Since

n = nn Mn +

|n|−1 

nj Mj + n|n| M|n|

j =n

and n − M|n| = nn Mn +

|n|−1 

  nj Mj + n|n| − 1 M|n| ,

j =n

if we combine (1.42), (3.7) and invoke Corollary 1.6.7, Lemmas 3.2.1 and 3.2.5 we obtain that    n |Kn | = n − M|n| Kn−M|n|  ⎛  ⎞  n−1    j   = ⎝ ψMn ⎠ sn Mn Ksn Mn  j  j =1    = sn Mn Ksn Mn    ⎛ ⎞  sn  −1  l−1 sn −1     i l ⎠  ⎝ = rn Mn DMn + rn Mn KMn   l=0 i=0  l=0 ⎛  ⎞  sn  −1   l  = ⎝ rn ⎠ Mn KMn   l=0    ≥ Mn KMn  2 ≥ Mn−1 ≥

2 Mn

R2

. 

The proof is complete. Lemma 3.2.12 Let n=

ji s  

nk Mk ,

i=1 k=li

where 0 ≤ l1 ≤ j1 ≤ l2 − 2 < l2 ≤ j2 ≤ . . . ≤ ls − 2 < ls ≤ js .

3.2 Vilenkin-Fejér Kernels

133

Then n |Kn (x)| ≥

Ml2i R6

  , for x ∈ Ili +1 eli −1 + eli ,

where R = supn mn . r Proof Let n = i=1 sni Mni , where n1 > n2 > · · · > nr ≥ 0 and x ∈ Ili +1 eli −1 + eli . By combining (1.42) in Lemma 1.6.5, Corollary 1.6.7 and (3.8) in Lemma 3.2.4, we obtain that DMli = 0 and Dsnk Msn = Ksnk Msn = 0, snk > li . k

k

Since sn1 > sn2 > · · · > snr ≥ 0 we find that n(k) = n −

k 

sni Mni

i=1

=

s 



nk+1

sni Mni ≤

i=k+1

(mi − 1)Mi

i=0

= mnk+1 Mnk+1 − 1 ≤ Mnk . According to Lemma 3.2.5 we have that mr mr i−1  i−1              Mk Ds M  | |K n n ≥ sli Mli Ksli Mli  − sk Mk Ksk Mk − k k r=1 k=lr

r=1 k=lr

=: I1 − I2 − I3 .   Let x ∈ Ili +1 eli −1 + eli and 1 ≤ sli ≤ mli − 1. By using (3.7) in Lemma 3.2.4 we get that     I1 = sli Mli Ksli Mli  ≥ Ml2i −1 . It is easy to see that k  s=0

ns Ms ≤

k  s=0

(ms − 1) Ms = mk Mk − m0 M0 ≤ Mk+1 − 2.

(3.26)

134

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

Since ji−1 ≤ li − 2, if we use the estimates above we obtain that I2 ≤

l i −2

 ns Ms Kn

s Ms

 (x)

(3.27)

s=0

≤

l i −2

ns Ms

s=0

(ns Ms + 1) 2

  li −2 mli −2 − 1 Mli −2  ≤ (ns Ms + 1) 2 s=0     mli −2 − 1 Mli −2 mli −2 − 1 Mli −2 ≤ Mli −1 + (li − 1) 2 2 ≤

Ml2i −1 2

−

Mli −2 Mli −1 Ml −1 (li − 1) + i . 2 2

For I3 we have that I3 ≤

l i −2

l i −2 i −2  l 2  ≤ n M ≤ M nk Mk (x) k k li −2 k Mk

 Mk Dn

k=0

k=0

(3.28)

k=0

Ml2i −1   − Mli −2 . ≤ Mli −2 Mli −1 − 1 ≤ Mli −2 Mli −1 − Mli −2 ≤ 2 By combining (3.26)–(3.28) we obtain that n |Kn (x)| ≥ I1 − I2 − I3 2 Mli −1 Mli −2 Ml −1 (li − 1) Mli −1 − i − + Mli −2 2 2 2 2 Ml −1 (li − 1) Mli −1 Mli −2 − i + Mli −2 . ≥ 2 2

≥ Ml2i −1 −

Ml2i −1

+

Suppose that li ≥ 5. Then n |Kn (x)| ≥ I1 − I2 − I3 ≥

Ml2 Ml2 Mli −1 Mli −2 ≥ 5i ≥ 6i . 4 R R

Let 2 ≤ li ≤ 4. According to the fact that li − 1 ≤ Mli −2 we easily get that n |Kn (x)| ≥ I1 − I2 − I3 ≥ Mli −2 ≥

Ml2i R

≥

Ml2i R6

.

3.2 Vilenkin-Fejér Kernels

135

In the case when li = 0 or li = 1 then it coincide l0 = n and the estimate follows from Lemma 3.2.11. The proof is complete.  Corollary 3.2.13 Let 2 < n ∈ N+ and qn = M2n + M2n−2 + . . . + M2 + M0 . Then  M2    qn−1 Kqn−1 (x) ≥ 2k for x ∈ I2k+1 e2k−1 + el2k and R = sup mn , 6 R n where k = 0, 1, . . . , n. Lemma 3.2.14 Let x ∈ INk,l , k = 0, . . . , N − 2, l = k + 1, . . . , N − 1. Then  |Kn (x − t)| dμ (t) ≤ IN

CMl Mk . nMN

(3.29)

CMk , MN

(3.30)

Let x ∈ INk,N , k = 0, . . . , N − 1. Then  |Kn (x − t)| dμ (t) ≤ IN

where C is an absolute constant. Proof Let x ∈ INk,l for 0 ≤ k < l ≤ N − 1 and t ∈ IN . Since x − t ∈ INk,l by combining Lemma 3.2.1 and (3.12) in Corollary 3.2.7 we obtain that  n |Kn (x − t)| dμ (t) ≤ C IN

 l  Mi i=0

≤ Cμ(IN )

IN

  KM (x − t) dμ (t) i

l  CMk Ml Mi Mk ≤ MN i=0

so that  |Kn (x − t)| dμ (t) ≤ IN

and (3.29) is proved.

CMk Ml nMN

136

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

Let x ∈ INk,N . Then, by applying Lemma 3.2.1 and (3.12) in Corollary 3.2.7, we have that 

 |n|  n |Kn (x − t)| dμ (t) ≤ Mi IN

IN

i=0

  KM (x − t) dμ (t) . i

(3.31)

Let 

  x =  0, . . . , 0, xk = 0, 0 . . . , 0, xN , xN+1 , xq , . . . , x|n|−1 , . . . , t = 0, . . . , 0, xN , . . . , xq−1 , tq = xq , tq+1 , . . . , t|n|−1 , . . . ,

for some q = N, . . . , |n| − 1. By using Lemmas 3.2.1 and 3.2.3 in (3.31) it is easy to see that  IN

C |Kn (x − t)| dμ (t) ≤ Mi n q−1 i=0

≤

 Mk dμ (t)

(3.32)

IN

CMk Mq CMk ≤ . nMN MN

Let 

  x = 0, . . . , 0, xk = 0, 0, . . . , 0, xN , xN+1 , xq , . . . , x|n|−1 , . . . ,   t = 0, . . . , 0, xN , xN+1 . . . , x|n|−1 , . . . .

If we apply again Lemmas 3.2.1 and 3.2.3 in (3.31), we obtain that  IN

 |n|−1 C CMk |Kn (x − t)| dμ (t) ≤ Mi Mk dμ (t) ≤ . n MN IN

(3.33)

i=0

By combining (3.32) and (3.33) also (3.30) is proved so the proof is complete. The next lemma is a simple consequence of Lemma 3.2.14 when n ≥ MN : Lemma 3.2.15 Let x ∈ INk,l , k = 0, . . . , N − 1, l = k + 1, . . . , N. Then  |Kn (x − t)| dμ (t) ≤ IN

where C is an absolute constant.

CMl Mk for n ≥ MN , MN2



3.3 Approximation of Vilenkin-Fejér Means

137

3.3 Approximation of Vilenkin-Fejér Means Our first result of this Section reads: Theorem 3.3.1 Let 1 ≤ p < ∞, f ∈ Lp (Gm ) and n ∈ N. Then σn f − f p ≤ Cp ωp (1/MN , f ) + Cp

N−1  s=0

Ms ωp (1/Ms , f ) , MN

where Cp is an absolute constant depending only on p. Proof Let f ∈ Lp (Gm ), 1 ≤ p < ∞ and MN < n ≤ MN+1 . Then σn f − f p (3.34) ≤ σn f − σn SMN f p + σn SMN f − SMN f p + SMN f − f p   = σn SMN f − f p + SMN f − f p + σn SMN f − SMN f p ≤ Cp ωp (1/MN , f ) + σn SMN f − SMN f p . By routine calculations we get that σn SMN f − SMN f

(3.35)

N 1 1 = Sk SMN f + n n

n 

M

k=1

N 1 1 Sk f + n n

M

=

k=1

n 

SMN f − SMN f

k=MN +1

N 1 n − MN SMN f − SMN f Sk f + n n

M

=

k=1

MN MN σMN f − SMN f n n  MN  = SMN σMN f − SMN f n   MN = SMN σMN f − f . n =

Sk SMN f − SMN f

k=MN +1

138

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

By using (3.35) we find that σn SM f − SM f = N N p



MN n

p

  SM σM f − f N N p

(3.36)

  ≤ SMN σMN f − f p ≤ σMN f − f p . Moreover,  σMN f (x) − f (x) = 

Gm

= IN

+

(f (x − t) − f (x)) KMN (t) dμ(t)

(3.37)

(f (x − t) − f (x)) KMN (t) dμ(t)

N−1 s −1   m s=0 ns =1

IN (ns es )

(f (x − t) − f (x)) KMN (t) dμ(t)

=: I + I I. If we apply Lemma 3.2.1 and generalized Markdowns’s inequality (see Remark 1.5.2), we get that  I p ≤ IN

f (x − t) − f (x)p 

≤ ωp (1/MN , f ) IN

MN − 1 dμ(t) 2

(3.38)

MN − 1 dμ(t) 2

≤ ωp (1/MN , f ) . and I I p ≤ CMs

N−1 s −1   m s=0 ns =1

≤ CMs

N−1 s −1   m s=0 ns =1

≤C

IN (ns es )

N−1  s=0

f (x − t) − f (x)p dμ(t)

(3.39)

ωp (1/Ms , f ) dμ(t) IN (ns es )

Ms ωp (1/Ms , f ) . MN

The proof is complete by just combining (3.34)–(3.39).



3.3 Approximation of Vilenkin-Fejér Means

139

This result immediately implies the following two corollaries: Corollary 3.3.2 Let 1 ≤ p < ∞, f ∈ Lp (Gm ) and n ∈ N. Then σn f − f p → 0, as n → ∞. Corollary 3.3.3 Let f ∈ lip (α, p) , i.e.   ωp (1/Mn , f ) = O 1/Mnα , as n → ∞. Then ⎧ ⎨ O (1/MN ) , if α > 1, σn f − f p = O (N/MN ) , if α = 1,  ⎩  O 1/MNα , if α < 1. Theorem 3.3.4 Let 1 ≤ p < ∞, f ∈ Lp (Gm ) and σM f − f = o (1/Mn ) , as n → ∞. n p Then f is a constant function. Proof Since Mn −1 1  σMn f − SMn f = k f(k) ψk , Mn k=0

by using Minkowski’s integral inequality we get that M −1 n  k f(k) ψk ≤ Mn σMn f − f p + Mn SMn f − f p k=0

p

≤ 2Mn σMn f − f p → 0, as n → ∞.

Let 0 ≤ j < Mn . Then j f(j ) =

 ψj (x) Gm

M n −1 k=0

k f(k) ψk (x) dμ(x).

140

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

By using Hõlder’s inequality, we obtain that ⎛    j f(j ) ≤ ⎝

⎞1/p p M −1 n     k f(k) ψk (x) dμ(x)⎠ → 0, as n → ∞.   Gm  k=0

It follows that j f(j ) = 0 and f(j ) =



f(0) , 0,

if if

j = 0, j = 0. 

The proof is complete.

3.4 Almost Everywhere Convergence of Vilenkin- Fejér Means First, we state the following result concerning pointwise convergence: Theorem 3.4.1 If x ∈ Gm and f ∈ L1 (Gm ) is continuous at x, then lim σm f (x) = f (x) .

m→∞

Proof It is easy to see that  σn f (x) − f (x) =

(f (t) − f (x)) Kn (x − t) dt IN (x)



+

(f (t) − f (x)) Kn (x − t) dt. Gm \IN (x)

By using the localization principle (Lemma 1.9.7) we get that  (f (t) − f (x)) Dn (x − t) dt → 0, n → ∞. Gm \IN (x)

Hence, if we use the regularity of the Fejér means, we find that  (f (t) − f (x)) Kn (x − t) dt → 0, n → ∞. Gm \IN (x)

(3.40)

3.4 Almost Everywhere Convergence of Vilenkin- Fejér Means

141

On the other hand,          (f (t) − f (x)) Kn (x − t) dt     IN (x)   ≤ sup |f (u) − f (x)| |Kn (x − t)| dt u∈IN (x)

≤ sup

Gm

|f (u) − f (x)| .

u∈IN (x)

Since f is continuous at x, the last term is small enough if N is large enough and we can conclude that (3.40) holds so the proof is complete.  Now, we will prove that the maximal operator of Fejér means with respect to the Vilenkin-Fourier series is of weak-(1,1) type. In particular, the following is true: Theorem 3.4.2 Let f ∈ L1 (Gm ). Then * + sup y μ σ ∗ f > y ≤ f 1 . y>0

Proof By Theorem 2.4.2 we obtain that the proof will be complete if we show that 

 ∗  σ f  dμ ≤ f 1 ,

(3.41)

I

for every function f, which satisfy conditions in (2.10), where I denotes the support of the function f. Without lost the generality, we may assume that f is a function with support I and μ (I ) = MN . We may assume that I = IN . It is easy to see that  σn f =

Kn (x − t)f (t)dμ (t) = 0 for n ≤ MN IN

Therefore, we can suppose that n > MN . Hence,         ∗  σ f (x) = sup  Kn (x − t)f (t)dμ (t) .  n>MN   IN

142

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

Let t ∈ IN and x ∈ IN . Then x − t ∈ IN and if we apply Lemma 3.2.10, we get that     ∗  σ f (x) dμ(x) ≤ |Kn (x − t) f (t)| dμ (t) dμ (x) sup n>MN IN

IN

 

IN

sup |Kn (x − t) f (t)| dμ (x) dμ (t)

≤ IN IN

n>MN





|f (t)| dμ (t)

≤

sup |Kn (x − t)| dμ (x) n>MN

IN

IN





|f (t)| dμ (t)

≤

sup |Kn (x)| dμ (x) n>MN

IN

IN

≤ C f 1 . 

so (3.40) holds and the proof is complete. Theorem 3.4.3 Let f ∈ L1 . Then σn f → f a.e., as n → ∞. Proof Since Sn P = P for every P ∈ P, according to regularity of Fejér means we obtain that σn P → P

a.e.,

as

n → ∞,

where P is dense in the space L1 (see Proposition 1.8.1). On the other hand, by using Theorems 3.4.2 and 1.5.9, we obtain almost everywhere convergence of Fejér means: σn f → f a.e., as n → ∞. The proof is complete.



3.4 Almost Everywhere Convergence of Vilenkin- Fejér Means

143

Now we introduce the notion of Lebesgue points and Vilenkin-Lebesgue points. Definition 3.4.4 A point x is called a Lebesgue point of an integrable function f if 1 h→0 h



x+h

lim

|f (t) − f (x)| dμ(t) = 0.

x

It is known that almost every point x is a Lebesgue point of f ∈ L1 and the Fejér means σnT f of the trigonometric Fourier series of f ∈ L1 converges to f at each Lebesgue point. We will below prove an analogical theorem for the Fejér means σn f with respect to Vilenkin systems (see Theorem 3.4.9). First we need to give the following definition: Definition 3.4.5 We introduce the operator WA defined by WA f (x) =:

A−1 

Ms

m s −1  rs =1

s=0

|f (t) − f (x)| dμ(t).

IA (x−rs es )

A point x ∈ Gm is called a Vilenkin-Lebesgue point of the function f ∈ L1 (Gm ) if lim WA f (x) = 0.

A→∞

We define VA by VA f (x) =:

A 

Ms

s=0

=

m s −1  rs =1

f (t)dμ(t) IA (x−rs es )

ms −1  A  Ms  DMA (x − rs es − t)f (t)dμ(t) MA Gm s=0

rs =1

⎞ ⎛  m A s −1  M s ⎝ DMA (x − rs es − t)⎠ f (t)dμ(t) = MA Gm rs =1

s=0

 =

YA (x − t)f (t)dμ(t), Gm

where YA (x) =

ms −1 A  Ms  DMA (x − rs es ). MA s=0

rs =1

144

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

It is obvious that lim WA f (x) = 0

A→∞

if and only if

lim VA |f − f (x)|(x) = 0.

A→∞

Moreover, we define V ∗ by V ∗ f (x) = sup |VA f (x)|. A∈N

Theorem 3.4.6 Let N ∈ N. Then  sup |YA (x)| dμ ≤ C < ∞, Gm \IN A>N

where C is an absolute constant. Proof Let A > N and x ∈ INk,l , k = 0, . . . , N − 2, l = k + 1, . . . , N − 1. Then it is easy to prove that x − rs es ∈ Gm \ IN . By using Corollary 1.6.7 (Paley’s Lemma) we get that DMA (x − rs es ) = 0 and    A  s −1  Ms m   |YA (x)| =  DMA (x − rs es ) = 0 for A > N.  s=0 MA rs =1 

(3.42)

By using again Corollary 1.6.7 (Paley’s Lemma) we can conclude that  DMA (x − rk ek ) =

MA , x ∈ IA (rk ek ), 0, x ∈ Gm \ IA (rk ek ).

Hence,  Ms  DMA (x − rk ek ) MA  Ms , x ∈ IA (rk ek ), = 0, x ∈ Gm \ IA (rk ek ).

|YA (x)| =

(3.43)

3.4 Almost Everywhere Convergence of Vilenkin- Fejér Means

145

By combining (1.5), (3.42) and (3.43) we find that  sup |YA (x)| dμ(x) =

Gm \IN A>N

+

N−2  N−1 

 sup |YA (x)| dμ(x)

k,l k=0 l=k+1 j =l+1 xj =0 IN A>N

N−1  k=0

=

j−1 N−1  m

sup |YA (x)| dμ(x)

INk,N A>N

N−1  k=0

≤C

sup |YA (x)| dμ(x) IA (rk ek ) A>N

N−1  k=0

Mk < C < ∞. MN 

The proof is complete.

We apply this lemma to prove the following important theorem for the maximal operator V ∗ : Theorem 3.4.7 Let f ∈ L1 (Gm ). Then the operator V ∗ is of weak-type (1, 1), i.e., * + sup y μ V ∗ f > y ≤ f 1 .

y>0

Proof Since A−1  ∗ V f ≤ Cf ∞ sup 1 Ms ≤ Cf ∞ , ∞ A∈N MA s=0

we obtain that V ∗ f is bounded from L∞ (Gm ) to L∞ (Gm ). According to Theorem 2.4.2 we obtain that the proof will be complete if we show that   ∗  V f  dμ ≤ cf 1 , (3.44) I

for every function f which satisfy the conditions in (2.10), where I denotes the support of the function f. Without lost the generality, we may assume that f is a function with support I and μ (I ) = MN . We may assume that I = IN . It is easy to see that Vn f = 0 when n ≤ MN .

146

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

Therefore, we can suppose that n > MN . Hence,         ∗  V f (x) = sup  Yn (x − t)f (t)dμ (t) .  n>MN   IN Let t ∈ IN and x ∈ IN . Then x − t ∈ IN and if we apply Lemma 3.4.6, we get that 

  ∗ V f (x) dμ(x) ≤





IN

|Yn (x − t) f (t)| dμ (t) dμ (x)

sup n>MN IN

 

IN

sup |Yn (x − t) f (t)| dμ (x) dμ (t)

≤ IN IN

n>MN





|f (t)| dμ (t)

= IN

sup |Yn (x)| dμ (x) n>MN IN

≤ C f 1 , 

so (3.44) holds and the proof is complete. Theorem 3.4.8 Let f ∈ L1 (Gm ) . Then lim WA f (x) = 0

A→∞

a.e. x ∈ Gm .

Proof It is easy to see that lim WA f (x) = 0

A→∞

for every Vilenkin polynomial. Hence, since the Vilenkin polynomials are dense in L1 (Gm ), Theorems 3.4.7 and 1.5.9 imply the proof.  Theorem 3.4.9 Let f ∈ L1 (Gm ). Then lim σn f (x) = f (x)

n→∞

for all Vilenkin-Lebesgue points of f .

3.5 Approximate Identity

147

Proof By combining (3.4) in Lemma 3.2.3 and (3.12) in Corollary 3.2.7, we get that |n|

C |σn f (x) − f (x)| ≤ MA n A=0

 |f (t) − f (x)||KMA (x − t)|dμ(t) Gm

|n| s −1   m C |f (t) − f (x)| dμ(t) MA Ms ≤ n IA (x−rs es ) A=0

s=0

rs =1

|n|

C MA WA f (x) → 0, as n → ∞. n

≤

A=0



The proof is complete. As a corollary, we obtain Theorem 3.4.3. Corollary 3.4.10 Let f ∈ L1 (Gm ). Then lim σn f (x) = f (x) a.e. on Gm .

n→∞

3.5 Approximate Identity The properties established in Corollary 3.2.9 ensure that the kernels of the Fejér means {KN }∞ N=1 form an approximate identity. The following definition plays a central role to study norm convergence: ∞ Definition 3.5.1 The family {n }∞ n=1 ⊂ L (Gm ) forms an approximate identity provided that

 n (x)dμ(x) = 1

(A1) Gm

(A2)

 |n (x)| dμ(x) < ∞

sup n∈N Gm



(A3)

Gm \IN

|n (x)| dμ(x) → 0, as n → ∞, for any N ∈ N+ .

The term “approximate identity” is used because of the fact that n ∗ f → f in any reasonable sense.

as n → ∞

148

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

Note that the Dirichlet kernels {DN }∞ N=1 does not form an approximate identity, but the kernels {KN }∞ of the Fejér means forms an approximate identity. N=1 Next, we prove an important result, which will be used to obtain norm convergence of some well-known and general summability methods in this and the next Chapter: Theorem 3.5.2 Suppose that f ∈ Lp (Gm ) for some 1 ≤ p < ∞ and that the family ∞ {n }∞ n=1 ⊂ L (Gm )

forms an approximate identity. Then n ∗ f − f p → 0 as n → ∞. The same holds for p = ∞ if f is uniformly continuous. Proof Let ε > 0. By combining Corollary 1.9.5 and (A2) condition we get that sup f (· − t) − f (·)p sup n 1 < ε/2,

t ∈IN

n∈N

whenever N is large enough. By applying Minkowski’s integral inequality and (A1) and (A3) conditions, we find that  n ∗ f − f p = n (t)(f (· − t) − f (·))dμ(t) 

Gm

|n (t)| f (· − t) − f (·)p dμ(t)

≤ 

Gm

=  +

IN

p

|n (t)| f (· − t) − f (·)p dμ(t)

Gm \IN

|n (t)| f (· − t) − f (·)p dμ(t)

≤ sup f (· − t) − f (·)p sup n 1 t ∈IN

n∈N

+ sup f (· − t) − f (·)p t ∈Gm



Gm \IN

|n (t)|dμ(t)

< ε/2 + ε/2 < ε if n is large enough. The proof is complete.



3.5 Approximate Identity

149

Theorem 3.5.3 Suppose that f ∈ L1 (Gm ) and that the family ∞ {n }∞ n=1 ⊂ L (Gm )

forms an approximate identity. In addition, let (A4)

lim

sup |n | = 0 for any N ∈ N+ .

n→∞ G \I m N

(a) If the function f is continuous at t0 , then n ∗ f (t0 ) → f (t0 ) as n → ∞. (b) If the functions {n }∞ n=1 are even and the left and right limits f (t0 − 0) and f (t0 + 0) do exist and are finite, then n ∗ f (t0 ) → L as n → ∞, where L =:

f (t0 + 0) + f (t0 − 0) . 2

Proof It is evident that     |n ∗ f (t0 ) − f (t0 )| =  n (t)(f (t0 − t) − f (t0 ))dμ(t) Gm     ≤  n (t)(f (t0 − t) − f (t0 ))dμ(t) IN     n (t)f (t0 − t)dμ(t) +    + 

Gm \IN

Gm \IN

  n (t)f (t0 )dμ(t)

=: I + I I + I I I. Let f be continuous at t0 . For any ε > 0 there exists N such that I ≤ sup |f (t0 + t) − f (t0 ))| sup n 1 < ε/2. t ∈IN

n∈N

150

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

Using (A4) condition, we get that I I ≤ sup |n | f 1 → 0, as n → ∞. Gm \IN

We conclude from (A3) that  I I I ≤ |f (t0 )|

Gm \IN

|n (t)|dμ(t) → 0, as n → ∞.

Thus part (a) is proved. For the proof of part (b), we first note that  n ∗ f (t0 ) =

n (t)f (t0 − t)dμ(t) 

Gm

=

n (t) Gm

f (t0 − t) + f (t0 + t) dμ(t) 2

and  n ∗ f (t0 ) − L = 

n (t)

f (t0 + t) − f (t0 + 0) dμ(t) 2

n (t)

f (t0 − t) − f (t0 − 0) dμ(t). 2

Gm

+ Gm

Hence, if we use part (a), we immediately get the proof of part (b) so the proof is complete.  There is an essential difference between the Vilenkin-Fejér kernels and trigonometric Fejér kernels. Remark 3.5.4 If we consider the Fejér kernels with respect to the trigonometric system, we have that KnT (x) =

1 n



sin(nx/2) sin(x/2)

2

and 0 ≤ KnT (x) ≤ min(n, π 2 (nx 2 )

−1

).

Moreover, 

π −π

KnT (x)dx = 1, KnT (x) ≥ 0,

and conditions (A1), (A2), (A3) and (A4) hold.

(3.45)

3.6 Final Comments and Open Questions

151

Conditions (A4) and (3.45) do not hold for the Vilenkin-Fejér kernels. Indeed, by using Lemma 3.2.1, for any e0 ∈ In (e0 ) ⊂ Gm \In , (n ∈ N+ ) and for any k ∈ N+ , we get that     M0  |KMk (e0 )| =  1 − r0 (e0 )      M0   = 1 − exp (2πı/m0)  =

1 1 ≥ , 2 sin(π/m0 ) 2

so that lim

sup

k→∞ In (e0 )⊂Gm \In

    KM (x) ≥ lim KM (e0 ) k k k→∞

≥

1 > 0, for any n ∈ N+ . 2

Hence (A4) and (3.45) are not true for the Fejér kernels with respect to the Vilenkin system. However, in some publications you can find that some researchers use such an estimate (for details see [169]).

3.6 Final Comments and Open Questions (1) Lemma 3.2.1 and Corollary 3.2.2 with respect to the Vilenkin system is due to Gát [111] and for the Walsh system see the book [295]. (2) The proof of Lemma 3.2.3 for bounded Vilenkin system can be found in Tephnadze [345, 346]. (3) Lemma 3.2.4 is proved by Blahota and Tephnadze [38], but here we gave a completely different and simpler proof. In the same paper Blahota and Tephnadze [38] also proved Lemma 3.2.5. (4) Corollary 3.2.7 is proved in the book of Agaev, Vilenkin, Dzhafarly and Rubinshtein [3] (see also Tephnadze [355]), Theorem 3.2.8 can be found in Pál and Simon [260] and Corollary 3.2.9 in this form was proved in Nadirashvili, Tephnadze and Tutberidze [233]. (5) The proof of Lemma 3.2.11 can be found in Persson and Tephnadze [270]. (6) The analogy of Lemma 3.2.12 for the Walsh system was proved in Tephnadze [354] and Corollary 3.2.13 is a simple consequence of it. A similar lower estimate was proved in Blahota, Gát and Goginava [40] and [41]. (7) The proofs of Lemmas 3.2.14 and 3.2.15 is due to Tephnadze [345, 346] (see also Blahota, Tephnadze [38]). (8) Theorems 3.3.1 and 3.3.2, Corollary 3.3.3 and Theorem 3.3.4 can be found in [3]. Similar results concerning the Walsh system are proved in [295].

152

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

(9) Theorem 3.4.1, where the pointwise convergence of Fejér means was investigated, can be found in [3]. (10) Theorem 3.4.2 and Theorem 3.4.3 can be found in [295] for the Walsh system and in [3] for the Vilenkin system. These results with the proofs given in this book can be found in Nadirashvili, Tephnadze and Tutberidze [233]. (11) The definition of Vilenkin-Lebesgue points and the results concerning almost everywhere convergence of Vilenkin-Fejér means of integrable functions given in Sect. 3.4 can be found in Goginava and Gogoladze [134]. A similar definition of Walsh-Lebesgue points and the convergence of Walsh-Fejér means in Walsh-Lebesgue points was proved by Weisz [395]. (12) The definition of approximation identity (see Definition 3.5.1), Theorem 3.5.2 and Theorem 3.5.3, where norm and almost everywhere convergence of approximation identity are investigated, can be found in Muscalu and Schlag [230] (see also [233]). For Remark 3.5.4 see Nadirashvili, Tephnadze and Tutberidze [233]. (13) For the trigonometric system Fejér [94] proved uniform convergence of his means for any continuous functions. In particular, if f ∈ C([−π, π]) then T σn f − f → 0, as n → ∞, C

where σnT f is n-th Fejér mean with respect to the trigonometric system. Lebesgue [208] proved that the Fejér means converge to f almost everywhere for every integrable function f ∈ L1 (Gm ), i.e. that lim σ T f (x) n→∞ n

= f (x) a. e. on Gm .

Convergence of the Fejér means with respect to the trigonometric system in Lebesgue points can be found in [464]: lim σ T f (x) n→∞ n

= f (x)

for all Lebesgue points of f ∈ L1 ([−π, π]). Another generalization was given by Marcinkiewicz [221] and Zygmund [463], where they consider a.e strong convergence. Divergence of Fejér means with respect to the trigonometric system on the sets of measure zero can be found in Karagulian [186]. (14) Uniform convergence and (C, 1) summability of the Fourier series of continuous functions with respect to the Walsh-Kaczmarz system was studied by Skvortcov [322]. Gát [108] proved that, for any integrable function, the Fejér means with respect to the Walsh-Kaczmarz system converge almost everywhere to the function. He showed that the maximal operator of WalshKaczmarz-Fejér means is of weak type (1, 1) and of type (p, p) for all 1 < p ≤ ∞. On the other hand, the maximal operator σnκ f is not of (1, 1) type.

3.6 Final Comments and Open Questions

153

(15) In [377] Toledo established an iteration for the L1 (Gm )-norm of WalshFejér kernels and proved some properties of this sequence, including that its supremum is exactly equal to 17/15. Fridli [101] used the dyadic modulus of continuity to characterize the set of functions in the space Lp (Gm ), whose Vilenkin-Fejér means converge at a given rate. Divergence of Fejér means with respect to the Vilenkin system on the sets of measure zero was proved by Karagulian [186]. In particular, for any set E ⊂ Gm , with μ(E) = 0, there exists a bounded function f ∈ L1 (Gm ) such that σn f (x) diverge at any x ∈ E. The next open problems can be considered as important basic knowledge to be able to prove some new results concerning convergence and divergence of some summability methods and their maximal operators with respect to bounded Vilenkin systems on the martingale Hardy spaces: Open Problem Let n=

ji s  

nk Mk ,

i=1 k=li

where 0 ≤ l1 ≤ j1 ≤ l2 − 2 < l2 ≤ j2 ≤ . . . ≤ ls − 2 < ls ≤ js . Does there exist an absolute constant C such that nKn ≤ C

ji s  

  ( nk − 1) Mk KMk 

(3.46)

i=1 k=li

+C

s  s          Mli KMli  + C Mji KMji ? i=1

i=1

Remark 3.6.1 An analogical positive result for the Walsh system can be found in Tephnadze [358], but in this case nk − 1 = 0 for all k ∈ N+ and the first term of the right-hand side of inequality (3.46) is equal to zero. To obtain a similar estimate for the bounded Vilenkin system, the method of Tephnadze [358] and a little more precise estimates in the proof can be used.

154

3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces

Open Problem Let n =

s

ji

i=1

k=li

nk Mk , where

0 ≤ l1 ≤ j1 ≤ l2 − 2 < l2 ≤ j2 ≤ . . . ≤ ls − 2 < ls ≤ js . Does there exist an absolute constant C > 0 such that   n |Kn (x)| ≥ CMl2i for x ∈ Ili +1 eli −1 + eli and n |Kn (x)| ≥ C ( nk − 1) Mk2 , for x ∈ Ik+1 (ek−1 + ek ) , where li ≤ k ≤ ji , i = 1, . . . , s? Remark 3.6.2 The analogical result for the Walsh system can be found in Tephnadze [358], but the second lower estimate is not considered for this case. The first lower estimate for the bounded Vilenkin system is proved in Lemma 3.2.12. Open Problem Find the optimal constant C, such that Kn 1 < C, for all n ∈ N, where Kn is the Fejér kernel with respect to the Vilenkin system. Remark 3.6.3 Since the Walsh system is an example of Vilenkin systems, according to a result of Toledo [377], we have that C ≥ 15 17 . Open Problem Derive the optimal constant C, such that Knκ 1 < C < ∞ for all n ∈ N, where Knκ is the Fejér kernel with respect to the Kaczmarz system. Open Problem Find an appropriate (A4) condition for the family ∞ {n }∞ n=1 ⊂ L (Gm )

of approximate identities which provides almost everywhere convergence of n ∗ f to the function f so that this condition is fulfilled also for the Fejér kernel with respect to the Vilenkin system.

3.6 Final Comments and Open Questions

155

Open Problem Consider the kernel of some classical summability methods with respect to the Walsh-Kaczmarz system. An open problem is to investigate in each case if they satisfy the conditions (A1)–(A3). Moreover, in each positive case investigate whether it is possible to find an appropriate (A4) condition which provides almost everywhere convergence of n ∗ f to the function f and that this condition is also fulfilled for the Fejér kernel with respect to the Walsh-Kaczmarz system. Open Problem For any integrable function f ∈ L1 (Gm ) find a suitable definition of Kaczmarz-Lebesgue points and prove that lim σ κ f (x) n→∞ n

= f (x)

for all Kaczmarz-Lebesgue points of f , where σnκ f is the n-th Fejér mean with respect to the Kaczmarz system.

Chapter 4

Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

4.1 Introduction The n-th Nörlund and Riesz logarithmic means are defined by Ln f =:

n−1 1  Sk f ln n−k k=0

and n 1  Sk f , Rn f =: ln k k=1

respectively, where ln =:

n  1 k=1

k

.

It is known that the Nörlund logarithmic means have better approximation properties than the partial sums and that the Riesz logarithmic means are better than Fejér means in the same sense. In [114], Gát and Goginava proved some convergence and divergence properties of the Nörlund logarithmic means of continuous or integrable functions. Moreover, Gát and Goginava [115] proved that for each measurable function satisfying

φ (u) = o u log1/2 u , as u → ∞,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 L.-E. Persson et al., Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series, https://doi.org/10.1007/978-3-031-14459-2_4

157

158

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

there exists an integrable function f such that  φ (|f (x)|) dμ (x) < ∞ Gm

and that there exists a set with positive measure such that the Nörlund logarithmic means of the function diverges on this set. It follows that weak-(1,1) type inequality does not hold for the maximal operator L∗ of the Nörlund logarithmic means defined by L∗ f =: sup |Ln f | , n∈N

but there exists an absolute constant Cp such that ∗ L f ≤ Cp f p , when f ∈ Lp , p > 1. p L∗# , defined Moreover, if we consider the following restricted maximal operator . by   .∗# f =: sup LMn f  , (Mk =: m0 . . . mk−1 , k = 0, 1 . . .), L n∈N

then * ∗ + .# f > y ≤ c f 1 , f ∈ L1 (Gm ), y > 0. yμ L Hence, if f ∈ L1 (Gm ), then LMn f → f, a.e. on Gm . If we consider the maximal operator R ∗ of the Riesz logarithmic means defined by R ∗ f =: sup |Rn f | , n∈N

then * + yμ R ∗ f > y ≤ c f 1 , f ∈ L1 (Gm ), y > 0. Moreover, for any f ∈ L1 (Gm ), we have that lim Rn f (x) = f (x)

n→∞

for all Vilenkin-Lebesgue points of f .

4.1 Introduction

159

The boundedness of the maximal operator of the Riesz logarithmic means does not hold from the space L1 (Gm ) to the space L1 (Gm ). However, Rn f − f p → 0,

as

n → ∞, (f ∈ Lp (Gm ), 1 ≤ p < ∞).

Another well-known summability method is the so called (C, α)-means (or Cesàro means) σnα , which are defined by 1  α−1 An−k Sk f, Aαn n

σnα f =:

k=1

where Aα0 =: 0,

Aαn =:

(α + 1) . . . (α + n) , n!

α = −1, −2, . . .

It is well-known that for α = 1 this summability method coincides with the Fejér summation and for α = 0, we just have the partial sums of the Vilenkin-Fourier series. Moreover, if we consider the maximal operator of the Cesàro means σ α,∗ , defined by   σ α,∗ f =: sup σnα f  n∈N

for 0 < α ≤ 1, then * + yμ σ α,∗ f > y ≤ c f 1 , f ∈ L1 (Gm ), y > 0. The boundedness of the maximal operator of the Cesàro means does not hold from the space L1 (Gm ) to the space L1 (Gm ). However, α σ f − f → 0, n p

as

n → ∞, (f ∈ Lp (Gm ), 1 ≤ p < ∞).

The n-th Nörlund mean tn and T mean Tn of the Fourier series of f are defined by 1  tn f =: qn−k Sk f Qn

(4.1)

1  qk Sk f, Qn

(4.2)

n

k=1

and n−1

Tn f =:

k=0

160

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

respectively, where (qk , k ∈ N) is a sequence of non-negative numbers and Qn =:

n−1 

qk .

k=0

The representations  tn f (x) =

f (t) Fn (x − t) dμ (t) Gm

and  Tn f (x) =

f (t) Fn−1 (x − t) dμ (t)

Gm

play central roles in the sequel, where 1  qn−k Dk Qn

(4.3)

1  qk Dk Qn

(4.4)

n

Fn =:

k=1

and Fn−1 =:

n

k=1

are called the kernels of the Nörlund and T means, respectively. The Nörlund means are generalizations of the Fejér, Cesàro and Nörlund logarithmic means and the T means is a generalization of the Riesz logarithmic means. The Nörlund and T summations are general summability methods, which satisfy the conditions (A1)–(A3) in Definition 3.5.1. This means that all Nörlund and T means are approximation identities. According to all these facts it is of prior interest to study the behavior of operators related to Nörlund and T means of Fourier series with respect to orthonormal systems. Móricz and Siddiqi [227] investigated the approximation properties of some special Nörlund means of Walsh-Fourier series of Lp functions in norm. In particular, they proved that if f ∈ Lp (Gm ), 1 ≤ p < ∞, n = Mj + k, 1 ≤ k ≤ Mj (n ∈ N+ ) and (qk , k ∈ N) is a sequence of non-negative numbers, such that n−1 nα−1  α qk = O(1), for some 1 < α ≤ 2, Qαn k=0

4.1 Introduction

161

then



  n−1 Cp  1 1 tn f − f p ≤ Mi qn−Mi ωp , f + Cp ωp ,f , Qn Mi Mj i=0

when (qk , k ∈ N) is non-decreasing, while  



n−1  Cp   1 1 Qn−Mj +1 − Qn−Mj+1 +1 ωp tn f −f p ≤ , f +Cp ωp ,f , Qn Mi Mj i=0

when (qk , k ∈ N) is non-increasing. Let us define the maximal operator t ∗ of the Nörlund means by t ∗ f =: sup |tn f | . n∈N

If (qk , k ∈ N) is non-increasing and satisfying the conditions 1 =O Qn



1 nα

 , as n → ∞

(4.5)

and

qn − qn+1 = O

1 n2−α

 , as n → ∞,

(4.6)

then * + yμ t ∗ f > y ≤ c f 1 , f ∈ L1 (Gm ), y > 0.

(4.7)

Inequality (4.7) also holds for every maximal operator of the Nörlund means with non-increasing sequence (qk , k ∈ N), which satisfy the weaker conditions nα = O (1) , as n → ∞ Qn

(4.8)

Qn → 0, as n → ∞, for any ε > 0. nα+ε

(4.9)

and

When the sequence (qk , k ∈ N) is non-increasing, then weak-(1,1) type inequality (4.7) holds for every maximal operator of Nörlund means.

162

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

The boundedness of the maximal operator of the Nörlund means does not hold from the space L1 (Gm ) to the space L1 (Gm ). However, tn f − f p → 0,

n → ∞, (f ∈ Lp (Gm ), 1 ≤ p < ∞).

as

Moreover, if (qk , k ∈ N) is non-decreasing and satisfying the condition qn−1 =O Qn

 1 , as n → ∞, n

(4.10)

or (qk , k ∈ N) is non-increasing, then for any f ∈ L1 (Gm ), we have that lim tn f (x) = f (x)

n→∞

for all Vilenkin-Lebesgue points of f . Let us define the maximal operator T ∗ of T means by T ∗ f =: sup |Tn f | . n∈N

If (qk , k ∈ N) is non-decreasing and satisfying the condition 1 =O Qn

 1 , as n → ∞, n

(4.11)

or if (qk , k ∈ N) is non-increasing, then * + yμ T ∗ f > y ≤ c f 1 , f ∈ L1 (Gm ), y > 0. The boundedness of the maximal operator of Nörlund means does not hold from the space L1 (Gm ) to the space L1 (Gm ). However, Tn f − f p → 0,

as

n → ∞, (f ∈ Lp (Gm ), 1 ≤ p < ∞).

4.2 Well-Known and New Examples of Nörlund and T Means We define Bn and Bn−1 means as the class of Nörlund and T means, respectively, with monotone and bounded sequence (qk , k ∈ N), such that 0 < q∞ < ∞, where q∞ =: lim qn . n→∞

4.2 Well-Known and New Examples of Nörlund and T Means

163

If the sequence (qk , k ∈ N) is non-decreasing, we have that nq0 ≤ Qn ≤ nq∞ . In the case when the sequence (qk , k ∈ N) is non-increasing, then nq∞ ≤ Qn ≤ nq0 . In both cases, we can conclude that the conditions (4.10) and (4.11) are fulfilled. Well-known examples of Nörlund and T means with a monotone and bounded sequence (qk , k ∈ N) are the Fejér means σn , defined by 1 Sk f. n n

σn f =:

k=1

It is evident that also in this case (4.10) and (4.11) are fulfilled. The Cesàro means σnα (sometimes also denoted (C, α)) and its inverse σnα,−1 are defined by 1  α−1 An−k Sk f Aαn n

σnα f =:

k=1

and 1  α−1 Ak Sk f, Aαn n−1

σnα,−1 f =:

k=0

respectively, where Aα0 =: 0,

Aαn =:

(α + 1) . . . (α + n) , n!

α = −1, −2, . . .

It is well-known that Aαn =

n 

Aα−1 n−k

(4.12)

k=0

and Aαn − Aαn−1 = Aα−1 and Aαn ∼ nα . n

(4.13)

164

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Obviously, for qn = Aα−1 n , |qn − qn+1 | = O (1) , nα−2

 1 q0 =O , Qn nα

as

n → ∞,

(4.14)

as

n → ∞,

(4.15)

as

n → ∞.

(4.16)

and qn−1 =O Qn

 1 , n

Let Vnα denote the Nörlund mean, where

qk = (k + 1)α−1 ,

k ∈ N,

0 1,

(4.18)

Sn ψ1 → ψ1 , as n → ∞.

(4.19)

Sn ψ1 = and

On the other hand,    qn−1 ψ1   lim |tn ψ1 − ψ1 | = lim  n→∞ n→∞ Qn     qn−1    = A > 0, = lim  n→∞ Qn 

168

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

which means that the Nörlund summability method (4.1) generated by (qk , k ≥ 0) is not regular. The proof is complete.  Corollary 4.3.3 (a) Let the sequence (qk , k ∈ N) be non-increasing. Then the Nörlund summability method generated by (qk , k ∈ N) is regular. (b) Let the sequence (qk , k ∈ N) be non-decreasing. Then the Nörlund summability method generated by (qk , k ∈ N) is not always regular. Proof (a) Let the sequence (qk , k ∈ N) be non-increasing. Then qn−1 qn−1 1 ≤ = → 0, as n → ∞ Qn nqn−1 n and (4.17) is fulfilled. According to Theorem 4.3.2, we conclude that in this case the Nörlund summability method is regular so (a) is proved. (b) To prove part (b), we construct a Nörlund method with non-decreasing coefficients (qk , k ∈ N) which is not regular. Let (qk = 2k , k ∈ N). Then Qn =

n−1  2k = 2n − 1 ≤ 2n k=0

so that 2n−1 qn−1 2n−1 1 ≥ n =  0, as n → ∞ = n Qn 2 −1 2 2 and (4.17) does not hold. By again using Theorem 4.3.2, in this case we can conclude that the Nörlund summability method is not always regular, so also part (b) is proved.  Theorem 4.3.4 Let (qk , k ≥ 0) be a sequence of non-negative numbers with q0 > 0. Then the summability method (4.2) generated by (qk , k ≥ 0) is regular if and only if lim Qn = ∞.

n→∞

Proof Assume that (4.20) holds and let lim Sn f (x) = f (x) .

n→∞

(4.20)

4.4 Kernels of Nörlund Means

169

Then, for any ε > 0, there exists N0 , such that |Sn f (x) − f (x)| ≤ ε, when n ≥ N0 . Since (qk , k ≥ 0) is a sequence of non-negative numbers we obtain that ⎛ ⎞ N0 n−1  1 ⎝ ⎠ qk |(Sk f (x) − f (x))| |tn f (x) − f (x)| ≤ + Qn k=0

k=N0 +1

n QN0 ε  ≤c + qk ≤ ε, Qn−1 Qn

as n → ∞,

k=N0 +1

which means that the summability method (4.2) is regular. Assume now instead that (4.20) does not hold, i.e. lim Qn = A < ∞.

n→∞

By combining (4.18) and (4.19), we get that      q0 ψ1     = lim  q0  = q0 > 0, lim |tn ψ1 − ψ1 | = lim    n→∞ n→∞ Qn n→∞ Qn  A 

so the summability method (4.2) is not regular.

Corollary 4.3.5 All summability methods which were mentioned in the previous Sections, such as Fejér, Nörlund and Riesz logarithmic means, Cesàro and inverse Cesàro means, βnα f, βnα,−1 f, Unα f, Unα,−1 f, Vnα f, Vnα,−1 f , are all regular summability methods.

4.4 Kernels of Nörlund Means Now we study kernels of Nörlund means with respect to Vilenkin systems. If we invoke Abel transformations for aj = Aj − Aj −1 , j = 1, . . . , n, n  j =1 n  j =MN

aj bn−j = An b0 +

n−1  Aj (bj − bj +1 ),

(4.21)

j =1

aj bn−j = An b0 − AMN −1 bn−MN +

n−1  j =MN

Aj (bj − bj +1 ),

(4.22)

170

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

when bj = qj , aj = 1 and Aj = j for j = 0, 1, . . . , n, then (4.21) and (4.22) give the following identities: Qn =

n−1 

qj =

j =0 n−1 

n n−1     qn−j · 1 = qn−j − qn−j −1 j + q0 n, j =1

qn−j =

j =MN

(4.23)

j =1

n−1 

qn−j · 1

j =MN

=

n−1    qn−j − qn−j −1 j + q0 n − (MN − 1)qn−MN . j =MN

Moreover, if we use the Abel transformations (4.21) and (4.22) for bj = qn−j , aj = Dj and Aj = j Kj for any j = 0, 1, . . . , n − 1, we get the identities: ⎛ ⎞ n−1    1 ⎝ Fn = qn−j − qn−j −1 j Kj + q0 nKn ⎠ Qn

(4.24)

j =1

and n 1  qn−j Dj Qn

(4.25)

j =MN

⎛

=

1 ⎝ Qn

n−1  



⎞

qn−j − qn−j −1 j Kj + q0 nKn − qn−MN (MN − 1)KMN −1 ⎠ .

j =MN

Analogously, if we use the Abel transformations (4.21) and (4.22) for bj = qj , aj = Sj and Aj = j σj for any j = 0, 1, . . . , n − 1, we get that ⎞ ⎛ n−1  1 ⎝  tn f = qn−j − qn−j −1 j σj f + q0 nσn f ⎠ Qn

(4.26)

j =1

and n 1  qn−j Sj f Qn j =MN

⎞ ⎛ n−1  1 ⎝   = qn−j − qn−j −1 j σj f + q0 nKn − qn−MN (MN − 1)σMN −1 f ⎠ . Qn j =MN

4.4 Kernels of Nörlund Means

171

First we consider Nörlund kernels with respect to Vilenkin systems, which are generated by non-decreasing sequences. Lemma 4.4.1 Let (qk , k ∈ N) be a sequence of non-decreasing numbers, satisfying condition (4.10). Then there exists an absolute constant c, such that ⎫ ⎧ |n|  ⎬ c ⎨ |Fn | ≤ Mj KMj  . ⎭ n⎩ j =0

Proof By using (4.10), we get that ⎛ ⎞ ⎛ ⎞ n−1 n−1     1   1 ⎝ ⎝ qn−j − qn−j −1  + q0 ⎠ ≤ qn−j − qn−j −1 + q0 ⎠ Qn Qn j =1

j =1

≤

c qn−1 ≤ . Qn n

Hence, in view of (4.10) if we apply (3.12) in Corollary 3.2.7 and use the equality (4.24), we obtain that ⎛ ⎞⎞ |n| n−1     1   ⎝ |Fn | ≤ ⎝ qn−j − qn−j −1  + q0 ⎠⎠ Mi KMi  Qn ⎛

⎛

j =1

i=0

j =1

i=0

⎛ ⎞⎞ |n| n−1     1  ⎝ =⎝ qn−j − qn−j −1 + q0 ⎠⎠ Mi KMi  Qn ≤

|n| |n|   c   qn−1  Mi KMi  ≤ Mi KMi  . Qn n i=0

i=0



The proof is complete.

We also state an analogical estimate, but now without any restriction like (4.10). Lemma 4.4.2 Let n ≥ MN and (qk , k ∈ N) be a sequence of non-decreasing numbers. Then, there exists an absolute constant c, such that   ⎧ ⎫   |n| n ⎨ ⎬     1   c     q D M K . ≤ n−j j  j Mj Q ⎭  n  MN ⎩ j =MN

j =0

172

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Proof Let MN − 1 ≤ j ≤ n. In view of (3.12) in Corollary 3.2.7, we find that |j | |n| |n|    1       1 c  Kj  ≤ Ml KMl  ≤ Ml KMl  ≤ Ml KMl  . j MN − 1 MN l=0

l=0

l=0

Since the sequence (qk , k ∈ N) is non-decreasing, we get that qn−MN (MN − 1) =

M N −1 j =0

qn−MN ≤

M N −1

qn−MN +j ≤ Qn

j =0

and n−1    qn−j − qn−j −1  j + q0 n + qn−M (MN − 1) N j =MN

≤

n−1    qn−j − qn−j −1  j + q0 n + qn−M (MN − 1) N j =1

=

n−1    qn−j − qn−j −1 j + q0 n + qn−MN (MN − 1) j =1

= Qn + qn−MN (MN − 1) ≤ 2Qn . By using (4.25), we can conclude that     n  1     q D n−j j  Q  n j =MN   ⎛ ⎞   n−1    1   ⎝ qn−j − qn−j −1 j Kj + q0 nKn + qn−MN (MN − 1)KMN −1 ⎠ =    Qn j =MN ⎞⎞ ⎛ ⎛ |n| n−1       1 qn−j − qn−j −1  j + q0 n⎠⎠ c ⎝ Mi KMi  ≤⎝ Qn MN j =MN

≤

i=0

|n|   c  Mi KMi  . MN i=0

The proof is complete.



4.4 Kernels of Nörlund Means

173

Next, we prove some important estimates which are analogous to those of the Fejér means and which will be used to study Nörlund means generated by nondecreasing sequences (qk , k ∈ N) and their maximal operators on the martingale Hardy spaces in the next Chapters. Lemma 4.4.3 Let x ∈ INk,l , k = 0, . . . , N − 2, l = k + 1, . . . , N − 1 and (qk , k ∈ N) be a sequence of non-decreasing numbers satisfying the condition (4.10). Then  |Fn (x − t)| dμ (t) ≤ IN

cMl Mk . nMN

If x ∈ INk,N , k = 0, . . . , N − 1, then  |Fn (x − t)| dμ (t) ≤ IN

cMk . MN

Here c is an absolute constant. Proof Let x ∈ INk,l for 0 ≤ k < l ≤ N −1 and t ∈ IN . First, we observe that x −t ∈ INk,l . Next, if we apply Lemma 4.4.1 and invoke (3.2) and (3.3) in Lemma 3.2.3, then we obtain that 

|n|

IN

c |Fn (x − t)| dμ (t) ≤ Mi n

 IN

i=0

c ≤ n



  KM (x − t) dμ (t) i

(4.27)

l  cMk Ml Mi Mk dμ (t) ≤ nMN

IN i=0

and the first estimate is proved. Now, let x ∈ INk,N . Since x − t ∈ INk,l for t ∈ IN , by applying Lemmas 3.2.1 and 4.4.1 combined with (3.2) and (3.3) in Lemma 3.2.3 we can conclude that 

|n|

|Fn (x − t)| dμ (t) ≤ IN

c Mi n



i=0

≤

c n

|n|−1  i=0

IN

  KM (x − t) dμ (t) i

 Mk dμ (t) ≤

Mi IN

(4.28)

cMk . MN

Hence, the proof of the second estimate is complete by just combining (4.27) and (4.28). 

174

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

By applying Lemma 4.4.3, we obviously get the following: Lemma 4.4.4 Let n ≥ MN , x ∈ INk,l , k = 0, . . . , N − 1, l = k + 1, . . . , N and (qk , k ∈ N) be a non-decreasing sequence satisfying condition (4.10). Then, there exists an absolute constant c, such that  cMl Mk |Fn (x − t)| dμ (t) ≤ . MN2 IN Next, we state an analogical estimate, but now without any restriction like (4.10). Lemma 4.4.5 Let x ∈ INk,l , k = 0, . . . , N − 1, l = k + 1, . . . , N and (qk , k ∈ N) be a non-decreasing sequence. Then  IN

    n  1     dμ (t) ≤ cMl Mk , q D − t) (x n−j j Q  MN2  n j =MN 

where c is an absolute constant. Proof Let x ∈ INk,l for 0 ≤ k < l ≤ N − 1 and t ∈ IN . It is evident that x − t ∈ INk,l . Since n ≥ MN , if we combine Lemmas 3.2.1 and 4.4.2 and invoke (3.2) and (3.3) in Lemma 3.2.3, we readily obtain that  IN

    n  1    qn−j Dj (x − t) dμ (t) Q  n j =MN 

c  Mi MN



c  Mi MN



|n|

≤

IN

i=0 l

≤

i=0

(4.29)

  KM (x − t) dμ (t) i

Mk dμ (t) ≤ IN

cMk Ml . MN2

Now, let x ∈ INk,N . Since x−t ∈ INk,N according to Lemma 1.6.9 we can conclude that      n  1    qn−j Dj (x − t) dμ (t) (4.30) Q IN  n  j =MN

c ≤ Qn

n  j =MN

 qn−j

Mk dμ (t) IN

4.4 Kernels of Nörlund Means

= =

175

cQn−MN Qn

 Mk dμ (t) IN

cQn−MN Mk cMk ≤ . Qn MN MN 

Hence, the proof is complete by just combining estimates (4.29) and (4.30).

Now we prove a lemma which is very important for our further investigation to prove norm convergence in Lebesgue spaces of Nörlund means generated by nondecreasing sequences (qk , k ∈ N). Lemma 4.4.6 Let (qk , k ∈ N) be a sequence of non-decreasing numbers. Then, for any n, N ∈ N+ ,  Fn (x)dμ(x) = 1,

(4.31)

Gm

 |Fn (x)| dμ(x) ≤ c < ∞,

sup

(4.32)

n∈N Gm



Gm \IN

|Fn (x)| dμ(x) → 0, as n → ∞,

(4.33)

where c is an absolute constant. Proof According to Corollary 1.6.8, we readily obtain (4.31). By using (3.17) in Corollary 3.2.9 combined with (4.23) and (4.24), we get that  |Fn (x)| dμ(x) Gm

 1  qn−j − qn−j −1 j Qn n−1

≤

j =1



  Kj (x) dμ(x) + q0 n Qn Gm

 |Kn (x)|dμ(x) Gm

 cq0 n c  qn−j − qn−j −1 j + < c < ∞, Qn Qn n−1

≤

j =1

so also (4.32) is proved. According to (3.18) in Corollary 3.2.9 and also (4.23) and (4.24), we find that  Gm \IN

 1  qn−j − qn−j −1 j Qn n−1

≤

|Fn (x)| dμ(x)

j =0

(4.34)  Gm \IN

  Kj (x) dμ(x)+ q0 n Qn

 Gm \IN

|Kn (x)|dμ(x)

176

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

 1  q0 nαn qn−j − qn−j −1 j αj + Qn Qn n−1

≤

j =0

=: I + I I. where αn → 0, as n → ∞. Since the sequence is non-decreasing, we can conclude that II =

q0 nαn ≤ αn → 0, as n → ∞. Qn

(4.35)

On the other hand, for any ε > 0 there exists N0 ∈ N, such that αn < ε, when n > N0 . Moreover,  1  qn−j − qn−j −1 j αj Qn n−1

I =

(4.36)

j =1

N0 n−1    1  1   qn−j − qn−j −1 j αj + qn−j − qn−j −1 j αj = Qn Qn j =1

j =N0 +1

=: I1 + I2 . Since the sequence is non-decreasing, we can conclude that |qn−j − qn−j −1 | < 2qn−1 so that 0   1  2qn−1 N0 qn−j − qn−j −1 j αj ≤ → 0, as n → ∞. (4.37) Qn Qn

N

I1 =

j =0

Furthermore, for any ε, I2 ≤

n−1  ε   qn−j − qn−j −1 j Qn j =N0 +1

 ε  qn−j − qn−j −1 j < ε, Qn n−1

≤

j =0

(4.38)

4.4 Kernels of Nörlund Means

177

By combining (4.34)–(4.38) we have proved that (4.33) holds so the proof is complete.  Next we prove a lemma which is very important in proving almost everywhere convergence of Nörlund means generated by non-decreasing sequences (qk , k ∈ N). Lemma 4.4.7 Let n ∈ N and (qk , k ∈ N) be a sequence of non-decreasing numbers. Then      n  1   sup  qn−j Dj (x) dμ(x) ≤ c < ∞, Qn  Gm \IN n>MN  j =MN

where c is an absolute constant. Proof Let n > MN and x ∈ INk,l , k = 0, . . . , N − 2, l = k + 1, . . . , N − 1. By using Lemma 4.4.2, (3.2) and (3.3), we get that     n l  1     1    ≤ q D Mi KMi (x) (x) n−j j Q  M N  n j =MN  i=0 l cMl Mk 1  Mi Mk ≤ ≤ MN MN i=0

and     |n| n  1     cMl Mk 1  sup  qn−j Dj (x) ≤ Mi KMi (x) ≤ . MN MN n>MN  Qn j =M  i=0 N

(4.39)

Let n > MN and x ∈ INk,N . By using Lemma 1.6.9 we can conclude that     n n   1   Qn−MN  ≤ c q D qn−j Mk ≤ Mk ≤ cMk , (x) n−j j Q  Q Qn n  n j =MN  j =MN so that     n  1   sup  qn−j Dj (x) ≤ cMk . n>MN  Qn j =M  N

(4.40)

Hence, we can conclude that estimates (4.39) and (4.40) are the same as the estimates (3.23) and (3.24) in Lemma 3.2.10. Therefore, if we follow the steps in

178

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

the proof of (3.25), we get that  Gm \IN

    n  1    sup  qn−j Dj (x) dμ ≤ c < ∞, Q n>MN  n j =M  N 

which finishes the proof.

We also need the following important identity for Nörlund means without any restriction on the sequence (qk , k ∈ N). Lemma 4.4.8 Let sn Mn < r ≤ (sn + 1) Mn , where 1 ≤ sn ≤ mn − 1. Then Qr Fr = Qr Dsn Mn − ψsn Mn −1

sn  Mn −2

  qr−sn Mn +l − qr−sn Mn +l+1 lK l

l=1

− ψsn Mn −1 (sn Mn − 1) qr−1 K sn Mn −1 + ψsn Mn Qr−sn Mn Fr−sn Mn . Proof Let sn Mn < r ≤ (sn + 1) Mn , where 1 ≤ sn ≤ mn − 1. It is easy to see that Qr Fr =

r 

qr−k Dk =

s n Mn

k=1

qr−l Dl +

l=1

r 

qr−l Dl =: I + I I.

(4.41)

l=sn Mn +1

We apply (1.41) in Lemma 1.6.4 and invoke Abel transformation to obtain that I =

sn  Mn −1

(4.42)

qr−sn Mn +l Dsn Mn −l

l=0

=

sn  Mn −1

qr−sn Mn +l Dsn Mn −l + qr−sn Mn Dsn Mn

l=1

= Dsn Mn

sn  Mn −1

qr−sn Mn +l − ψsn Mn −1

sn  Mn −1

l=0

  = Qr − Qr−sn Mn Dsn Mn − ψsn Mn −1

sn  Mn −2

qr−sn Mn +l D l

l=1

  qr−sn Mn +l − qr−sn Mn +l+1 lK l

l=1

− ψsn Mn −1 qr−1 (sn Mn − 1) K sn Mn −1 .

4.4 Kernels of Nörlund Means

179

According to the identity (1.40) in Lemma 1.6.4 we can rewrite I I as follows: II =

r−s n Mn 

(4.43)

qr−sn Mn −l Dl+sn Mn

l=1

= Qr−sn Mn Dsn Mn + ψsn Mn Qr−sn Mn Fr−sn Mn . The proof is complete by just combining identities (4.41), (4.42), and (4.43).



We also consider the kernels of Nörlund means with respect to the Vilenkin systems which are generated by non-increasing sequences (qk , k ∈ N), but now with some new restrictions on the indexes. Lemma 4.4.9 Let (qk , k ∈ N) be a sequence of non-increasing numbers satisfying the condition (4.11). Then ⎫ ⎧ |n|  ⎬ c ⎨ |Fn | ≤ Mj KMj  , ⎭ n⎩ j =0

where c is an absolute constant. Proof We have ⎛ ⎞ ⎛ ⎞ n−1 n−1     1 1   ⎝ ⎝ qn−j − qn−j −1  + q0 ⎠ ≤ − qn−j − qn−j −1 + q0 ⎠ Qn Qn j =1

j =1

≤

2q0 − qn−1 2q0 c ≤ ≤ . Qn Qn n

Hence, if we apply (3.12) in Corollary 3.2.7 and invoke equalities (4.23) and (4.24), then we get that ⎛

⎛ ⎞⎞ |n| n−1       1 qn−j − qn−j −1  + q0 ⎠⎠ ⎝ |Fn | ≤ ⎝ Mi KMi  Qn ⎛

j =1

i=0

⎛ ⎞⎞ |n| n−1       1 ⎝ =⎝ − qn−j − qn−j −1 + q0 ⎠⎠ Mi KMi  Qn j =1

i=0

180

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

≤

|n| |n|   2q0    2q0 − qn−1  Mi KMi  ≤ Mi KMi  Qn Qn i=0

i=0

|n|

≤

  c Mi KMi  . n i=0



The proof is complete.

We will use the next result in this Chapter for proving norm convergence in Lebesgue spaces of Nörlund means generated by a non-increasing sequence (qk , k ∈ N). Corollary 4.4.10 Let (qk , k ∈ N) be a sequence of non-increasing numbers satisfying the condition (4.11). Then, for any n, N ∈ N+ ,  Fn (x)dμ(x) = 1,

(4.44)

Gm

 |Fn (x)| dμ(x) ≤ C < ∞,

sup

(4.45)

n∈N Gm



Gm \IN

|Fn (x)| dμ(x) → 0, as n → ∞,

(4.46)

where C is an absolute constant. Proof If we compare the estimation of Kn in Lemma 3.2.7 or Fn in Lemma 4.4.1 with the estimation of Fn in Lemma 4.4.9, we find that they are quite the same. Hence, the proof is analogous to those of Corollary 3.2.9 and Lemma 4.4.6, so we leave out the details.  For the almost everywhere convergence of Nörlund means generated by nonincreasing sequences (qk , k ∈ N), we will use the nest corollary. Corollary 4.4.11 Let (qk , k ∈ N) be a sequence of non-increasing numbers satisfying the condition (4.11). Then  sup |Fn | dμ ≤ c < ∞,

Gm \IN n>MN

where c is an absolute constant. Proof The proof is analogous to that of Lemma 3.2.10 so we omit the details.



We will also investigate another class of Nörlund means generated by nonincreasing sequences (qk , k ∈ N) with weaker restrictions on the indexes.

4.4 Kernels of Nörlund Means

181

Lemma 4.4.12 Let (qk , k ∈ N) be a sequence of non-increasing numbers, satisfying conditions (4.5) and (4.6) for 0 < α < 1. Then ⎧ ⎫ |n| ⎬ cα ⎨ α  |Fn | ≤ α Mj KMj  , ⎭ n ⎩

(4.47)

j =0

where cα is a constant depending only on α. Proof According to the fact that (qk , k ∈ N) is a sequence of non-negative and non-increasing numbers we have two cases: 1. limk→∞ qk ≥ c > 0, 2. limk→∞ qk = 0, In the first case we obtain that (4.10) and (4.11) are satisfied. Since the case n = O (1) , as n → ∞, Qn has already been considered in Lemma 4.4.1, we can exclude it. Hence, we may assume that qn = o(1), as n → ∞.

(4.48)

By combining (4.6) and (4.48), we get that qn =

∞ 

(ql − ql+1 )

l=n

≤

∞  cα cα ≤ 1−α 2−α l n l=n

and Qn =

n−1 n   cα ql ≤ ≤ cα nα . 1−α l l=0

l=1

Let Mn < k ≤ Mn+1 . It is easy to see that     Qk DsMn  ≤ cMnα DsMn 

(4.49)

and     (sMn − 1) qk−1 KsMn −1  ≤ cα k α−1 Mn KsMn −1    ≤ cα Mnα KsMn −1  .

(4.50)

182

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Let n = sn1 Mn1 + sn2 Mn2 + · · · + snr Mnr , n1 > n2 > · · · > nr and n(k) = snk+1 Mnk+1 + · · · + snr Mnr , 1 ≤ snl ≤ ml − 1, l = 1, . . . , r, where snk ∈ Zmk , k = 1, . . . , r. By combining (4.49), (4.50) and Lemma 4.4.8, we have that sn1 Mn1 −1   

α−2       n(1) + l  |lKl | |Qn Fn | ≤ cα Mnα1 Dsn1 Mn1  + cα   l=1

      + cα Mnα1 Ksn1 Mn1 −1  + cα Qn(1) Fn(1)  . By repeating this process r times we find that |Qn Fn | ≤ cα

r      Mnαk Dsnk Mnk 

(4.51)

k=1

+ cα

s M −1

nk r nk  

k=1

n(k) + l

α−2

|lKl |

l=1

r      + cα Mnαk Ksnk Mnk −1  k=1

=: I + I I + I I I. According to the fact that r ≤ |n|, if we combine Corollary 1.6.7 and Lemma 3.2.1 and invoke (1.42) in Lemma 1.6.5 we obtain that I ≤ cα

|n|    Mkα Dsk Mk  k=1

≤ cα

|n|    Mkα DMk  k=1

|n|    ≤ cα Mkα KMk  k=1

(4.52)

4.4 Kernels of Nörlund Means

183

and I I I ≤ cα

r      Mnα−1 K − M D M  nk snk Mnk nk snk Mnk  k

(4.53)

k=1

≤ cα

r r           Mnαk Ksnk Mnk  + cα Mnαk Dsnk Mnk  k=1

≤ cα

k=1

|n| 

  Mkα KMk  .

k=0

Next, we can rewrite I I as I I = cα

+ cα

r 

snk+1 Mnk+1 −1

k=1

l=1

r 

snk Mnk −1



α−2 |lKl | n(k) + l

(4.54)



α−2 |lKl | n(k) + l



k=1 l=snk+1 Mnk+1

= : I I1 + I I2 . For I I1 we find that I I1 ≤ cα

r 

snk+1 Mnk+1 −1



snα−2 Mnα−2 k+1 k+1

k=1

≤ cα

|n| 

Mkα−2

k=1

= cα

|n| 

|n| 

Mkα−2

|n|  k=0

|lKl |

k M i −1  

|lKl |

i=1 l=Mi−1

Mkα−2

k=1

≤ cα

M k −1  l=1

k=1

≤ cα

|lKl |

l=1

k  i=1

Mkα−1

k  j =0

Mi

i 

  Mj KMj 

j =0

Mj |KMj |

(4.55)

184

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

= cα

|n| 

Mj |KMj |

j =0

≤ cα

|n| 

|n| 

Mkα−1

k=j

  Mjα KMj  .

j =0

Moreover, I I2 ≤ cα

r 

snk Mnk −1



l α−2 |lKl |

(4.56)

k=1 l=snk+1 Mnk+1

≤ cα

Mi+1 −1

nk r  

Miα−2

k=1 i=nk+1 +1

≤ cα

nk r  

Miα−2 Mi

nk r  

|n|  i=1

≤ cα

|n| 

Miα−1

i    Mj KMj  j =0

Miα−1

k=1 i=nk+1 +1

≤ cα

|lKl |

l=Mi

k=1 i=nk+1 +1

= cα



i 

  Mj KMj 

j =0

i    Mj KMj  j =0

  Mjα KMj  .

j =0

Since by assumption (4.5), |Qn | ≥ cα nα , for some cα > 0 the proof of (4.47) follows by just combining (4.51)–(4.56).  Our next lemma is very important for our further investigation in this Chapter to prove norm convergence in Lebesgue spaces of Nörlund means generated by nonincreasing sequences (qk , k ∈ N), which satisfy conditions (4.5) and (4.6). Corollary 4.4.13 Let 0 < α ≤ 1 and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying the conditions (4.5) and (4.6). Then for any n, N ∈ N+ ,  Fn (x)dμ(x) = 1,

(4.57)

Gm



sup n∈N Gm

|Fn (x)| dμ(x) ≤ Cα < ∞,

(4.58)

4.4 Kernels of Nörlund Means

185

 Gm \IN

|Fn (x)| dμ(x) → 0, as n → ∞,

(4.59)

where Cα is a constant depending only on α. Proof According to Corollary 1.6.8, we can see that (4.57) holds. By combining (3.5) in Lemmas 3.2.3 and 3.2.7, we can conclude that  |Fn (x)| dμ (x) ≤ Gm

 |n|   1  α KM (x) dμ (x) M l l α n Gm l=0

≤

|n| 1  α Ml ≤ Cα < ∞, nα l=0

and also (4.58) holds. Now x ∈ INk,l , k = 0, . . . , N − 2, l = k + 1, . . . , N − 1. By using (3.3) in Lemmas 3.2.3 and 3.2.7, we get that |Fn (x)| ≤

l l  cMlα Mk c  α  c  α  ≤ K M M M ≤ . (x) M k s s s nα nα nα s=0

(4.60)

s=0

k,q

Let x ∈ Iq+1 ⊂ INk,N , for some N ≤ q ≤ n − 1, that is   x = 0, . . . , 0, xk = 0, 0 . . . , xN−1 = 0, . . . , xq−1 = 0, xq = 0, . . . , x|n|−1 , . . . . Then q−1 cMk Mqα c  α M M ≤ . k i nα nα

|Fn (x)| ≤

(4.61)

i=0

k,|n|

Let x ∈ INk,N , where x ∈ I|n| , that is   x = 0, . . . , 0, xk = 0, xk+1 = 0, . . . , xN = 0, . . . , x|n|−1 = 0, . . . . Then |Fn (x)| ≤

|n|−1 α cMk M|n| c  α M M ≤ . k i nα nα i=0

(4.62)

186

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

If we combine (4.61) and (4.62), we can conclude that  |Fn (x)| dμ(x)

(4.63)

INk,N

=

k,q

|n|−1 

k,|n|

|Fn (x)| dμ(x)

I|n|

cMk Mqα−1 nα

q=N

≤

|Fn (x)| dμ(x) +

Iq+1

q=N

≤



|n|−1 

+

α−1 cMk M|n|

nα

c(|n| − N)Mk MNα−1 . α M|n|

Hence, if we apply (4.60) and (4.63), we find that  Gm \IN

=

|Fn (x)| dμ(x)

N−2  N−1 



mj−1



k,l k=0 l=k+1 xj =0,j ∈{l+1,...,N−1} IN

≤ c

N−2  N−1  k=0 l=k+1

|Fn (x)| dμ(x) +

N−1  k=0

INk,N

|Fn (x)| dμ(x)

 MNα−1 ml+1 . . . mN−1 Mlα Mk + c (|n| − N)M k α MN nα M|n| N−1 k=0

=: I + I I. Moreover, I =

N−2  N−1  k=0 l=k+1

≤c

N−2  k=0

≤c

N−2  k=0

≤

Mlα−1 Mk α M|n|

(N − k)Mkα α M|n| |n| − k 2(|n|−k)/p

c(|n| − N) → 0, as n → ∞, 2(|n|−N)/p

4.4 Kernels of Nörlund Means

187

and II ≤

c(|n| − N) → 0, as n → ∞. 2(|n|−N)/p

By combining the last three estimates, we see that also (4.59) holds so the proof is complete.  The following Lemma will be needed for the almost everywhere convergence of Nörlund means generated by non-increasing sequences (qk , k ∈ N). Lemma 4.4.14 Let n ∈ N and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying the conditions (4.5) and (4.6). Then      n  1    sup  qn−j Dj (x) dμ(x) ≤ C < ∞, (4.64) Gm \IN n>MN  Qn j =M  N where c is an absolute constant. Proof Let n > MN and x ∈ INk,l , k = 0, . . . , N − 2, l = k + 1, . . . , N − 1. By combining Lemmas 3.2.1 and 4.4.12, we get that     n  1   cMlα Mk  ≤ q D , (x) n−j j  Qn  MNα   j =MN so that     n  1   cMlα Mk sup  qn−j Dj (x) ≤ . MNα n>MN  Qn j =M  N

(4.65)

Let n > MN and x ∈ INk,N . By using Lemma 1.6.9, we can conclude that     n n   1    ≤ c q D qn−j Mk ≤ cMk (x) n−j j Q  Q n j =M  n j =MN  N so that     n  1   sup  qn−j Dj (x) ≤ cMk . n>MN  Qn j =M  N

(4.66)

188

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

By combining (1.5), (4.65) and (4.66), we obtain that  Gm \IN

=

+

    n   1   sup  qn−j Dj (x) dμ(x) Q n>MN  n j =M  N 

mj−1

N−2  N−1 



sup |Fn (x)| dμ(x)

k,l k=0 l=k+1 xj =0,j ∈{l+1,...,N−1} IN n>MN

N−1  k=0

≤c

sup |Fn (x)| dμ(x)

INk,N n>MN

N−2  N−1  k=0 l=k+1

≤c

N−2  N−1  k=0 l=k+1

≤c

N−1  l=k+1

 Mk ml+1 . . . mN−1 Mlα Mk +c α MN MN MN N−1 k=0

 Mk Mlα−1 Mk +c α MN MN N−1 k=0

Mkα + c ≤ C < ∞, MNα

so (4.64) holds and the proof is complete.



We prove the following estimations, which will be useful to investigate Nörlund means and their maximal operators in the martingale Hardy spaces in the next Chapters: Lemma 4.4.15 Let 0 < α ≤ 1 and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying the conditions (4.5) and (4.6). Then  |Fm (x − t)| dμ (t) ≤ IN

cα Mlα Mk , for x ∈ INk,l , mα MN

where k = 0, . . . , N − 2, l = k + 2, . . . , N − 1. Moreover,  cα Mk |Fm (x − t)| dμ (t) ≤ , for x ∈ INk,N , k = 0, . . . , N − 1. M N IN

(4.67)

(4.68)

Here Cα is a constant depending only on α. Proof Let x ∈ INk,l , where k = 0, . . . , N − 2, l = k + 2, . . . , N − 1. Since x − t ∈ INk,l , for t ∈ IN if we apply Lemma 4.4.12 combined with (3.2) and (3.3)

4.4 Kernels of Nörlund Means

189

in Lemma 3.2.3, we can conclude that |Fm (x − t)| ≤

l  cα  α  Mi KMi (x − t) α m

(4.69)

i=0

cα  α Mi Mk mα l

≤

i=0

≤

cα Mlα Mk . mα

Now let x ∈ INk,l for some 0 ≤ k < l ≤ N − 1. Since x − t ∈ INk,l for t ∈ IN and m ≥ MN , from (4.69) we readily obtain that 

Cα Mlα Mk , mα MN

|Fm (x − t)| dμ (t) ≤ IN

so (4.67) holds. Let x ∈ INk,N , k = 0, . . . , N − 1. Then, by applying Lemma 4.4.12, we have that 

|m|

|Fm (x − t)| dμ (t) ≤ IN

Cα  α Mi mα i=0

 IN

  KM (x − t) dμ (t) . i

(4.70)

Let x ∈ INk,N , k = 0, . . . , N − 1, t ∈ IN and xq = tq , where N ≤ q ≤ |m| − 1. By using Lemma 3.2.1 and estimate (4.70), we get that 

Cα  α |Fm (x − t)| dμ (t) ≤ α Mi m q−1

IN

i=0

≤

Cα Mk Mqα mα MN

 Mk dμ (t)

(4.71)

IN

≤

Cα Mk . MN

Let x ∈ INk,N , k = 0, . . . , N − 1, t ∈ IN and xN = tN , . . . , x|m|−1 = t|m|−1 . By again applying Lemma 3.2.1 and (4.70), we have that  |Fm (x − t)| dμ (t) ≤ IN

 |m|−1 Cα  α Cα Mk M Mk dμ (t) ≤ . i α m MN IN

(4.72)

i=0

The proof of (4.68) follows from (4.71) and (4.72), so the proof is complete.



By applying Lemma 4.4.15 for n ≥ MN , we obtain the following useful result.

190

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Corollary 4.4.16 Let 0 < α < 1, m ≥ MN and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying the conditions (4.5) and (4.6). Then there exists an absolute constant Cα depending only on α, such that  |Fm (x − t)| dμ (t) ≤

Cα Mlα Mk MN1+α

IN

for x ∈ INk,l ,

where k = 0, . . . , N − 1, l = k + 2, . . . , N. Remark 4.4.17 For some sequences (qk , k ∈ N) of non-increasing numbers, conditions (4.5) and (4.6) can be true or false independently. In some proofs in the next Chapter, we will use weaker conditions than (4.5) and (4.6), which gives a chance to prove such results for a wider class of Nörlund means with non-increasing sequences (qn , n ∈ N): Lemma 4.4.18 Let 0 < α ≤ 1 and (qn , n ∈ N) be a sequence of non-increasing numbers satisfying the conditions (4.8) and (4.9). Then |Qn Fn | ≤ Cα

⎧ |n| ⎨ ⎩

Mjα ϕMj

j =0

⎫  ⎬ KM  , j ⎭

(4.73)

where lim

ϕj

j →∞ j ε

= 0, for every ε > 0.

(4.74)

Proof Let 0 < α ≤ 1 and (qk , k ≥ 0) satisfy the conditions (4.8) and (4.9). Since nα ϕn ≥ Qn ≥ nqn−1 we obtain that qn−1 ≤ nα−1 ϕn ,

(4.75)

where ϕn satisfies condition (4.74). By using Abel transformation we get that Qn =

n−2    qj − qj +1 j + qn−1 (n − 1) + q0 j =1

and n−2    qj − qj +1  j ≤ Qn ≤ nα ϕn . j =1

(4.76)

4.4 Kernels of Nörlund Means

191

Next we aim to prove   qj − qj +1  ≤ cα j α−2 ϕj ,

(4.77)

but assume on the contrary that   qj − qj +1  ≥ j α−2 ϕj δj for all j ∈ N, where δj is any function, such that lim δj = ∞.

(4.78)

j →∞

Under condition (4.78) there exists an increasing sequence (αk , k ≥ 0) , such that αk+1 ≥ 2αk and δαk ↑ ∞. Hence, αk+1 +1

αk+1 +1     qj − qj +1  j ≥ cα ϕα δα j α−1 k k

j =αk

(4.79)

j =αk αk+1  +2

≥ cα ϕαk δαk

x α−1 dx αk

≥

cα ϕαk δαk α αk+1 x |αk ≥ cα ϕαk δαk αkα . α

By combining (4.75) and (4.79) we get that Qαk+1 +3   (αk+1 + 3)α ϕαk+1 + 3 αk+1 +1 

≥

j =1

 qj − qj +1  j

  ≥ cα δαk → ∞, as k → ∞. (αk+1 + 3)α ϕαk+1 + 3

This is a contradiction to the condition (4.76), that is Qn = O (1) , as n → ∞. nα ϕn

(4.80)

It follows that (4.77) holds. Moreover, it is easy to see that     Qk DsMn  ≤ cα Mnα ϕMn DsMn 

(4.81)

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4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

and     (sMn − 1) qk−1 KsMn −1  ≤ cα ϕk k α−1 Mn KsMn −1    ≤ cα Mnα ϕMn KsMn −1  .

(4.82)

Let n = sn1 Mn1 + sn2 Mn2 + · · · + snr Mnr , n1 > n2 > · · · > nr , n(k) = snk+1 Mnk+1 + · · · + snr Mnr , 1 ≤ snl ≤ ml − 1, l = 1, . . . , r. By combining (4.77)–(4.82) and Lemma 4.4.8, we have that     |Qn Fn | ≤ cα Mnα1 ϕMn1 Dsn1 Mn1  + cα

sn1 Mn1 −1 

α−2   (1)  n +l  ϕ (1) |lKl |   n +l

 l=1

    + cα Mnα1 ϕMn1 Ksn1 Mn1 −1    + cα Qn(1) Fn(1)  . By repeating this process r-times we get that |Qn Fn | ≤ cα

r 

    Mnαk ϕMnk Dsnk Mnk 

k=1

+ cα

r  k=1

+ cα

r 

snk Mnk −1



n(k) + l

α−2

ϕn(k) +l |lKl |

l=1

    Mnαk ϕMnk Ksnk Mnk −1 

k=1

=: I + I I + I I I. By using (1.42) in Lemma 1.6.5, Corollary 1.6.7, Lemmas 3.2.1 and 3.2.4, we find that I ≤ Cα

|n|  k=1

  Mkα ϕMk DMk 

4.4 Kernels of Nörlund Means

193

≤ Cα

|n| 

  Mkα ϕMk KMk 

k=1

and I I I ≤ Cα

r 

    Mnα−1 ϕ K − D M  M n s M s M n n n n n k k k k k k k

k=1

≤ Cα

r 

    Mnαk ϕMnk KMnk 

k=1

≤ Cα

|n| 

  Mkα ϕMk KMk  .

k=1

Moreover, I I = cα

nk r  



α−2 n(k) + l ϕn(k) +l |lKl |

MA −1 sA 

k=1 A=1l=sA−1 MA−1

= cα

r n k+1 

α−2 ϕn(k) +l |lKl | n(k) + l

MA −1 sA 

k=1 A=1l=sA−1 MA−1

+ cα

r 

nk 

MA −1 sA 

k=1 A=nk+1 +1l=sA−1 MA−1

≤ cα

r 

Mnα−2 ϕ k+1 Mnk+1

k=1

+ cα

r 



α−2 n(k) + l ϕn(k) +l |lKl |

nk+1 sA MA −1  

|lKl |

A=1l=sA−1 MA−1 nk 

sA  MA −1

MAα−2 ϕMA

k=1 A=nk+1 +1

|lKl |

l=sA−1 MA−1

= I I1 + I I2 . By applying again Lemma 1.6.5 for I I1 , we get that I I1 ≤ cα

r 

Mnα−2 ϕ k+1 Mnk+1

k=1

≤ cα

n1  k=1

Mkα−2 ϕMk

nk+1 sA MA −1  

A 

  Mj KMj 

A=1l=sA−1 MA−1 j =0

k  A=1

MA

A  j =0

  Mj KMj 

194

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

≤ cα

|n| 

Mkα−1 ϕMk

j =0

k=0

= cα

|n| 

Mj |KMj |

j =0

≤ cα

|n| 

k  Mj |KMj |

|n|  ϕMk Mkα−1 k=j

  ϕMj Mjα KMj  .

j =0

By using Lemma 1.6.5 for I I2 , we have similarly that I I2 ≤ cα

r 

nk 

MAα−1 ϕMA

A 

k=1 A=nk+1 +1

≤ cα

|n| 

MAα−1 ϕMA

A=1

≤ cα

  Mj KMj 

j =0 A 

  Mj KMj 

j =0

|n|    ϕMj Mjα KMj  . j =0

The proof is complete by just combining the estimates above in the same way as in the proof of Lemma 4.4.12.  Lemma 4.4.19 Let 0 < α ≤ 1 and (qn , n ∈ N) be a sequence of non-increasing numbers satisfying the conditions (4.8) and (4.9). If r ≥ MN , then  |Fr (x − t)| dμ (t) ≤ IN

cα Mlα ϕMl Mk cα Mlα ϕMl Mk ≤ , x ∈ INk,l , (4.83) r α MN MN1+α

where k = 0, . . . , N − 2, l = k + 1, . . . , N − 1 and  cα Mk |Fr (x − t)| dμ (t) ≤ , x ∈ INk,l , MN IN where k = 0, . . . , N − 1. Proof Let x ∈ INk,l . Then, by applying (3.2) in Lemma 3.2.3, we have that KMn (x) = 0, when n > l > k. Suppose that k < n ≤ l. Using Lemma 3.2.1, we get that   KM (x) ≤ cMk . n

(4.84)

4.5 Kernels of T Means

195

Let n ≤ k < l. Then, by applying again Lemma 3.2.1, we can conclude that   KM (x) = (Mn + 1) /2 ≤ cMk . n If we now apply Lemma 4.4.18 and (4.84), we obtain that Qr |Fr (x)| ≤ cα

l 

  MAα ϕMA KMA (x)

(4.85)

A=0

≤ cα

l 

MAα ϕMA Mk

A=0

≤ cα Mlα ϕMl Mk . Let x ∈ INk,l , for some 0 ≤ k < l ≤ N − 1. Since x − t ∈ INk,l for t ∈ IN and r ≥ MN , it follows from (4.85) that  |Fr (x − t)| dμ (t) ≤ IN

Cα Mlα ϕMl Mk . r α MN

so the first inequality in (4.83) is proved and the second one follows by using the condition r ≥ MN . Let x ∈ INk,l , k = 0, . . . , N − 1. Then, by using Lemma 1.6.9 we have that  IN

1  |Fr (x − t)| dμ (t) ≤ qn−m Qn n

m=1

 |Dm (x − t)| dμ (t) IN

cMk 1  qn−m MN Qn n

≤

m=1

≤

cMk , MN

so also (4.4.19) is proved and the proof is complete.



4.5 Kernels of T Means In this Section we investigate kernels of T means with respect to Vilenkin systems. By applying the Abel transformations for aj = Aj − Aj −1 , j = 1, . . . , n,

196

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

we find that n−1 

aj bj = An−1 bn−1 − A0 b1 +

j =1

n−2 

Aj (bj − bj +1 )

(4.86)

j =1

and n−1 

aj bj = An−1 bn−1 − AMN −1 bMN +

j =MN

n−2 

Aj (bj − bj +1 ).

(4.87)

j =MN

If we invoke bj = qj , aj = 1 and Aj = j for j = 0, 1, . . . , n, then (4.86) and (4.87) give the following identities: n−1 n−1 n−2      qj − qj +1 j + qn−1 (n − 1) Qn =: qj = q0 + qj = q0 + j =0

j =1

(4.88)

j =1

and n−1 

n−2  

qj =:

j =MN

 qj − qj +1 j + qn−1 (n − 1) − (MN − 1)qMN .

j =MN

Moreover, if we use that D0 = K0 = 0, for any x ∈ Gm and invoke the Abel transformations (4.86) and (4.87) for bj = qj , aj = Dj and Aj = j Kj for any j = 0, 1, . . . , n − 1 we get the identities Fn−1 =

1  qj Dj Qn n−1

j =0

⎛ ⎞ n−2  1 ⎝  qj − qj +1 j Kj + qn−1 (n − 1)Kn−1 ⎠ = Qn j =1

(4.89)

4.5 Kernels of T Means

197

and n−1 1  qj Dj Qn

(4.90)

j =MN

⎛

⎞ n−2    1 ⎝ = qj − qj +1 j Kj + qn−1 (n − 1)Kn−1 − qMN (MN − 1)KMN −1 ⎠ . Qn j =MN

Analogously, if we use that S0 f = σ0 f = 0 for any x ∈ Gm and apply the Abel transformations (4.86) and (4.87) for bj = qj , aj = Sj and Aj = j σj for any j = 0, 1, . . . , n − 1 we get the identities 1  Tn f = qj Sj f Qn n−1

(4.91)

j =0

⎞ ⎛ n−2  1 ⎝  qj − qj +1 j σj f + qn−1 (n − 1)σn−1 f ⎠ = Qn j =1

and n−1 1  qj Sj f Qn j=MN

⎞ ⎛ n−2  1 ⎝   qj − qj+1 j σj f + qn−1 (n − 1)σn−1 f − qMN (MN − 1)σMN −1 f ⎠ . = Qn j=MN

First we consider the kernels of T means with respect to Vilenkin systems generated by non-increasing sequences (qk , k ∈ N). Lemma 4.5.1 Let (qk , k ∈ N) be a sequence of non-increasing numbers satisfying condition (4.11). Then, for some constant c, we have that ⎧ ⎫ |n|   c ⎨ ⎬    −1  Mj KMj  . Fn  ≤ ⎭ n⎩ j =0

198

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Proof By using (4.88), we get that ⎛ ⎞ ⎛ ⎞ n−2 n−2       1 ⎝ 1 qj − qj +1  + qn−1 ⎠ ≤ ⎝ q0 + qj − qj +1 + qn−1 ⎠ q0 + Qn Qn j =1

j =1

≤

2q0 c ≤ . Qn n

Hence, if we apply 3.12 and use the equality (4.89), we can conclude that ⎛ ⎛ ⎞⎞ |n| n−1       1    −1  ⎝  ⎝ ⎠ ⎠ qn−j − qn−j −1 + qn−1 Mi KMi  Fn  ≤ q0 + Qn j =1

≤

i=0

|n|   2q0  Mi KMi  Qn i=0

|n|

≤

  c Mi KMi  . n i=0



so the proof is complete.

Lemma 4.5.2 Let x ∈ INk,l , k = 0, . . . , N − 2, l = k + 1, . . . , N − 1 and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying condition (4.11). Then, for some c > 0,      n−1  1   cMl Mk  qj Dj (x − t) dμ (t) ≤ . (4.92) Q nMN IN  n  j =MN

Let x ∈ INk,N , k = 0, . . . , N − 1. Then, for some c > 0,  IN

    n−1  1   cMk  qj Dj (x − t) dμ (t) ≤ . Q MN  n j =MN 

(4.93)

4.5 Kernels of T Means

199

Proof Let x ∈ INk,l , for 0 ≤ k < l ≤ N − 1 and t ∈ IN . First, we observe that x − t ∈ INk,l . Next, we apply Lemmas 3.2.1 and 4.5.1 to obtain that  IN

     |n| n−1  1     c   KM (x − t) dμ (t) dμ ≤ q D M − t) (t) (x j j i i Q  n IN  n j =MN  i=0 c ≤ n ≤



l  Mi Mk dμ (t) IN i=0

cMk Ml nMN

and (4.92) is proved. Now, let x ∈ INk,N . Since x − t ∈ INk,N for t ∈ IN , by combining again Lemmas 3.2.1 and 4.5.1, we have that       |n| n−1    1   c KM (x − t) dμ (t)   qj Dj (x − t) dμ (t) ≤ Mi i Q n IN  n j =M IN  i=0 N ≤

 |n|−1 c Mi Mk dμ (t) n IN i=0

≤

cMk , MN

so also (4.93) is proved and the proof is complete.



Next, we generalize Lemma 4.5.1 but now without any restriction like condition (4.11). Lemma 4.5.3 Let (qk , k ∈ N) be a sequence of non-increasing numbers and n > MN . Then, for some c > 0, ⎫  ⎧    |n| n−1 ⎨ ⎬   1   c  qj Dj (x) ≤ Mj KMj (x) ,  Qn ⎭ MN ⎩   j =MN j =0 where c is an absolute constant.

200

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Proof Since the sequence (qk , k ∈ N) is non-increasing we can readily conclude that ⎛ ⎞ n−2    1 ⎝ qj − qj +1  + qn−1 ⎠ qMN + (4.94) Qn ⎛

j =MN

⎞ n−2    1 ⎝ qj − qj +1 + qn−1 ⎠ qMN + ≤ Qn j =MN

=

2qMN 2qMN c ≤ ≤ . Qn QMN +1 MN

If we apply Abel transformation (4.90) and (4.94) combined with (3.12) in Corollary 3.2.7, we get that     n−1  1     q D j j Q  n j =MN  ⎛ ⎞ n−2  1 ⎝   qj − qj +1 j Kj + qn−1 (n − 1)Kn−1 − qMN (MN − 1)KMN −1 ⎠ = Qn ⎛

j =MN

⎞ |n| n−2       c ⎝   ⎠ ≤ Mi KMi  qMN + qj − qj +1 + qn−1 Qn j =MN

≤

i=0

|n|   c  Mi KMi  . MN i=0

The proof is complete.



Lemma 4.5.4 Let x ∈ INk,l , k = 0, . . . , N − 1, l = k + 1, . . . , N and (qk , k ∈ N) be a sequence of non-increasing numbers. Then, for some c > 0,  IN

    n−1  1     dμ (t) ≤ cMl Mk . q D − t) (x j j Q  MN2  n j =MN 

4.5 Kernels of T Means

201

Proof Let x ∈ INk,l for 0 ≤ k < l ≤ N − 1 and t ∈ IN . First, we observe that x − t ∈ INk,l . Next, we apply Lemmas 3.2.1 and 4.5.3 to obtain that  IN

    n−1  1     dμ (t) q D − t) (x j j Q   n j =MN  |n|

c  ≤ Mi MN

 IN

i=0

≤ ≤

c MN



(4.95)

  KM (x − t) dμ (t) i

l  Mi Mk dμ (t) IN i=0

cMl Mk . MN2

Now, let x ∈ INk,N . Since x − t ∈ INk,N for t ∈ IN , by using Lemma 1.6.9 we have that |Di (x − t)| ≤ Mk and  IN

    n−1  1    qj Dj (x − t) dμ (t) Q  n j =MN  |n|

≤

c  qi Qn i=0

≤

(4.96)

 |Di (x − t)| dμ (t) IN

|n|−1  c  qi Mk dμ (t) Qn IN i=0

≤

cMk . MN

According to (4.95) and (4.96) the proof is complete.



The next lemma will be very important in this Chapter to prove norm convergence in Lebesgue spaces of T means generated by non-increasing sequences (qk , k ∈ N).

202

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Lemma 4.5.5 Let (qk , k ∈ N) be a sequence of non-increasing numbers. Then, for any n, N ∈ N+ ,  Gm

Fn−1 (x)dμ(x) = 1,



sup n∈N Gm



Gm \IN

(4.97)

   −1  Fn (x) dμ(x) ≤ c < ∞,

(4.98)

   −1  Fn (x) dμ(x) → 0, as n → ∞,

(4.99)

where c is an absolute constant. Proof According to Corollary 1.6.8 we easily obtain (4.97). By using (3.17) in Corollary 3.2.9 combined with (4.88) and (4.89), we get that  Gm

   −1  Fn (x) dμ(x)

 1  ≤ qj − qj +1 j Qn n−2



j =0

  Kj  dμ + qn−1 (n − 1) Gm

 |Kn−1 |dμ Gm

 c  cqn−1 (n − 1) qj − qj +1 j + ≤ c < ∞. Qn Qn n−2

≤

j =0

Thus (4.98) is proved. By using (3.18) in Corollary 3.2.9 and inequalities (4.88) and (4.89), we can conclude that     −1  (4.100) Fn (x) dμ(x) Gm \IN

 1  qj − qj +1 j ≤ Qn n−2

j =0

+

qn−1 (n − 1) Qn

 Gm \IN

  Kj (x) dμ(x)

 Gm \IN

|Kn−1 (x)|dμ(x)

 1  qn−1 (n − 1)αn−1 qj − qj +1 j αj + ≤ Qn Qn n−2

j =0

=: I + I I, where αn → 0, as n → ∞.

4.5 Kernels of T Means

203

Since the sequence (qk , k ∈ N) is non-increasing, we can conclude that II =

qn−1 (n − 1)αn−1 ≤ αn−1 → 0, as n → ∞. Qn

Moreover, for any ε > 0 there exists N0 ∈ N, such that αn < ε when n > N0 . Furthermore,  1  qj − qj +1 j αj Qn n−2

I =

j =0

=

N0 n−2    1  1   qj − qj +1 j αj + qj − qj +1 j αj Qn Qn j =0

j =N0 +1

= I1 + I2 . The sequence is non-increasing and therefore |qj − qj +1 | < 2q0 and I1 ≤

2q0 N0 → 0, as n → ∞ Qn

I2 =

n−2  1   qj − qj +1 j αj Qn

and

j =N0 +1

n−2  ε   qj − qj +1 j ≤ Qn j =N0 +1

 ε  qj − qj +1 j < ε. Qn n−2

≤

j =0

Thus I2 → 0, so that I → 0, as n → ∞. Hence, the proof of (4.99) follows from (4.100). The proof is complete.  The next two lemmas are very important for our further investigations in this Chapter to prove almost everywhere convergence of T means generated by nonincreasing sequences (qk , k ∈ N).

204

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Corollary 4.5.6 Let (qk , k ∈ N) be a sequence of non-increasing numbers. Then 

    n−1  1    sup  qj Dj (x) dμ(x) ≤ c < ∞, Q Gm n>MN  n j =M  N

where c is an absolute constant. Proof If we compare the estimation of Kn in Lemma 3.2.7 and     n−1  1    qn−j Dj (x) Q  n j =MN  in Lemma 4.4.7 with the estimation of     n−1  1    qj Dj (x) Q  n j =MN  in Lemma 4.5.3, we find that they are quite the same. Hence, the proof is step by step analogously to that of Lemma 3.2.10, so we omit the details.  Lemma 4.5.7 Let (qk , k ∈ N) be a sequence of non-increasing numbers satisfying condition (4.11). Then      sup Fn−1  dμ ≤ c < ∞, Gm n>MN

where c is an absolute constant. Proof The estimation of the Nörlund kernels Fn in Corollary 4.4.2 and the estimation of T kernels Fn−1 in Lemma 4.5.1 are similar. Therefore, the proof is analogous to that of Lemma 4.4.7, so we omit the details.  In the next lemmas we investigate T means with respect to Vilenkin systems generated by non-decreasing sequences (qk , k ∈ N). Lemma 4.5.8 Let (qk , k ∈ N) be a sequence of non-decreasing numbers satisfying the condition (4.10). Then, for some c > 0, ⎫ ⎧ |n|   c ⎨ ⎬    −1  Mj KMj  . Fn  ≤ ⎭ n⎩ j =0

4.5 Kernels of T Means

205

Proof By condition (4.10), ⎛ ⎞ n−2  1 ⎝  qj − qj +1  + qn−1 + q0 ⎠ Qn ⎛

(4.101)

j =0

⎞ n−2    1 ⎝ qj +1 − qj + qn−1 + q0 ⎠ = Qn j =0

≤

2qn−1 + q0 Qn

≤

3qn−1 c ≤ . Qn n

If we now apply (4.90) combined with (3.12) in Lemma 3.2.7 and (4.101), we get that ⎛ ⎛ ⎞⎞ |n| n−1          −1  ⎝ 1 ⎝ qj − qj +1  + qn−1 + q0 ⎠⎠ Mi KMi  Fn  ≤ Qn j =1

i=0

|n|   3qn−1  Mi KMi  ≤ Qn i=0

≤

c n

|n| 

  Mi KMi  .

i=0



The proof is complete.

Lemma 4.5.9 Let x ∈ INk,l , k = 0, . . . , N − 2, l = k + 1, . . . , N − 1 and (qk , k ∈ N) be a sequence of non-decreasing numbers satisfying condition (4.10). Then  IN

  cMl Mk  −1  . Fn (x − t) dμ (t) ≤ nMN

Let x ∈ INk,N , k = 0, . . . , N − 1. Then  IN

  cMk  −1  . Fn (x − t) dμ (t) ≤ MN

Here c is an absolute constant. Proof The proof is quite analogously to the proof of Lemma 4.5.2, so we leave out the details. 

206

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

We also point out the following useful consequence of Lemma 4.5.9: Lemma 4.5.10 Let n ≥ MN , x ∈ INk,l , k = 0, . . . , N − 1, l = k + 1, . . . , N and (qk , k ∈ N) be a sequence of non-decreasing numbers satisfying condition (4.10). Then    cMl Mk   −1 . Fn (x − t) dμ (t) ≤ MN2 IN Now we state some corollaries which easily follows Lemma (4.5.8) and which will be used to prove norm convergence of T means generated by non-decreasing sequences (qk , k ∈ N). Corollary 4.5.11 Let (qk , k ∈ N) be a sequence of non-decreasing numbers, satisfying condition (4.10). Then for any n, N ∈ N+ ,  Gm

Fn−1 (x)dμ(x) = 1,



sup n∈N Gm



Gm \IN

   −1  Fn (x) dμ(x) ≤ c < ∞,

   −1  Fn (x) dμ(x) → 0, as n → ∞,

where c is an absolute constant. Proof By using Lemma 4.5.8, the proof is step by step analogous to that of Corollary 4.4.2, so we leave out the details.  Corollary 4.5.12 Let (qk , k ∈ N) be a sequence of non-decreasing numbers, satisfying condition (4.10). Then, for any n, N ∈ N+ , 

    n−1  1    sup  qj Dj (x) dμ(x) ≤ c < ∞. Q Gm n>MN  n j =M  N

Proof The proof follows by applying Lemma 4.5.8.



Finally we study some special subsequences of the kernels of Nörlund and T means. Lemma 4.5.13 Let n ∈ N. Then FMn (x) = DMn (x) − ψMn −1 (x)F −1 Mn (x)

(4.102)

4.5 Kernels of T Means

207

and −1 FM (x) = DMn (x) − ψMn −1 (x)F Mn (x). n

(4.103)

Proof By using (1.41) in Lemma 1.6.4 we get that n 1  qMn −k Dk (x) QMn

M

FMn (x) =

k=1

Mn −1 1  qk DMn −k (x) QMn

=

k=0

Mn −1   1  qk DMn (x) − ψMn −1 (x)D j (x) QMn

=

k=0

= DMn (x) − ψMn −1 (x)F −1 Mn (x) Hence, (4.102) is proved. Identity (4.103) can be proved analogously. The proof is complete.  The next four corollaries will be used to prove norm convergence and almost everywhere convergence of subsequences of Nörlund and T means. Corollary 4.5.14 Let (qk , k ∈ N) be a sequence of non-decreasing numbers. Then, for any n, N ∈ N+ and some c > 0,  Gm

−1 FM (x)dμ(x) = 1, n



sup n∈N Gm



Gm \IN

   −1  FMn (x) dμ(x) ≤ c < ∞,

   −1  FMn (x) dμ(x) → 0, as n → ∞.

Proof According to (4.103), the proof is a direct consequence of Lemma 4.4.6 and Corollaries 1.6.7 and 1.6.8.  Corollary 4.5.15 Let (qk , k ∈ N) be a sequence of non-decreasing numbers. Then, for any n, N ∈ N+ and some c > 0, 

    n −1  1 M   sup  qj Dj (x) dμ(x) ≤ c < ∞. Gm n>N  Qn j =M  N

208

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Proof In view of (4.103) the proof follows by applying Corollary 1.6.7 (Paley’s Lemma) and Lemma 4.4.7.  Corollary 4.5.16 Let (qk , k ∈ N) be a sequence of non-increasing numbers. Then, for any N ∈ N+ and some c > 0,  FMn (x)dμ(x) = 1, Gm



  FM (x) dμ(x) ≤ c < ∞, n

sup n∈N Gm



Gm \IN

  FM (x) dμ(x) → 0, as n → ∞. n

Proof By (4.102), the proof is a consequence of Lemma 4.5.5 and Corollaries 1.6.7 and 1.6.8.  Corollary 4.5.17 Let (qk , k ∈ N) be a sequence of non-increasing numbers. Then, for any N ∈ N+ and some c > 0,     Mn  1    sup  qj DMn −j (x) dμ(x) ≤ c < ∞. Gm n>N  Qn j =M  N



Proof Here we again use 4.5.13 together with Corollary 1.6.7 and Lemma 4.5.6 and the proof follows in a similar way as above. 

4.6 Norm Convergence of Nörlund and T Means in Lebesgue Spaces First we consider norm convergence of Nörlund means with respect to Vilenkin systems. Theorem 4.6.1 Let f ∈ Lp (Gm ) for 1 ≤ p < ∞ and (qk , k ∈ N) be a sequence of non-decreasing numbers. Then tn f − f p → 0, as n → ∞. Proof According to Lemma 4.4.6, we conclude that the conditions (A1), (A2) and (A3) in Theorem 3.5.2 are fulfilled, which implies the stated norm convergence. 

4.6 Norm Convergence of Nörlund and T Means in Lebesgue Spaces

209

Theorem 4.6.2 Let f ∈ Lp (Gm ) for 1 ≤ p < ∞ and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying the condition (4.11). Then tn f − f p → 0, as n → ∞. Proof By Corollary 4.4.10, the conditions (A1), (A2) and (A3) in Theorem 3.5.2 are fulfilled and stated norm convergence follows.  Corollary 4.6.3 Let f ∈ Lp (Gm ) for 1 ≤ p < ∞. Then σn f − f p → 0, as n → ∞, Bn f − f p → 0, as n → ∞, βnα f − f p → 0, as n → ∞. Proof Since σn f and βnα f are Nörlund means generated by non-decreasing sequences (qk , k ∈ N), the corresponding norm convergences are direct consequences of Theorem 4.6.1. If the means Bn f are generated by a non-decreasing sequence (qk , k ∈ N), then this result follows from Theorem 4.6.1, if (qk , k ∈ N) is non-increasing, then it follows from Theorem 4.6.2 and (4.11).  Now, we consider subsequences of Nörlund means generated by non-increasing sequences without any restrictions like (4.11) on the sequence (qk , k ∈ N). Theorem 4.6.4 Let f ∈ Lp (Gm ) for 1 ≤ p < ∞ and (qk , k ∈ N) be a sequence of non-increasing numbers. Then tMn f − f p → 0, as n → ∞. Proof According to Corollary 4.5.16, we conclude that conditions (A1), (A2) and (A3) in Theorem 3.5.2 are fulfilled and the claimed norm convergence is proved.  Since σnα f, Vnα f and Unα f are Nörlund means generated by non-increasing sequences (qk , k ∈ N), we have the following consequence of Theorem 4.6.4. Corollary 4.6.5 Let f ∈ Lp (Gm ) for 1 ≤ p < ∞. Then α σM f − f p → 0 as n → ∞, n α VM f − f p → 0 as n → ∞, n α f − f p → 0 as n → ∞. UM n

In the next theorem, we consider weaker conditions then (4.11) which are sufficient to obtain almost everywhere convergence of Nörlund means generated by non-increasing sequences (qk , k ∈ N).

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4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Theorem 4.6.6 Let f ∈ Lp (Gm ) for 1 ≤ p < ∞ and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying conditions (4.5) and (4.6). Then tn f − f p → 0 as n → ∞. Proof By Corollary 4.5.16, the conditions (A1), (A2) and (A3) in Definition 3.5.1 are fulfilled and by using Theorem 3.5.2 the claimed norm convergence is proved.  By this theorem, we get stronger result for σnα f and Vnα f means than those in Corollary 4.6.5. Corollary 4.6.7 Let f ∈ Lp (Gm ) for 1 ≤ p < ∞. Then σnα f − f p → 0 as n → ∞, Vnα f − f p → 0 as n → ∞. Proof First we note that σnα f and Vnα f are Nörlund means generated by nonincreasing sequences (qk , k ∈ N). On the other hand, estimates (4.14) and (4.15) for σnα f means and for Vnα f means imply that the conditions (4.5) and (4.6) are satisfied. Hence, Theorem 4.6.6 implies the statements in the corollary.  Now we consider norm convergence of T means. Theorem 4.6.8 Let f ∈ Lp (Gm ) for 1 ≤ p < ∞ and (qk , k ∈ N) be a sequence of non-increasing numbers. Then Tn f − f p → 0 as n → ∞. Proof According to Corollary 4.5.11, conditions (A1), (A2) and (A3) in Definition 3.5.1 are satisfied so also this norm convergence follows from Theorem 3.5.2.  Theorem 4.6.9 Let f ∈ Lp (Gm ) for 1 ≤ p < ∞ and (qk , k ∈ N) be a sequence of non-decreasing numbers satisfying condition (4.11). Then Tn f − f p → 0 as n → ∞. Proof Conditions (A1), (A2) and (A3) in Definition 3.5.2 hold again by Lemma 4.4.6. Hence, norm convergence follows from Theorem 3.5.2. 

4.7 Almost Everywhere Convergence of Nörlund and T Means

211

According to Theorems 4.6.8 and 4.6.9, we get the following result for T means: Corollary 4.6.10 Let f ∈ Lp (Gm ) for Bn−1 f . Then Bn−1 f − f p → 0 as n → ∞, σnα,−1 f − f p → 0 as n → ∞, Vnα,−1 f − f p → 0 as n → ∞, Unα,−1 f − f p → 0 as n → ∞. Proof If (qk , k ∈ N) in Bn−1 f is non-decreasing and satisfies condition (4.11), then the statement follows from Theorem 4.6.9, if it is non-increasing, the statement is a direct consequence of Theorem 4.6.8. Analogously, the summability methods σnα,−1 f, Vnα,−1 f and Unα,−1 f are T means generated by non-increasing sequences (qk , k ∈ N) and the stated norm convergence follows from Theorem 4.6.8.  Now, we consider subsequences of T means generated by non-decreasing sequences, but without any restrictions on the sequence (qk , k ∈ N). Theorem 4.6.11 Let f ∈ Lp (Gm ) for Bn−1 f and (qk , k ∈ N) be a sequence of non-decreasing numbers. Then TMn f − f p → 0 as n → ∞. Proof Corollary 4.5.14 yields that conditions (A1), (A2) and (A3) in Theorem 3.5.2 hold.  Since βnα,−1 f are T means generated by non-decreasing sequence (qk , k ∈ N), we have Corollary 4.6.12 Let f ∈ Lp (Gm ) for Bn−1 f . Then α,−1 βM f − f p → 0 as n → ∞. n

4.7 Almost Everywhere Convergence of Nörlund and T Means In this section we investigate almost everywhere convergence of Nörlund and T Means.

212

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Theorem 4.7.1 Let tn f be the Nörlund means and Fn be the corresponding Nörlund kernels defined by (4.3) such that  IN

    n  1    sup  qn−k Dk (x) dμ (x) ≤ c < ∞. Q n>MN  n k=M +1  N

If the maximal operator t ∗ of Nörlund means is bounded from Lp1 to Lp1 for some 1 < p1 ≤ ∞, then the operator t ∗ is of weak-type (1, 1), i.e., for all f ∈ L1 (Gm ), * + sup yμ t ∗ f > y ≤ f 1 . y>0

Proof In view of Theorem 2.4.2, the proof is complete if we prove that 

 ∗ t f (x) dμ(x) ≤ cf 1

(4.104)

I

for every function f satisfying conditions in (2.10), where I denotes the support of the function f. Without lost the generality we may assume that f is a function with support I and μ (I ) = MN . We may also assume that I = IN . It is easy to see that tn f = 0 when n ≤ MN . Therefore, we can suppose that n > MN . Moreover, Sn f = 0 for n ≤ MN , ⎞ ⎛ MN 1 ⎝ qn−k Sk f (x)⎠ = 0 Qn k=0

and  IN

⎞ ⎛ MN 1 ⎝ qn−k Dk (x − t)⎠ f (t)dμ (t) = 0. Qn k=0

4.7 Almost Everywhere Convergence of Nörlund and T Means

213

Hence,    ∗  1 t f (x) ≤ sup   Q n n>MN  IN    1  + sup   Q n n>MN  IN    1  = sup   Q n n>MN  IN

 ⎛ ⎞  M N   ⎝ qn−k Dk (x − t)⎠ f (t)dμ (t)  k=0 

(4.105)

    ⎝ qn−k Dk (x − t)⎠ f (t)dμ (t)  k=MN +1   ⎞ ⎛  n    ⎝ qn−k Dk (x − t)⎠ f (t)dμ (t) .   k=MN +1 ⎛

n 

⎞

Let t ∈ IN and x ∈ IN . Then x − t ∈ IN and (4.105) implies that  IN

 ≤ IN

∗  t f (x) dμ(x)   ⎛ ⎞    n   1   ⎝ ⎠ sup qn−k Dk (x − t) f (t) dμ (t) dμ (x) Q n>MN  n k=MN +1 

  ≤ IN IN

IN

  ⎞ ⎛   n   1  ⎝ sup  qn−k Dk (x − t)⎠ f (t) dμ (x) dμ (t) n>MN  Qn k=M +1  N

and, furthermore,  IN

 ∗ t f (x) dμ(x) ≤

  IN IN

  ⎛ ⎞   n   1   ⎝ ⎠ sup  qn−k Dk (x) f (t) dμ (x) dμ (t) n>MN  Qn k=M +1  N 

= f 1 IN

 ⎛ ⎞   n   1   ⎝ ⎠ sup  qn−k Dk (x)  dμ (x) Q n>MN  n k=M +1  N

≤ c f 1 . Thus (4.104) holds so the proof is complete.



214

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

By using the same technique, we obtain the following results. Theorem 4.7.2 Let tn f be Nörlund means and Fn be the corresponding Nörlund kernels defined by (4.3) such that  sup |Fn (t) |dμ (t) < C < ∞. n>MN IN

If the maximal operator t ∗ of the Nörlund means is bounded from Lp1 to Lp1 for some 1 < p1 ≤ ∞, then, for all f ∈ L1 (Gm ), * + sup yμ t ∗ f > y ≤ f 1 . y>0

Theorem 4.7.3 Let Tn f be T means and Fn−1 be the corresponding Nörlund kernels defined by (4.4) such that  IN

    n  1    sup  qk Dk (x) dμ (x) ≤ C < ∞. n>MN  Qn k=M +1  N

If the maximal operator T ∗ of the T means is bounded from Lp1 to Lp1 for some 1 < p1 ≤ ∞, then for all f ∈ L1 (Gm ), * + sup yμ T ∗ f > y ≤ f 1 . y>0

Theorem 4.7.4 Let Tn f be T means and Fn−1 be the corresponding kernels defined by (4.4) such that 

sup |Fn−1 (t) |dμ (t) ≤ C < ∞.

n>MN IN

If the maximal operator T ∗ of the Nörlund means is bounded from Lp1 to Lp1 for some 1 < p1 ≤ ∞, then, for all f ∈ L1 (Gm ), * + supyμ T ∗ f > y ≤ f 1 . y>0

Next, we present some applications concerning almost everywhere convergence of some summability methods. The study of almost everywhere convergence is one of the most difficult topics in Fourier analysis.

4.7 Almost Everywhere Convergence of Nörlund and T Means

215

By using Theorem 1.5.9, Proposition 1.8.1 and the results in our previous Sections, we in particular obtain the following almost everywhere convergence results. Theorem 4.7.5 Let f ∈ L1 and tn be regular Nörlund means with non-decreasing sequences (qk , k ∈ N). Then tn f → f a.e., as n → ∞. Proof Since Sn P = P , for every P ∈ P, according to regularity of Nörlund means with non-decreasing sequence (qk , k ∈ N), we obtain that tn P → P

a.e.,

as

n → ∞,

where P ∈ P and P is dense in the space L1 (see Proposition 1.8.1). On the other hand, by combining Lemma 4.4.7 and Theorem 4.7.1, we obtain that the maximal operator t ∗ of the Nörlund means with non-decreasing sequence (qk , k ∈ N) is bounded from the space L1 to the space weak − L1 , that is, * + sup yμ t ∗ f > y ≤ f 1 . y>0

Hence, according to Theorem 1.5.9, we obtain the claimed almost everywhere convergence of Nörlund means.  In particular, since σn f and βnα f are regular Nörlund means with non-decreasing sequence (qk , k ∈ N) we have Corollary 4.7.6 Let f ∈ L1 . Then σn f → f

a.e., as n → ∞,

βnα f → f

a.e., as n → ∞.

and

Next we consider almost everywhere convergence of Nörlund means with nonincreasing sequences (qk , k ∈ N). Theorem 4.7.7 Let f ∈ L1 and tn be Nörlund means with non-increasing sequence (qk , k ∈ N) satisfying condition (4.11). Then tn f → f a.e., as n → ∞.

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4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Proof The proof is quite analogous to that of Theorem 4.7.5 if we apply Corollary 4.4.11 and Theorem 4.7.1 so we omit the details.  Corollary 4.7.8 Let f ∈ L1 and tn be Nörlund means with monotone and bounded sequence (qk , k ∈ N). Then Bn f → f

a.e., as n → ∞.

Proof When the sequence (qk , k ∈ N) is non-decreasing the statement follows from Theorem 4.7.5. Moreover, if the sequence (qk , k ∈ N) is non-decreasing, then we have regularity. Furthermore, condition (4.11) is also fulfilled. Hence, the stated convergence follows from Theorem 4.7.7.  Theorem 4.7.9 Let f ∈ L1 and tn be Nörlund means with non-increasing sequence (qk , k ∈ N) satisfying conditions (4.5) and (4.6). Then tn f → f a.e., as n → ∞. Proof The proof is similar to the proof of Theorem 4.7.5 if we apply Lemma 4.4.14 and Theorem 4.7.1, so we omit the details.  Corollary 4.7.10 Let f ∈ L1 and 0 < α < 1. Then σnα f → f

a.e., as n → ∞

and Vnα f → f

a.e., as n → ∞.

Proof By using (4.12), (4.13), (4.14) and (4.15), we obtain that both summability methods σnα and Vnα are Nörlund means with non-increasing sequences (qk , k ∈ N) satisfying conditions (4.5) and (4.6). Hence, the proof is complete by just using Theorem 4.7.9.  Theorem 4.7.11 Let f ∈ L1 and tn be Nörlund means with non-increasing sequence (qk , k ∈ N). Then tMn f → f

a.e., as n → ∞.

Proof By applying Corollary 4.5.17 and Theorem 4.7.1, the proof is analogous to that of Theorem 4.7.5. 

4.7 Almost Everywhere Convergence of Nörlund and T Means

217

Since Unα are regular Nörlund means with non-increasing sequences (qk , k ∈ N), we can state Corollary 4.7.12 Let f ∈ L1 . Then α UM f →f n

a.e., as n → ∞.

Theorem 4.7.13 Let f ∈ L1 and Tn be regular T means with non-increasing sequence (qk , k ∈ N). Then Tn f → f

a.e., as n → ∞.

Proof According to the regularity of T means with non-increasing sequence (qk , k ∈ N), we obtain that Tn P → P

a.e.,

as

n → ∞,

where P ∈ P. By combining Corollary 4.5.6 and Theorem 4.7.3, we can conclude that * + sup yμ T ∗ f > y ≤ f 1 . y>0

Hence the proof follows by just using Theorem 1.5.9.



Since σnα,−1 f, Vnα,−1 f and Unα,−1 f are regular T means with non-increasing sequences (qk , k ∈ N), we get Corollary 4.7.14 Let f ∈ L1 . Then σnα,−1 f → f

a.e., as n → ∞,

Vnα,−1 f → f

a.e., as n → ∞,

Unα,−1 f → f

a.e., as n → ∞.

Theorem 4.7.15 Let f ∈ L1 and Tn be T means with non-decreasing sequence (qk , k ∈ N) satisfying condition (4.10). Then Tn f → f

a.e., as n → ∞.

Proof Since almost all points are Vilenkin-Lebesgue points if we invoke Corollary 4.5.12, we immediately get the proof. 

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4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Corollary 4.7.16 Let f ∈ L1 and Bn−1 be T means with monotone and bounded sequences (qk , k ∈ N). Then Bn−1 f → f

a.e., as n → ∞.

Proof When the sequence (qk , k ∈ N) is non-increasing , it follows from Theorem 4.7.13. Moreover, if the sequence (qk , k ∈ N) is non-decreasing and, in addition, it is bounded, then we have regularity and also condition (4.10) is fulfilled so the convergence statement follows from Theorem 4.7.15.  Theorem 4.7.17 Let f ∈ L1 and Tn be T means with non-decreasing sequence (qk , k ∈ N). Then TMn f → f

a.e., as n → ∞.

Proof The proof is analogous to the proof of Theorem 4.7.5 so we leave out the details.  Since βnα f is a regular T mean with non-increasing sequence (qk , k ∈ N), we can finally state Corollary 4.7.18 Let f ∈ L1 . Then α βM f →f n

a.e., as n → ∞.

4.8 Convergence of Nörlund and T Means in Vilenkin-Lebesgue Points Our first main result concerning convergence of Nörlund means reads as follows. Theorem 4.8.1 (a) Let (qk , k ∈ N) be a sequence of non-decreasing numbers. If the function f ∈ L1 (Gm ) is continuous at a point x, then tn f (x) → f (x), as n → ∞. Furthermore, lim tn f (x) = f (x)

n→∞

for all Vilenkin-Lebesgue points of f ∈ L1 (Gm ).

4.8 Convergence of Nörlund and T Means in Vilenkin-Lebesgue Points

219

(b) Let (qk , k ∈ N) be a sequence of non-increasing numbers satisfying the condition (4.11). If the function f ∈ L1 (Gm ) is continuous at a point x, then tn f (x) → f (x), as n → ∞. Moreover, lim tn f (x) = f (x)

n→∞

for all Vilenkin-Lebesgue points of f ∈ L1 (Gm ). Proof (a) Let (qk , k ∈ N) be a non-decreasing sequence. Suppose that x is either a point of continuity of a function f ∈ L1 (Gm ) or a Vilenkin-Lebesgue point of the function f ∈ L1 (Gm ). According to Theorems 3.4.1 and 3.4.9, we can conclude that lim |σn f (x) − f (x)| = 0.

n→∞

Hence, by combining (4.23) and (4.26), we get that |tn f (x) − f (x)| (4.106) ⎞ ⎛ n−2  1 ⎝  ≤ qn−j − qn−j −1 j |σj f (x) − f (x)| + q0 n|σn f (x) − f (x)|⎠ Qn j =1

 1  q0 nαn qn−j − qn−j −1 j αj + =: I + I I, Qn Qn n−2

≤

j =0

where αn → 0, as n → ∞. Since the sequence (qk , k ∈ N) is non-decreasing, we obtain that I I ≤ αn → 0, as n → ∞. On the other hand, since αn converges to 0, we get that there exists an absolute constant A, such that αn ≤ A for any n ∈ N and for any ε > 0 there exists N0 ∈ N, such that αn < ε when n > N0 . Hence, I =

N0 n−1    1  1   qn−j − qn−j −1 j αj + qn−j − qn−j −1 j αj Qn Qn j =1

=: I1 + I2 .

j =N0 +1

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4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Since |qn−j − qn−j −1 | < 2qn−1

and αn ≤ A,

we obtain that 0   1  2AN0 qn−1 → 0, as n → ∞ qn−j − qn−j −1 j αj ≤ Qn Qn

N

I1 =

j =1

and n−1  1   qn−j − qn−j −1 j αj Qn

I2 =

j =N0 +1

n−1  ε   qn−j − qn−j −1 j ≤ Qn j =N0 +1

 ε  qn−j − qn−j −1 j < ε. Qn n−1

≤

j =0

We conclude that also I2 → 0, I → 0 and in view of (4.106), part (a) is proved. (b) Assume now that the sequence is non-increasing and satisfies condition (4.11). To prove convergence in Vilenkin-Lebesgue points we use the estimations (4.23) and (4.26) to obtain that  1  q0 nαn qn−j −1 − qn−j j αj + Qn Qn n−2

|tn f − f (x)| ≤

j =0

=: I I I + I V , where αn → 0, as n → ∞. It is evident that IV ≤

q0 nαn ≤ cαn → 0, as n → ∞. Qn

(4.107)

4.8 Convergence of Nörlund and T Means in Vilenkin-Lebesgue Points

221

Moreover, for any ε > 0 there exists N0 ∈ N such that αn < ε when n > N0 . It follows that  1  qn−j −1 − qn−j j αj Qn n−2

j =1

=

N0 n−2    1  1   qn−j −1 − qn−j j αj + qn−j −1 − qn−j j αj Qn Qn j =1

j =N0 +1

=: I I I1 + I I I2 . Since the sequence is non-increasing, we can conclude that |qn−j − qn−j −1 | < 2q0. Hence, I I I1 ≤

2q0N0 → 0, as n → ∞ Qn

and I I I2 ≤

n−2  1   qn−j −1 − qn−j j αj Qn j =N0 +1

≤

n−2  ε(n − 1)   qn−j − qn−j −1 Qn j =N0 +1

≤

 ε(n − 1)  q0 − qn−N0 Qn

≤

2q0 ε(n − 1) < cε. Qn

Hence, also I I I → 0. Thus, according to (4.107), the proof of part (b) is also complete.  Corollary 4.8.2 Let f ∈ L1 (Gm ). Then, for all Lebesgue points of f ∈ L1 (Gm ), σn f → f, as n → ∞, Bn f → f, as n → ∞, βnα f → f, as n → ∞.

222

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Theorem 4.8.3 Let (qk , k ∈ N) be a sequence of non-increasing numbers. Then lim tMn f (x) = f (x)

n→∞

for all Lebesgue points of f ∈ L1 (Gm ). Proof By using Lemma 4.5.13 we get that  tMn f (x) =

f (t) Fn (x − t) dμ (t) Gm



=

 f (t) DMn (x − t) dμ (t) −

Gm

f (t) ψMn −1 (x − t)F −1 Mn (x − t)

Gm

=: I − I I. By applying Corollary 2.3.11, we get that I = SMn f (x) → f (x) for all Lebesgue points of f ∈ L1 (Gm ). Moreover, according to Proposition 1.2.1, we find that ψMn −1 (x − t) = ψMn −1 (x)ψ Mn −1 (t) so that  f (t) F −1 Mn (x − t)ψ Mn −1 (t)d(t).

I I = ψMn −1 (x) Gm

By combining Theorem 1.5.3 and Corollary 4.5.11, we have that the function f (t) F −1 Mn (x − t) ∈ L1 (Gm ), for any x ∈ Gm and I I are the Fourier coefficients of an integrable function. Hence, according to the Riemann-Lebesgue Lemma, we get that I I → 0, as n → ∞, for any x ∈ Gm . 

The proof is complete.

Corollary 4.8.4 Let f ∈ L1 (Gm ). Then, for all Lebesgue points of f ∈ L1 (Gm ), α σM f → f, as n → ∞, n α f → f, as n → ∞, VM n α f → f, as n → ∞. UM n

Now, we state the corresponding results for T means.

4.8 Convergence of Nörlund and T Means in Vilenkin-Lebesgue Points

223

Theorem 4.8.5 (a) Let (qk , k ∈ N) be a sequence of non-increasing numbers. If the function f ∈ L1 (Gm ) is continuous at a point x, then Tn f (x) → f (x), as n → ∞. Moreover, lim Tn f (x) = f (x)

n→∞

for all Vilenkin-Lebesgue points of f ∈ L1 (Gm ). (b) Let (qk , k ∈ N) be a sequence of non-decreasing numbers satisfying the condition (4.10). If the function f ∈ L1 (Gm ) is continuous at a point x, then Tn f (x) → f (x), as n → ∞. Furthermore, lim Tn f (x) = f (x)

n→∞

for all Vilenkin-Lebesgue points of f ∈ L1 (Gm ). Proof (a) Let (qk , k ∈ N) be a non-increasing sequence. Suppose that x is either point of continuity of function f ∈ L1 (Gm ) or a Vilenkin-Lebesgue point of function f ∈ L1 (Gm ). According to Theorems 3.4.1 and 3.4.9, we can conclude that lim |σn f (x) − f (x)| = 0.

n→∞

By combining (4.88) and (4.91) we get that |Tn f (x) − f (x)| ⎞ ⎛ n−2  1 ⎝  ≤ qj − qj +1 j |σj f (x) − f (x)| + qn−1 (n − 1)|σn−1 f (x) − f (x)|⎠ Qn j =0

 1  (n − 1)αn−1 q qj − qj +1 j αj + n−1 Qn Qn n−2

≤

j =0

=: I + I I,

where αn → 0, as n → ∞.

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4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Since the sequence (qk , k ∈ N) is non-increasing we can conclude that II =

qn−1 (n − 1)αn−1 ≤ αn−1 → 0, as n → ∞. Qn

Obviously, αn ≤ A < ∞ where n ∈ N and for any ε > 0 there exists N0 ∈ N, such that αn < ε when n > N0 . Moreover, N0 n−2    1  1   qj − qj +1 j αj + qj − qj +1 j αj =: I1 + I2 . I = Qn Qn j =0

j =N0 +1

Since the sequence (qk , k ∈ N) is non-increasing, we can conclude that |qj − qj +1 | < 2q0, 0   1  2q0 N0 qj − qj +1 j αj ≤ → 0, as n → ∞ Qn Qn

N

I1 =

j =0

and I2 =

n−2  1   qj − qj +1 j αj Qn j =N0 +1

≤

n−2  ε   qj − qj +1 j Qn j =N0 +1

 ε  qj − qj +1 j < ε. Qn n−2

≤

j =0

Therefore, also I → 0, so that (a) is proved. (b) Now we assume that the sequence (qk , k ∈ N) is non-decreasing and satisfies condition (4.10). By combining (4.88) and (4.91), we get that  1  qn−1 (n − 1)αn qj +1 − qj j αj + Qn Qn n−2

|Tn f (x) − f (x)| ≤

j =0

=: I I I + I V , where αn → 0, as n → ∞. Obviously, IV ≤

qn−1 (n − 1)αn ≤ αn → 0, as n → ∞. Qn

4.8 Convergence of Nörlund and T Means in Vilenkin-Lebesgue Points

225

For any ε > 0 there exists N0 ∈ N such that αn < ε/2 when n > N0 . We have that III =

N0 n−2    1  1   qj +1 − qj j αj + qj +1 − qj j αj Qn Qn j =0

j =N0 +1

=: I I I1 + I I I2 . Since the sequence is non-decreasing, we conclude that |qj +1 − qj | < 2qj +1 < 2qn−1 . Hence, I I I1 ≤

2q0N0 → 0, as n → ∞ Qn

I I I2 ≤

n−2  1   qj +1 − qj j αj Qn

and

j =N0 +1

≤

n−2  ε(n − 1)   qj +1 − qj Qn j =N0 +1

≤

 ε(n − 1)  qn−1 − qn−N0 Qn

≤

2qn−1 ε(n − 1) < ε. Qn

Thus I I I → 0 and the proof of (b) is complete.  Corollary 4.8.6 Let f ∈ L1 (Gm ). Then, for all Lebesgue points x of f ∈ L1 (Gm ), Bn−1 f (x) → f (x), as n → ∞,

σnα,−1 f (x) → f (x), as n → ∞,

Vnα,−1 f (x) → f (x), as n → ∞,

Unα,−1 f (x) → f (x), as n → ∞.

226

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Theorem 4.8.7 Let (qk , k ∈ N) be a sequence of non-decreasing numbers. Then lim TMn f (x) = f (x) for all Lebesgue points of f ∈ L1 (Gm ).

n→∞

Proof Use first identity in Lemma 4.5.13 to get that  TMn f (x) =

f (t) Fn−1 (x − t) dμ (t)

Gm



=

 f (t) ψMn −1 (x − t)F Mn (x − t)

f (t) DMn (x − t) dμ (t) − Gm

Gm

=: I − I I. By applying Corollary 2.3.11 we get that I = SMn f (x) → f (x) for all Lebesgue points of f ∈ L1 (Gm ). Moreover, according to Proposition 1.2.1, we find that ψMn −1 (x − t) = ψMn −1 (x)ψ Mn −1 (t) so that  I I = ψMn −1 (x)

f (t) F Mn (x − t)ψ Mn −1 (t)d(t). Gm

By combining Theorem 1.5.3 and Corollary 4.5.11 we find that the function f (t) F Mn (x − t) ∈ L1 , for any x ∈ Gm and I I are the Fourier coefficients of an integrable function. Thus the RiemannLebesgue Lemma implies that I I → 0 for any x ∈ Gm . 

The proof is complete.

Corollary 4.8.8 Let f ∈ L1 (Gm ). Then, for all Lebesgue points x of f ∈ L1 (Gm ), α,−1 βM f (x) → f (x), as n → ∞. n

4.9 Riesz and Nörlund Logarithmic Kernels and Means

227

4.9 Riesz and Nörlund Logarithmic Kernels and Means The kernels of n-th Riesz and Nörlund logarithmic means are defined by n−1 1  Dk f , Pn f =: ln k k=1

Hn f =:

1 ln

n−1  k=1

Dk f n−k

respectively, where ln =:

n−1  1 k=1

k

.

Since the Riesz means are examples of T means, if we apply Corollary 4.5.5 and Lemma 4.5.6, we immediately get the following results. Corollary 4.9.1 Let n ∈ N. Then, for any n, N ∈ N+ ,  Pn (x)dμ(x) = 1, Gm

 |Pn (x)| dμ(x) ≤ c < ∞,

sup n∈N Gm



Gm \IN

|Pn (x)| dμ(x) → 0, as n → ∞,

where c is an absolute constant. Corollary 4.9.2 Let n ∈ N. Then 

    n−1  1   D (x) j   dμ ≤ c < ∞, sup  j  Gm n>MN  Qn j =M N

where c is an absolute constant. By using Abel transformation for the kernel of Riesz logarithmic means, we obtain that Kn 1  Kj + . ln j +1 ln n−1

Pn =

j =1

(4.108)

228

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

The next lemma plays a crucial role to study the kernel of Riesz logarithmic means with respect to the Vilenkin systems. Lemma 4.9.3 Let x ∈ IN (xk ek + xl el ) , 1 ≤ xk ≤ mk − 1, 1 ≤ xl ≤ ml − 1, k = 0, . . . , N − 2, l = k + 1, . . . , N − 1. Then 

  Kj (x − t)

n 

IN j =M +1 N

j +1

dμ (t) ≤

cMk Ml . MN2

(4.109)

Let x ∈ IN (xk ek ) , 1 ≤ xk ≤ mk − 1, k = 0, . . . , N − 1. Then 

n  IN j =M +1 N

  Kj (x − t) j +1

dμ (t) ≤

cMk ln . MN

(4.110)

Proof Let x ∈ IN (xk ek + xl el ) , 1 ≤ xk ≤ mk − 1, 1 ≤ xl ≤ ml − 1, k = 0, . . . , N − 2, l = k + 1, . . . , N − 1. According to Lemma 3.2.14 we can conclude that    n n Kj (x − t)   cMk Ml dμ (t) ≤ j +1 (j + 1) j MN IN j =MN +1

j =MN +1

cMk Ml ≤ MN ≤

∞  j =MN +1



1 1 − j j +1



cMk Ml , MN2

so (4.109) is proved. Now let x ∈ IN (xk ek ) , 1 ≤ xk ≤ mk − 1, k = 0, . . . , N − 1. Then    n n Kj (x − t)   cMk cMk dμ (t) ≤ ≤ ln (4.111) j +1 MN (j + 1) MN IN j =MN +1

j =MN +1

and also (4.110) is proved so the proof is complete.



Now, we prove that Nörlund logarithmic means do not form approximate identity. In particular, we show that the kernels of these means are not uniformly integrable. Lemma 4.9.4 Let qA = M2A + M2A−2 + . . . + M0 .

4.9 Riesz and Nörlund Logarithmic Kernels and Means

229

Then, for some c > 0, Hq ≥ c log qA . A 1 Proof Set Nn =: ln Hn =

n−1  Dn−k k=1

k

.

Then we have that M2A−2 +...+M0 −1



NqA (x) =

k=1

+

M2A +...+M  0 −1 k=M2A−2 +...+M0

1 DM2A +M2A−2 +...+M0 −k (x) k

(4.112)

1 DM2A +M2A−2 +...+M0 −k (x) k

=: I + I I. Since k < M2A−2 + . . . + M0 , then DM2A +...+M0 −k (x) = DM2A (x) + r2A DM2A−2 +...+M0 −k (x). This gives that I = lqA−1 DM2A (x) + r2A NM2A−2 +...+M0 (x).

(4.113)

Moreover, by using Abel transformation, we get that II =

M2A +...+M  0 −1 k=M2A−2 +...+M0

1 DM2A +M2A−2 +...+M0 −k k

=

K1 M2A + . . . + M0 − 1

−

(M2A + . . . + M0 − 2) KM2A +...+M0 −2 M2A + . . . + M0 − 1

+

M2A +...+M  0 −2 k=M2A−2 +...+M0

M2A + M2A−2 + . . . + M0 − k KM2A +M2A−2 +...+M0 −k . k(k + 1)

230

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

Since (for details see Corollary 3.2.9) Kn 1 ≤ c, for all n ∈ N, we obtain that I I 1 ≤ +

2 M2A + . . . + M0 − 1

(4.114)

2 (M2A + . . . + M0 − 2) M2A + . . . + M0 − 1

+2

M2A +...+M  0 −2 k=M2A−2 +...+M0

M2A + M2A−2 + . . . + M0 − k < C < ∞. k(k + 1)

Hence, by (4.112), (4.113) and (4.114), we get that Nq ≥ lq DM + r2A Nq − c. A 1 A−1 2A A−1 1 We now discuss the right-hand side of this inequality, more exactly we give a lower bound for it. First, we consider the integral on the set Gm \ I2A :   =

Gm \I2A

Gm \I2A

 lq

A−1

 Nq

 DM2A (x) + r2A NqA−1 (x)dμ(x)

A−1

= NqA−1 1 −

 (x)dμ(x)



 Nq I2A

A−1

 (x)dμ(x)

Nq (0) = NqA−1 1 − A−1 . M2A Next, we note that on the set I2A we have that 

 lq I2A

A−1

 Nq (0) DM2A (x) + r2A NqA−1 (x)dμ(x) = lqA−1 − A−1 . M2A

It follows that 2NqA−1 (0) Nq ≥ Nq + lq − c. A 1 A−1 1 A−1 − M2A Moreover, according to a simple estimation, Nn (0) =

n−1  n−k k=1

k

n−1  1 =n − n + 1 = nln + O(n). k k=1

(4.115)

4.9 Riesz and Nörlund Logarithmic Kernels and Means

231

Therefore, since qA−1 ≤ M2A−2 (1 +

1 4 1 1 + + . . .) = M2A−2 ≤ M2A , 4 16 3 3

we obtain that 2 2qA−1 lqA−1 NqA−1 (0) = + o(1) lqA M2A M2A lqA ≤

(4.116)

2 lqA−1 + o(1). 3 lqA

Finally, by using (4.115) and (4.116), we conclude that Hq = 1 Nq A 1 A 1 lqA ≥

1 Nq + lqA−1 − 2 lqA−1 − o(1) A−1 1 lqA lqA 3 lqA

≥

1 Nq + 1 lqA−1 − o(1) A−1 1 lqA 3 lqA

≥

1 Nq + 1 − o(1) A−1 1 lqA 6

≥

lqA−1 Hq + 1 − o(1) A−1 1 lqA 6

≥

lqA−1 Hq + 1 . A−1 1 lqA 8

By iterating this estimate, we get A−1 A−1  lqk c  Hq ≥ 1 ≥ k ≥ cA ≥ c log qA . A 1 8 lqA 8A k=1

The proof is complete.

k=1



232

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

We finish this Section by presenting some convergence results. First we note that Corollaries 4.5.16 and 4.8.3 immediately implies the following: Corollary 4.9.5 Let n ∈ N. Then, for any n, N ∈ N+ ,  HMn (x)dμ(x) = 1,

(4.117)

Gm



  HMn (x) dμ(x) ≤ c < ∞,

sup

(4.118)

n∈N Gm



Gm \IN

  HMn (x) dμ(x) → 0, as n → ∞.

(4.119)

Since the Riesz means are examples of T means, applying Corollary 4.9.1 we immediately get the following results. Corollary 4.9.6 Let n ∈ N and f ∈ Lp (Gm ). Then Rn f − f p → 0 as n → ∞. Analogously to Theorem 1.9.2, by using Lemma 4.9.4, we obtain: Corollary 4.9.7 There exists f ∈ L1 (Gm ) such that Ln f − f 1  0, as n → ∞. Moreover, it is known that (see introduction of this Chapter) almost everywhere convergence does not hold for any f ∈ L1 (Gm ), but we have positive results for some subsequences of Nörlund logarithmic means. In particular, Theorem 4.6.4 and Corollary 4.9.5 implies Corollary 4.9.8 Let 1 ≤ p < ∞ and f ∈ Lp (Gm ). Then LMn f − f p → 0, as n → ∞. Next we investigate almost everywhere and pointwise convergence of Riesz and Nörlund logarithmic means. Since Riesz means are examples of T means, by applying Corollary 4.9.1 we immediately get the following results: Corollary 4.9.9 Let n ∈ N and f ∈ Lp (Gm ). Then lim Rn f (x) = f (x)

n→∞

for all Vilenkin-Lebesgue points x of f .

4.10 Final Comments and Open Questions

233

Corollary 4.9.10 Let f ∈ L1 (Gm ) be continuous at the point x. Then Rn f (x) → f (x), as n → ∞. Moreover, Theorem 4.8.3 immediately imply the following: Corollary 4.9.11 Let n ∈ N and f ∈ Lp (Gm ). Then lim LMn f (x) = f (x)

n→∞

for all Lebesgue points x of f . Corollary 4.9.12 Let f ∈ L1 (Gm ) be continuous at the point x. Then LMn f (x) → f (x), as n → ∞.

4.10 Final Comments and Open Questions (1) All the results in Sect. 4.3 (Theorem 4.3.2, Corollary 4.3.3, Theorem 4.3.4 and Corollary 4.3.5) can be found in Moore [226] and Tephnadze [355]. (2) Lemmas 4.4.1, 4.4.2, 4.4.3, 4.4.4 and 4.4.5 can be found in [272]. (3) Lemma 4.4.6 are due to Baramidze et al. [22]. (4) Lemma 4.4.7 was proved in Baramidze et al. [24]. (5) Lemma 4.4.8 and Lemma 4.4.9 were proved in [25]. (6) Corollary 4.4.10 is due to Baramidze et al. [22]. (7) The proof of Corollary 4.4.11 was presented in Baramidze et al. [24]. (8) Lemma 4.4.12 can be found in Blahota et al. [42]. (9) Corollary 4.4.13 is due to Baramidze et al. [22]. (10) The proof of Lemma 4.4.14 was given in Baramidze et al. [24]. (11) Lemma 4.4.15, Corollary 4.4.16, Remark 4.4.17 can be found in Blahota et al. [42]. (12) The proofs of Lemmas 4.4.18 and 4.4.19 were presented in [25]. (13) Lemmas 4.5.1, 4.5.2, 4.5.3 and 4.5.4 are due to Tutberidze [384]. (14) Lemma 4.5.5 was proved in Baramidze et al. [23]. (15) Corollary 4.5.6 can be found in Nadirashvili [232]. (16) The proof of Lemma 4.5.7 was presented in Nadirashvili [232]. (17) Lemmas 4.5.8, 4.5.9 and 4.5.10 are due to Tutberidze [384]. (18) The proof of Corollary 4.5.11 is given in Baramidze et al. [23]. (19) Lemma 4.5.13, Corollary 4.5.14 can be found in Baramidze et al. [23]. (20) Corollary 4.5.15 was presented in Nadirashvili [232]. (21) Corollary 4.5.16 was proved in Baramidze et al. [22]. (22) Corollary 4.5.17 can be found in Nadirashvili [232].

234

4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces

(23) All results in Sect. 4.6 concerning Nörlund means are presented and proved in Baramidze et al. [22] and concerning T means in Baramidze et al. [23]. (24) All the results in Sect. 4.7 concerning Nörlund means were proved by Baramidze et al. [24] and concerning T means by Nadirashvili [232]. (25) The proofs of Theorem 4.8.1, Corollary 4.8.2, Theorem 4.8.3 and Corollary 4.8.4 can be found in Baramidze et al. [22] (26) Theorem 4.8.5, Corollary 4.8.6, Theorem 4.8.7 and Corollary 4.8.8 are given and proved in Baramidze et al. [23]. (27) The proof of Corollary 4.9.1 can be found in Baramidze et al. [23]. (28) The proof of Corollary 4.9.2 can be found in Nadirashvili [232]. (29) The proof of Lemma 4.9.3 were given in Tephnadze [341]. (30) Lemma 4.9.4 can be found in [274]. (31) Corollary 4.9.5 is due to Baramidze et al. [22]. (32) The proof of Corollary 4.9.6 was given in Baramidze et al. [23]. (33) Corollary 4.9.7 can be found in [274]. (34) Corollary 4.9.8 is due to Baramidze et al. [23]. (35) Corollaries 4.9.9 and 4.9.10 are presented and proved in Baramidze et al. [23]. (36) Corollaries 4.9.11 and 4.9.12 is due to Baramidze et al. [23, 26]. (37) Nörlund logarithmic means was studied by Baramidze [19]. (38) The behavior of Cesàro means and some generalizations of this summability method with respect to Vilenkin and trigonomethric systems were studied by Akhobadze [4–7], Akhobadze and Zviadadze [8], Goginava [125], Shavardenidze [298], Tetunashvili [369–372]. In the two-dimensional case approximation properties of Cesàro means were considered by Nagy [234], Persson and Tepnadze [365–368], Tevzadze [373], Tutberidze [383] and Zhizhiashvili [458, 459, 461]. (39) In the two-dimensional case approximation properties of Nörlund means were investigated by Nagy (see [235–237]). (40) A general method of summation, the so called θ -summability, was considered by Butzer and Nessel [63], Bokor and Schipp [50], Schipp and Bokor [291], Schipp and Szili [293], Szili and Vertesi [338] (see also [308]), Weisz [422]– [425]. In particular, it was proved that if the kernel of these summability methods can be estimated by a non-increasing integrable function, then the θ -means of a function f ∈ L1 (R) converge almost everywhere to f. This convergence result was also proved for the θ -means of Vilenkin-Fourier series (see also Stein and Weiss [329]). As special cases they considered the Weierstrass, Picar, Bessel, Fejér, de La Vallee-Poussin and Riesz summations. Some general summability methods, which are called -means, were recently investigated by Blahota, Nagy [63] and Blahota, Nagy and Tephnadze [46].

4.10 Final Comments and Open Questions

235

The next two open problems have independent interest but the answer will give also basic information to derive some new results concerning convergence and divergence of some summability methods and their maximal operators with respect to bounded Vilenkin systems on the martingale Hardy spaces: Open Problem Let n > 2, n ∈ N, and qn = M2n + M2n−2 + . . . + M2 + M0 . Does there exist an absolute constant c > 0 such that  Pq

n−1

2  cM2k   (x) ≥ , for x ∈ I2k+1 e2k−1 + el2k , k = 0, 1, . . . , n, nM2n

where Pn =:

1 1  Dk , ln =: ln k k n−1

n−1

k=1

k=1

is the so-called Riesz logarithmic kernel with respect to bounded Vilenkin systems. Remark 4.10.1 In this case an analogous result is unknown also for the Walsh system. Remark 4.10.2 For the bounded Vilenkin system in [274] it was proved that for qnk = M2nk + M2nk −2 + M2 + M0 , there exist absolute positive constants c1 and c2 , such that c1 nk ≤ Hqnk ≤ c2 nk , 1

where 1  Dk , ln n−k n

Hn =:

ln =:

k=1

n−1  1 k=1

k

is the so-called Nörlund logarithmic kernel with respect to Vilenkin systems. We also note that an analogous result for the Walsh system can be found in Gàt and Goginava [114]. To obtain a similar type of lower estimates for the Riesz logarithmic kernels, a method of Gàt and Goginava [114] (see also [274]) can be used. Open Problem Does there exist necessary and sufficient condition for the indexes nk , k ∈ N, such that c1 ≤ Hnk 1 ≤ c2 , where c1 and c2 are absolute constants.

Chapter 5

Theory of Martingale Hardy Spaces

5.1 Introduction The theory of classical Hardy spaces defined on Rn is a very important topic of harmonic analysis and summability theory. These Hardy spaces are investigated in many books, for example in Duren [86], Stein [326, 327], Stein and Weiss [329], Lu [215], Uchiyama [385], Grafakos [148] and Weisz [429]. In the early 70s of the last century, with the development of the theory of classical Hardy spaces, the theory of martingale Hardy spaces was born. Until now, most of the important facts in harmonic analysis have been found to have their satisfactory counterparts in the martingale setting. For example, in martingale setting, the duality between H1 and BMO, and the Doob’s maximal inequality can be found in Garsia [105]. The Burkholder martingale transforms [57] can be considered as an analogue to the classical singular integral operators. On the other hand, the theory of martingale Hardy spaces has influenced the development of harmonic analysis. For example, the atomic decomposition of Hp , which is one of the most powerful tools in harmonic analysis nowadays, was first shown in a martingale setting by Herz [162]. Later, the theory of atomic decomposition of martingale spaces was investigated in Weisz [400]. The goodλ inequality, which is a useful tool to compare the integrability of two related measurable functions, was discovered by Burkholder and Gundy [58, 62] in a martingale setting. A much more simplified proof of T (b) theorem was given by Coifman et al. [71] via a martingale approach. The theory of martingale Hardy spaces was considered in the books [105, 213, 400, 423]. Moreover, the several applications of martingale theory to Fourier analysis were developed by Weisz [400, 423, 429]. The atomic decomposition, which is investigated in Sect. 5.2, plays an important role in this book. The first version of a similar decomposition can be found in Coifman and Weiss [70] for the classical case and in Herz [161] for the martingale © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 L.-E. Persson et al., Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series, https://doi.org/10.1007/978-3-031-14459-2_5

237

238

5 Theory of Martingale Hardy Spaces

case. The atomic decomposition is a useful characterization of the Hardy spaces by the help of which some boundedness results, duality theorems, inequalities and interpolation results can be proved. The atoms are relatively simple and easy to handle functions. The advantage of this decomposition is that many theorems can be proved by proving them only for atoms. In Sect. 5.3, the interpolation spaces between Hardy spaces are characterized. These results are due to Weisz [399, 400] and to classical Hardy spaces to Fefferman et al. [92] (see also Weisz [429]). In Sect. 5.4, we give sufficient conditions such that a σ -sublinear operator is bounded from the Hardy space Hp to Lp . These results play a central role in proving boundedness of summability means and their maximal operators. The results of this Section were proved in Weisz [429]. At the end of the Chapter, in Sect. 5.5, we give some examples of p-atoms and Hp -martingales for 0 < p ≤ 1, which help to prove sharpness of our main results in the next Chapters. In particular, we unify counterexamples of p-atoms which are given in papers of Blahota et al. [40, 41], Tephnadze [345–347, 352, 354, 355, 359, 363] and counterexamples of Hp -martingales given in Goginava [131], Tephnadze [342, 355, 357–360, 363], Persson et al. [271–274], Persson et al. [275, 276, 364] and Blahota et al. [38, 42, 44].

5.2 Martingale Hardy Spaces and Modulus of Continuity First we define the martingale Hardy spaces. Definition 5.2.1 For 0 < p ≤ ∞, the martingale Hardy space Hp consists of all Vilenkin martingales f = (fn , n ∈ N) for which f Hp =: f ∗ p < ∞, where f ∗ =: sup |fn | n∈N

was defined in Sect. 2.3. By (2.9), we immediately get the equivalence between Hp and Lp for p > 1. Corollary 5.2.2 If 1 < p ≤ ∞, then Hp ∼ Lp .

5.2 Martingale Hardy Spaces and Modulus of Continuity

239

Vilenkin-Fourier coefficients f(i) of a martingale f = (fn , n ∈ N) must be defined in a slightly different manner: f(i) =: lim



k→∞ Gm

fk ψ i dμ, (i ∈ N) .

If f ∈ L1 (Gm ), then limn→∞ fn = f in the L1 -norm and so 

 lim

k→∞ Gm

fk ψ i dμ =

Gm

f ψ i dμ.

Thus, the Vilenkin-Fourier coefficients of a martingale are the same as the ones of f ∈ L1 (Gm ). We can define the partial sums Sn f of a Vilenkin-Fourier series of a martingale f by Sn f =:

n−1 

f(k) ψk ,

(n ∈ N+ ) .

k=0

If f ∈ L1 (Gm ), then  Sn f (x) =

f (t) Dn (x − t) dμ (t) = (f ∗ Dn ) (x) . Gm

We have seen in Sect. 2.3 that the following important result holds: Lemma 5.2.3 If f ∈ L1 (Gm ), then the sequence (En f, n ∈ N) is a regular martingale and the maximal functions are also given by    1  f (x) = sup f (t) dμ (t) .  |I In (x) n∈N n (x)| ∗

Moreover, for 0 < p ≤ ∞, f Hp

    = sup SMn f . n∈N

p

The concept of modulus of continuity ωHp in martingale Hardy spaces Hp (p > 0) is defined by

ωHp

1 ,f Mn



=: f − SMn f Hp .

240

5 Theory of Martingale Hardy Spaces

We need to understand the meaning of the expression f − SMn f in the case when f is a martingale and SMn f is a function. Hence, we give an explanation in the following remark. Remark 5.2.4 Let 0 < p ≤ 1. Since SMn f = fn , for f = (fn , n ∈ N) ∈ Hp and 

   SMk fn , k ∈ N = SMk SMn f, k ∈ N   = SM0 f, . . . , SMn−1 f, SMn f, SMn f, . . . = (f0 , . . . , fn−1 , fn , fn , . . .) ,

we obtain that (f − SMn f, n ∈ N) is a martingale for which   f − SMn f k =



0, k = 0, . . . . , n, fk − fn , k ≥ n + 1.

(5.1)

Remark 5.2.5 Since f Hp ∼ f p when 1 < p < ∞, according the two-sided estimates (see Proposition 1.9.12) 1 ωp 2



1 ,f Mn



≤ f − SMn f p ≤ ωp



1 ,f Mn

 ,

we obtain that

ωHp

1 ,f Mn



 ∼ ωp

1 ,f Mn



and   1 f − SMn f ≤ EMn f, Lp ≤ f − SMn f . p p 2

(5.2)

5.3 Atomic Decomposition of the Martingale Hardy Spaces Hp

241

5.3 Atomic Decomposition of the Martingale Hardy Spaces Hp First we introduce the concept of p-atoms. Definition 5.3.1 A bounded measurable function a is a p-atom if there exist a Vilenkin interval I of the form In (x) such that  a dμ = 0, a∞ ≤ μ (I )−1/p , supp (a) ⊂ I. (5.3) I

Next, we prove the atomic decomposition for the martingale Hardy spaces Hp , where 0 < p ≤ 1. Theorem 5.3.2 A martingale f = (fn , n ∈ N) is in Hp (0 < p ≤ 1) if and only if there exist a sequence (ak , k ∈ N) of p-atoms and a sequence (μk , k ∈ N) of real numbers such that, for every n ∈ N, ∞ 

μk En a k = fn , a.e.,

(5.4)

k=0

where ∞ 

|μk |p < ∞.

k=0

Moreover, f Hp ∼ inf

∞ 

1/p |μk |

p

,

(5.5)

k=0

where the infimum is taken over all decompositions of f of the form (5.4). Proof Assume that f ∈ Hp and consider the following stopping times ρk with respect to the σ -algebras (Fn , n ∈ N), ( ' ρk =: inf n ∈ N : |fn | > 2k

(k ∈ Z) .

Let Fnk ∈ Fn−1 be the smallest set which contains {ρk = n} . In other words, if {ρk = n} ∈ Fn

242

5 Theory of Martingale Hardy Spaces

k k is decomposed into the disjoint union of Vilenkin intervals In,l ∈ Fn and In,l ∈ k Fn−1 denotes the Vilenkin interval which contains In,l , then

Fnk =



k In,l .

l

Define a new family of stopping times τk by ( ' k . τk (x) =: inf n ∈ N : x ∈ Fn+1 Then {ρk (x) = n} implies x ∈ Fnk , which implies that {τk (x) ≤ n − 1} . In other words ρk < τk on the set {ρk = ∞}. It is easy to see that μ (τk = ∞) ≤

∞ 

μ Fnk

(5.6)

n=1

≤

∞   n=1

∞ 

 k k μ In,l ≤R μ In,l

l

n=1

l

= Rμ (ρk =

∞) = Rμ f ∗ > 2k , where R = supn mn . The sequence of the stopping times τk is obviously nondecreasing. Since, by (5.7), p μ (τk = ∞) ≤ R2−kp f ∗ p → 0 as k → ∞, we can see that lim μ (τk = ∞) = 1.

k→∞

Thus lim τk = ∞ a.e.

k→∞

and so lim fnτk = fn

k→∞

a.e.

(n ∈ N) .

5.3 Atomic Decomposition of the Martingale Hardy Spaces Hp

243

On the other hand, lim τk = 0,

k→−∞

  because fnτk  ≤ 2k . Hence, fn =



 τ fn k+1 − fnτk .

(5.7)

k∈Z

It is easy to see that fn =



 τ  χ{τk ρ)

(ρ ≥ 0) ,

(5.10)

where λ is the Lebesgue and μ the Haar measure. Indeed, by the definition of f., we have that f.(μ (|f | > ρ)) ≤ ρ and, thus,   λ f. > ρ ≤ μ (|f | > ρ) . On the other hand, since f.is continuous on the right,    f. λ f. > ρ ≤ ρ and, therefore,   μ (|f | > ρ) ≤ λ f. > ρ . so that (5.10) holds. Note that if f. is continuous at a point t, then ρ = f.(t) is equivalent to t = μ (|f | > ρ) . The Lorentz space Lp,q (Gm ) consists of all measurable functions f for which

 f p,q =:

∞ 0

dt f.(t)q t q/p t

1/q < ∞ if 0 < p < ∞, 0 < q < ∞

and f p,∞ =: sup t 1/p f.(t) < ∞ if 0 < p < ∞. t >0

5.4 Interpolation Between Hardy Spaces Hp

247

In case of p = ∞, let Lp,∞ (Gm ) = L∞ (Gm ) . Proposition 5.4.1 One has Lp,p (Gm ) = Lp (Gm ) ,

Lp,∞ (Gm ) = weak − Lp (Gm )

(0 < p < ∞) .

Proof By (5.10), the first statement can be given directly if we replace q by p. To prove the second one, we note that f.(t) = ρ implies that   λ f. > ρ ≤ t. Thus,  1/p ρμ (|f | > ρ)1/p = ρλ f. > ρ ≤ t 1/p f.(t) and so f weak−Lp ≤ f p,∞ . On the other hand, for a given ε > 0, we can choose t such that f. is continuous in t and f p,∞ ≤ t 1/p f.(t) + ε. Set ρ = f.(t) . Then   λ f. > ρ = t and  1/p f p,∞ ≤ t 1/p f.(t) + ε = ρλ f. > ρ + ε ≤ f weak−Lp + ε which implies the equality in the reversed direction. The proof is complete.



One can prove that the Lorentz spaces Lp,q (Gm ) increase as the second exponent q increases, namely, for 0 < p < ∞ and 0 < q1 ≤ q2 ≤ ∞ one has Lp,q1 (Gm ) ⊂ Lp,q2 (Gm ). Now we generalize the definition of the Hardy spaces in a similar way. For 0 < p, q ≤ ∞ the Hardy-Lorentz spaces Hp,q (Gm ) consist of all martingales f for

248

5 Theory of Martingale Hardy Spaces

which f Hp,q =: f ∗ p,q < ∞. Of course, for p = q, we get back the Hardy spaces Hp (Gm ). Next we summarize some results without proofs from the books Bergh and Löfström [30] or Bennett and Sharpley [29]. Suppose that A0 and A1 are quasinormed spaces embedded continuously in a topological vector space A. One way to describe the real method of interpolation, the interpolation spaces between A0 and A1 are defined by means of the interpolating (Peetre) functional K(t, f, A0 , A1 ). For f ∈ A0 + A1 , this interpolating functional is defined by K (t, f, A0 , A1 ) =:

inf

f =f0 +f1

* + f0 A0 + t f1 A1 ,

where the infimum is taken over all choices of f0 and f1 such that f0 ∈ A0 , f1 ∈ A1 and f = f0 + f1 . The real interpolation space (A0 , A1 )θ,q is introduced as the space of all functions f in A0 + A1 such that

 f (A0 ,A1 )θ,q =:

∞

t −θ K (t, f, A0 , A1 )

0

q dt t

1/q 0

where 0 < θ < 1. We use the conventions (A0 , A1 )0,q = A0 and (A0 , A1 )1,q = A1 for each 0 < q ≤ ∞. Suppose that B0 and B1 are also quasi-normed spaces embedded continuously in a topological vector space B. A map T : A0 + A1 → B0 + B1 is said to be quasilinear from (A0 , A1 ) to (B0 , B1 ) if, for given a ∈ A0 + A1 and ai ∈ Ai with a0 + a1 = a, there exist bi ∈ Bi , (i = 0, 1), satisfying T a = b0 + b1 and bi Bi ≤ Ki ai Ai

(Ki > 0, i = 0, 1) .

(5.11)

5.4 Interpolation Between Hardy Spaces Hp

249

Lemma 5.4.2 If the operator T is linear and bounded from Ai to Bi , (i = 0, 1), then T is quasilinear. Proof The proof follows from the fact that bi = T ai (i = 0, 1).



The following theorem, the so called interpolation theorem, shows that the boundedness of a quasilinear operator is hereditary for the interpolation spaces. Theorem 5.4.3 (Interpolation Theorem) If 0 < q ≤ ∞, 0 ≤ θ ≤ 1 and T is a quasilinear map from (A0 , A1 ) to (B0 , B1 ), then T : (A0 , A1 )θ,q → (B0 , B1 )θ,q and, moreover, T a(B0 ,B1 )θ,q ≤ K01−θ K1θ a(A0, A1 )θ,q . where K0 and K1 are the constants in (5.11). The reiteration theorem below is another of the most important general results in the real interpolation theory. It says that the interpolation space of two interpolation spaces is also an interpolation space of the original spaces. Theorem 5.4.4 (Reiteration Theorem) Suppose that 0 ≤ θ0 , θ1 ≤ 1, θ0 = θ1 , 0 < q0 , q1 ≤ ∞ and Xi = (A0 , A1 )θi ,qi (i = 0, 1). If 0 < η < 1 and 0 < q ≤ ∞, then (X0 , X1 )η,q = (A0 , A1 )θ,q with equivalent norms, where θ = (1 − η) θ0 + ηθ1 . It is known that the real interpolation spaces of the Lp (Gm ) spaces are Lorentz spaces and that the real interpolation spaces of Lorentz spaces are Lorentz spaces, too. The key to understand this fact is the following result. Theorem 5.4.5 Let 0 < r < ∞ and 0 < θ < 1. If 0 < q < ∞ or q = ∞, r < p, then 

Lr , L∞

 θ,q

= Lp,q ,

1−θ 1 = . p r

250

5 Theory of Martingale Hardy Spaces

Proof First we prove the crucial equivalence   K t, f, Lr , L∞ ∼



tr

1/r f.(s) ds r

(5.12)

.

0

For a fixed t, take  f0 (x) =:

f (x) − f.(t r ) f (x) / |f (x)| , if |f (x)| > f.(t r ) , 0, else,

and f1 =: f − f0 . We define *  + E =: |f | > f. t r . It is easy to see that μ(E) ≤ t r and f.is constant on [μ(E), t r ]. Therefore,   K t, f, Lr , L∞ ≤ f0 r + t f1 ∞

 1/r   r r   . |f | − f t = dμ + t f. t r E



μ(E) 

=

 r f.(s) − f. t r ds

1/r

 +

0



  r r f. t ds

1/r

0 tr

=

tr

  r f.(s) − f. t r ds

0

1/r



tr

+

  r r f. t ds

0



tr

≤C

1/r f.(s)r ds

,

0

which shows the first part of (5.12). For the converse inequality assume that f = f0 + f1 with f0 ∈ Lr (Gm ) and f1 ∈ L∞ (Gm ). By using the inequality, μ (|f | > ρ0 + ρ1 ) ≤ μ (|f | > ρ0 ) + μ (|f | > ρ1 ) ,

1/r

5.4 Interpolation Between Hardy Spaces Hp

251

we obtain that, for any , 0 <  < 1, f.(s) ≤ f.0 ((1 − ) s) + f.1 (s) . Since f.1 is non-increasing, we can conclude that 

tr

1/r f.(s) ds r



tr

≤

0

r  f.0 ((1 − ) s) ds

1/r

 +

0



1/r f.1 (s) ds r

0 ∞

r f.0 ((1 − ) s) ds

≤

tr

1/r

+ t f.1 (0)

0

= (1 − )−1/r f0 r + t f1 ∞ . The proof of (5.12) is complete by letting ε → 0. First suppose that 0 < q < ∞. By (5.12), we have that

 f (Lr ,L∞ )θ,q =

∞

t

−θ

0

⎛  ⎝ ≤C

∞

q dt K t, f, L , L t 

t −θq

r



0

 =C

tr

∞

1/q (5.13)

q/r f.(s)r ds

0

∞

t

−θq/r



0

t

f.(s) ds

q/r

r

0

⎞1/q dt ⎠ t dt t

1/q .

Hence, we can conclude that



f (Lr ,L∞ )θ,q

∞

dt ≤C t f.(t)q t 0

 ∞ 1/q dt =C t q/p f.(t)q t 0

1/q

q/r−θq/r

= C f p,q . For the case r ≤ q this follows from Hardy’s inequality (1.33) but since f. is nonincreasing it holds also when r ≥ q, see Lemma 5.4.7 bellow and, in a more general form, Lemma 3.2 in [264].

252

5 Theory of Martingale Hardy Spaces

Conversely, by using (5.12) and (5.13) and the fact that f. is non-increasing, we obtain that  f (Lr ,L∞ )θ,q ≥ C

∞

t

−θq/r



0

 ≥C

t

f.(s) ds

q/r

r

0 ∞

t q/r−θq/r f.(t)q

0

dt t

dt t

1/q

1/q

= C f p,q . For q = ∞, we have that   f (Lr ,L∞ )θ,∞ = sup t −θ K t, f, Lr , L∞ t >0

≤ C sup t

−θ



1/r

tr

f.(s) ds r

0

t >0

= C sup t

(5.14)

−θ/r



t

f.(s)r s r/p s −r/p ds

1/r

0

t >0

≤ C f p,∞ sup t −θ/r



t

s −r/p ds

1/r

0

t >0

= C f p,∞ , because r < p. On the other hand, by (5.14) and the fact that f. is non-increasing, we have that f (Lr ,L∞ )θ,∞ ≥ C sup t −θ/r t >0



t

f.(s)r ds

1/r

0

≥ C sup t (1−θ)/r f.(t) t >0

= C f p,∞ . 

The proof is complete. Applying the reiteration theorem, we get the following general result.

Corollary 5.4.6 Suppose that 0 < η < 1 and 0 < p0 , p1 , q0 , q1 , q ≤ ∞. If p0 = p1 , then  p ,q  L 0 0 , Lp1 ,q1 η,q = Lp,q ,

1−η 1 η = + . p p0 p1

(5.15)

5.4 Interpolation Between Hardy Spaces Hp

253

In particular, for the case p0 = q0 , p1 = q1 , we have that  p  L 0 , Lp1 η,q = Lp ,

1−η 1 η = + . p p0 p1

Proof Let 0 < η < 1 and 0 < r ≤ p0 , p1 , q0 , q1 , q and 1 1 − θi = pi r Notice that

=

1 p



1−θ r .

(i = 0, 1) and θ = (1 − η) θ0 + ηθ1 .

If p0 = p1 , then Theorems 5.4.4 and 5.4.5 imply that

Lp0 ,q0 , Lp1 ,q1

 η,q

=

  Lr , L∞ θ

0 ,q0

  , Lr , L∞ θ

  = Lr , L∞ θ,q = Lp,q .

1 ,q1

η,q



The proof is complete.

The next lemma is an extension of Hardy’s inequality (1.33) to all 0 < q < ∞ and was proved by Riviere and Sagher [282]. A simple proof even in more general case can be found in Persson [264], see also Lemma 3.2 in [266]. The sharp constants in (5.16) and the reversed inequality are known in all cases, see [31]. Here we give another proof by using interpolation: Lemma 5.4.7 If f ≥ 0 is a non-increasing function on (0, ∞) and 0 < q ≤ ∞, 0 < s < q, then



∞ 1 0

t



q

t

f (u)du 0

t

s dt

t

1/q



∞

≤ Cq,s

f (t) t 0

Proof Consider the linear Hardy operator T defined by 1 Tf (t) =: t



q s dt

t

f (u)du. 0

Hölder’s inequality implies Tf (t) ≤ t −1/p f p . Hence (Tf )∼ (t) ≤ t −1/p f p

t

1/q .

(5.16)

254

5 Theory of Martingale Hardy Spaces

and we conclude that T =: Lp → L1,∞

(1 ≤ p ≤ ∞)

is bounded. Moreover, it is obvious that T maps L∞ into itself. Therefore, by applying Theorem 5.4.3 and Corollary 5.4.6, we get that

T : L1 , L∞

→ L1,∞ , L∞

θ,q

θ,q

and, thus, T : Lp,q → Lp,q

(1 < p < ∞, 0 < q ≤ ∞)

(5.17)

is also bounded. Since f is non-increasing, so is 1 v



v

f (u) du. 0

Thus f.(t) = f (t) and the non-increasing rearrangement of the function 1 v 1 t v 0 f (u) du at a point t is t 0 f (u) du. Therefore, (5.17) implies that



∞ 1 0

t



q

t

f (u)du

t

0

q/p dt



t

 ≤ Cq,p

∞

q q/p dt

f (t) t 0

1/q

t

which is exactly the desired inequality (5.16) with s = q/p < q.



The proof of the following theorem is based on the atomic decomposition. Theorem 5.4.8 Let f ∈ Hp (Gm ), y > 0 and 0 < p ≤ 1. Then f can be decomposed into the sum of two martingales g and h such that

 g∞ ≤ Cy

and hHp ≤ Cp

 {f ∗ >y}

f

Proof Choose N ∈ Z such that 2N−1 < y ≤ 2N . Set g =:

∞  N   k=−∞ j =0

μk,j,i a k,j,i

i

and h =:

∞  ∞   k=N+1 j =0 i

μk,j,i a k,j,i ,

 ∗ p

1/p dμ

.

5.4 Interpolation Between Hardy Spaces Hp

255

where the atomic decomposition was defined in the proof of Theorem 5.3.2. Then f = g + h. Inequality (5.8) and the fact that  τ f k  ≤ 2 k n

imply that |g| ≤ C0 2N+1 ≤ Cy, which is the first inequality. Moreover, by (5.5) and (5.10), ∞  ∞     μk,j,i p

hHp ≤ Cp

k=N+1 j =0 i

= 3p

∞ 

2kp μ (τk = ∞)

k=N+1 ∞ 

≤ Cp



2kp μ f ∗ > 2k

k=N+1 ∞ 

= Cp



(2p )k μ (2p )k < (f ∗ )p ≤ (2p )k+1

k=N+1

 ≤ Cp  ≤ Cp

 {f ∗ >2N+1 }

{f ∗ >y}

f∗

p

dμ

 ∗ p dμ, f

so also the second estimate is proved and the proof is complete.



Now the interpolation spaces between the Hardy spaces Hp (Gm ) can be identified. Theorem 5.4.9 If 0 < θ < 1, 0 < p0 ≤ 1 and 0 < q ≤ ∞, then (Hp0 , H∞ )θ,q = Hp,q ,

1−θ 1 = p p0

Proof Note that H∞ (Gm ) = L∞ (Gm ). Let f ∈ Hp,q (Gm ) and f8∗ be the nonincreasing rearrangement of f ∗ . Choose y in Theorem 5.4.8 such that y = f8∗ (t p0 ) for a fixed t ∈ [0, 1]. For this y let us denote the two martingales in Theorem 5.4.8

256

5 Theory of Martingale Hardy Spaces

by gt and ht . By the definition of the K-functional, we have that K(t, f, Hp0 , H∞ ) ≤ ht Hp + t gt H ∞ . 0

According to Theorem 5.4.8, we get that

 ht Hp ≤ C

f ∗ >f8∗ (t p0 )

0



t p0

=C

 ∗ p0 f dμ

f8∗ (x)p0 dx

1/p0

1/p0 .

0

Consequently, for 0 < q < ∞ 

∞

t

q/p0  p  ∞

q dt t 0 dt −θq p ht hp f8∗ (x) 0 dx t ≤C 0 t t 0 0

−θ

0



∞

=C

t (1−θ)q/p0

0

 t q/p0 1 dt . f8∗ (x)p0 dx t 0 t

By using inequality (5.16), we obtain that 

∞ 0

t −θ ht Hp

 ∞

q dt q dt ≤C = C f ∗ p,q . t (1−θ)q/p0 f8∗ (t)q 0 t t 0

Furthermore, 

∞

t −θ gt H∞

0

q dt ≤C t



∞ 0

 q dt . t (1−θ)q f8∗ t p0 t

Substituting u = t p0 , we can see that 

∞

t

1−θ

0

 ∞

q dt q du gt H∞ ≤C = C f ∗ p,q u(1−θ)q/p0 f8∗ (u)q t u 0

and so f Hp

0

,H∞



 θ,q

∞

= 0

q dt  t −θ K t, f, Hp0 , H∞ t

1/q ≤ C f Hp,q .

5.4 Interpolation Between Hardy Spaces Hp

257

If q = ∞, then sup t t >0

−θ

ht Hp ≤ C sup t 0

−θ



1/p0 f8∗ (x) dx p0

0

t >0

= C sup t

t p0

−θ/p0



t

f8∗ (x)p0 x p0 /p x −p0 /p dx

0

t >0

≤ C f ∗ p,∞ sup t −θ/p0 = C f ∗

1/p0



t

x −p0 /p dx

1/p0

0

t >0

p,∞

and sup t 1−θ gt H∞ ≤ C sup t 1−θ f8∗ (t p0 ) ≤ C sup t t >0

t >0

1−θ p0

t >0

f8∗ (t) ≤ C f ∗ p,∞ .

Then f Hp

0 ,H∞

 θ,∞

  = sup t −θ K t, f, Hp0 , H∞ ≤ C f Hp,∞ . t >0

To prove the converse, consider the sublinear operator T : f #−→ f ∗ . By the definition of the Hardy spaces, T : H∞ → L∞

and

T : Hp0 → Lp0

are bounded. Therefore, for all 0 < q ≤ ∞, we get from Theorems 5.4.3 and 5.4.5 that     T : Hp0 , H∞ θ,q → Lp0 , L∞ θ,q = Lp,q   is bounded, too. That is to say, f ∈ Hp0 , H∞ θ,q implies f Hp,,q = f ∗ p,q ≤ C f Hp



0 ,H∞ θ,q

. 

This completes the proof of the theorem.

By applying the reiteration theorem (Theorem 5.4.4), we get the following main result concerning real interpolation between Hp,q spaces: Theorem 5.4.10 Suppose that 0 < η < 1 and 0 < p0 , p1 , q0 , q1 , q ≤ ∞. If p0 = p1 , then 

Hp0 ,q0 , Hp1 ,q1

 η,q

= Hp,q ,

1−η 1 η = + . p p0 p1

258

5 Theory of Martingale Hardy Spaces

In particular, if p0 = q0 and p1 = q1 , then 

H p0 , H p1

 η,p

= Hp ,

1−η 1 η = + p p0 p1

= Lp ,

η 1 =1−η+ . p p1

and, for 1 < p1 ≤ ∞, 

H1 , Lp1

 η,p

The interpolation spaces between the classical Hardy spaces were identified first by Fefferman et al. [92]. Let X be a martingale space, Y be a measurable function space. Then T : X → Y is called σ -sublinear operator, if for any α ∈ C,   ∞  ∞       fk  ≤ |T (fk )| T   k=1

and

|T (αf )| = |α||T (f )|.

k=1

If this inequality holds for finitely many functions, then the operator is called sublinear. As a consequence of the above results, we get the following interpolation theorem concerning Hardy-Lorentz spaces, which will be used several times in this Chapter. Corollary 5.4.11 If a sublinear or linear operator V is bounded from Hp0 (Gm ) to Lp0 (Gm ) (resp. to Hp0 (Gm )) and from Lp1 (Gm ) to Lp1 (Gm ) (p0 ≤ 1 < p1 ≤ ∞), then it is also bounded from Hp,q (Gm ) to Lp,q (Gm ) (resp. to Hp,q (Gm )) for each p0 < p < p1 and 0 < q ≤ ∞. Proof If the operator V is linear, then the result follows from Theorem 5.4.3 and Theorem 5.4.10. If V is sublinear, then for a = a0 + a1 , a0 ∈ Hp0 (Gm ), a1 ∈ Lp1 (Gm ), we have that |V a| ≤ |V a0 | + |V a1 | . Choose bi ∈ Lpi (Gm ) (i = 0, 1) such that |V a| = b0 + b1 and 0 ≤ bi ≤ |V ai |. Then bi pi ≤ |V ai |pi ≤ Cpi ai Hp , i

which shows that the operator |V | is quasilinear. Therefore the proof follows by just applying Theorem 5.4.3 

5.5 Bounded Operators on Hp Spaces

259

5.5 Bounded Operators on Hp Spaces In this Section, we give some sufficient conditions for an operator to be bounded from Hp to Lp . By using the atomic characterization of the Hardy spaces, the next two theorems can be easily proved. Theorem 5.5.1 Suppose that an operator T is σ -sublinear and for some 0 < p0 ≤ 1,  |T a|p0 dμ ≤ Cp < ∞

(5.18)

I

for every p0 -atom a, where I denotes the support of the atom. If T is bounded from Lp1 to Lp1 , (1 < p1 ≤ ∞) , then Tf p0 ≤ Cp0 f Hp .

(5.19)

0

Moreover, if p0 < 1, then we have the weak (1,1) type estimate sup ρ μ {x ∈ Gm : |Tf (x)| > ρ} ≤ C f 1

(5.20)

ρ>0

for all f ∈ L1 . Proof Suppose that a is a p0 -atom, with support I. By (5.18), Hölder’s inequality and the Lp1 boundedness of T , we obtain that    |T a|p0 dμ = |T a|p0 dμ + |T a|p0 dμ Gm

I

I

⎞p0 /p1 ⎛  |I |1−p0 /p1 + Cp0 ≤ ⎝ |T a|p1 dμ⎠ I

⎛ ⎞p0 /p1  |I |1−p0 /p1 + Cp0 ≤ Cp0 ⎝ |a|p1 dμ⎠ I

p0 /p1 1−p /p  |I | 0 1 + Cp0 = Cp0 . ≤ Cp0 |I |−p0 /p1 |I | Since T is σ -sublinear, Theorem 5.3.2 implies that  |T a|p0 dμ ≤ Gm

∞  k=0

p

p

|μk |p0 T ak p00 ≤ Cp0 f H0p , 0

260

5 Theory of Martingale Hardy Spaces

which proves (5.19). If p0 < 1, then we get from Corollary 5.4.11 that the operator T is bounded from H1,∞ to L1,∞ . Thus, by Lemma 2.3.5, sup ρ μ {x ∈ Gm : |Tf (x)| > ρ} = Tf L1,∞

ρ>0

≤ C f H1,∞ ≤ C f 1 , 

which proves (5.20). The proof is complete.

Note that the σ -sublinearity cannot be omitted (see Bownik et al. [51, 52, 447]). Theorem 5.5.2 Suppose that the operator T is σ -sublinear and, for some 0 < p0 < 1, * + sup ρ p0 μ x ∈ I : |Tf (x)| > ρ ≤ Cp0 < +∞

ρ>0

for every p0 -atom a, where I denotes the support of the atom. If T is bounded from Lp1 to Lp1 , (1 < p1 ≤ ∞) then Tf weak−Lp0 ≤ Cp0 f Hp . 0

Moreover, if f ∈ L1 , then sup ρ μ {x ∈ Gm : |Tf (x)| > ρ} ≤ f 1 .

ρ>0

Proof It is easy to see that  sup ρ p0 μ {x ∈ I : |Tf (x)| > ρ} ≤ ρ>0

|T a|p0 dμ I

 ≤ Cp0

|T a|

p1

p0 /p1 dμ

|I |1−p0 /p1

Gm

≤ Cp0 and so sup ρ p0 μ {|T a| > ρ} ≤ Cp0

ρ>0

(5.21)

5.5 Bounded Operators on Hp Spaces

261

for every cube p0 -atom a. By Theorem 5.3.2, f ∈ Hp0 has an atomic decomposition ∞ 

f =

μk a k .

k=0

For ρ > 0 we define gk =: |T ak |χ{|T ak |≤ρ/|μk |} ,

hk =: |T ak |χ{|T ak |>ρ/|μk |}

and Eρ =:

∞ 

{hk = 0}.

k=0

Since, in view of (5.21),  9 |μk |p0 ρ μ{hk = 0} = μ |T ak | > ≤ Cp0 p , |μk | ρ 0 we have that μ(Eρ ) ≤ Cp0 ρ −p0

∞ 

|μk |p0 .

k=0

On the other hand,  gk 1 = 

{|T ak |≤ρ/|μk |} ρ/|μk |

≤ 0

≤ Cp0



|T ak | dμ

μ(|T ak | > t) dt

ρ |μk |

1−p0 .

Hence μ

)∞ 

: |μk ||T ak | > ρ

) ≤ μ(Eρ ) + μ x ∈ Eρ :

k=0

≤ μ(Eρ ) + μ

)∞ 

∞  k=0

: |μk ||T ak | > ρ

k=0

|μk |gk > ρ

k=0

≤ Cp0 ρ −p0

∞ 

|μk |p0 .

:

262

5 Theory of Martingale Hardy Spaces

Hence, by using the σ -sublinearity of T and Theorem 5.3.2, we have proved the first inequality of the theorem. The second one can be proved analogously by interpolation as in the proof of the previous theorem so we omit the details. The proof is complete. 

5.6 Examples of p-Atoms and Hp Martingales In this Section, we give important examples of p-atoms and martingales f ∈ Hp (Gm ), which will be used many times to prove sharpness of the results proved in the next Chapters. Example 5.6.1 Let 0 < p ≤ 1, R = supn∈N mn and (αk , k ∈ N) be an increasing sequence of integers. Then the function 1/p−1

ak =:

Mαk R



DMαk +1 − DMα k

is a p-atom and ak Hp ≤ 1.

(5.22)

Proof Since  supp(ak ) = Iαk ,

ak dμ = 0 Iαk

and 1/p−1

ak ∞ ≤

Mαk R

Mαk +1 ≤ Mα1/p = (μ(supp ak ))−1/p , k

we conclude that ak is a p-atom and (5.22) follows from Theorem 5.3.2. The proof is complete.  Example 5.6.2 Let 0 < p ≤ 1, (nk , k ∈ N) be an increasing sequence of integers and ak = DM2nk +1 − DM2nk . Then )  ak (i) =

1, i = M2nk , . . . , M2nk +1 − 1, 0, otherwise,

(5.23)

5.6 Examples of p-Atoms and Hp Martingales

263

and ⎧ ⎪ ⎨ Di − DM2nk , i = M2nk + 1, . . . , M2nk +1 − 1, Si ak = ak , i ≥ M2nk +1 , ⎪ ⎩ 0, otherwise.

(5.24)

Moreover, 1−1/p

ak Hp ≤ RM2nk

,

(5.25)

where R = supn∈N mn . Proof The proof follows by using Example 5.6.1 in the case when αk = 2nk . We leave out the details.  The next two estimates will be used frequently. Proposition 5.6.3 Let Mk ≤ n < Mk+1 , f ∈ Hp for some fixed 0 < p < ∞ and n ∈ N. If Sn f denotes the n-th partial sum with respect to the Vilenkin system, then the following estimate holds ∗ Sn f Hp ≤ Cp . S# f p + Cp Sn f p . S#∗ f was defined in (2.2). where . Proof The martingale (Sn f ) can be written as     SMl Sn f, l ≥ 1 = SM0 f, . . . , SMk f, Sn f, . . . , Sn f, . . . . It immediately follows that Sn f Hp

  ≤ Cp sup SMl f  + Cp Sn f p 0≤l≤k p

∗ ≤ Cp . S# f p + Cp Sn f p . The proof is complete.



Proposition 5.6.4 Let Mk ≤ n < Mk+1 , f ∈ Hp for some fixed 0 < p < ∞ and n ∈ N. If σn f denotes the n-th Fejér means with respect to the Vilenkin system, then the following estimate holds ∗ ∗ σn f Hp ≤ Cp . σ# p + Cp σn f p , S# f p + Cp . where . σ#∗ f was defined in (3.1). S#∗ f was defined in (2.2) and .

264

5 Theory of Martingale Hardy Spaces

Proof Since   SMl σn f, l ∈ N+ =

 M0 σM0 f (n − M0 )SM0 f Mk σMk f (n − Mk )SMk f + ,..., + , σn f, σn f, . . . n n n n we can complete the proof as in the previous Proposition.



Also the next five examples of martingales will be used many times to prove sharpness of our main results. Example 5.6.5 Let 0 < p ≤ 1, R = supn∈N mn , (λk , k ∈ N) be a sequence of real numbers such that ∞ 

|λk |p ≤ Cp < ∞

(5.26)

k=0

and (ak , k ∈ N) be a sequence of p-atoms defined by 1/p−1

ak =:

M2αk R



DM2αk +1 − DM2αk ,

(5.27)

where (αk , k ∈ N) is an increasing sequence. Then f = (fn , n ∈ N) ∈ Hp , where 

fn =:

λk ak

{k:2αk c > 0 as k → ∞.

(6.72)

k→∞

Proof (a) By using Theorem 6.6.6 and obvious estimates, we find that Sn f − f H1 ≤ Sn f − SMk f H + SMk f − f H 1 1   = Sn SMk f − f H + SMk f − f H 1 1

   1 ≤ v (n) + v ∗ (n) + 1 ωH1 ,f Mk 

  1 ∗ ≤ c v (n) + v (n) ωH1 ,f , Mk and (6.71) is proved. The second statement in part (a) follows from this estimate and the given assumption. (b) Under the conditions of part (b), there exists a sequence {αk , k ∈ N} ⊂ {nk , k ∈ N} such that v (αk ) + v ∗ (αk ) ↑ ∞ as k → ∞

(6.73)

316

6 Vilenkin-Fourier Coefficients and Partial Sums in Martingale Hardy Spaces

and 2  v (αk ) + v ∗ (αk ) ≤ v (αk+1 ) + v ∗ (αk+1 ) .

(6.74)

Let f = (fn , n ∈ N) be a martingale from Lemma 5.6.6, where λk =

1 . v (αk ) + v ∗ (αk )

In view of (6.73) and (6.74) we conclude that (5.36) is satisfied and by using Lemma 5.6.6 we obtain that f ∈ H1 . According to (5.37) we find that * + ⎧ 1 ⎨ v(αk )+v ∗ (αk ) , if j ∈ M|αk | , . . . , M|αk |+1 − 1 , k ∈ N, . . . ∞ * + f(j ) = M|αk | , . . . , M|αk |+1 − 1 . if j ∈ / ⎩ 0, k=0

(6.75) By using (5.39) we get that Sαk f =

k−1 DM − DM  |α |+1 |α | i

i=1

v (αi )

+ v∗

i

(αi )

+

Dαk −M|α

k|

v (αk ) + v ∗ (αk )

.

(6.76)

According to Theorem 5.3.2 we find that ∞  f − SM f ≤ n H 1

1 v (αi ) + v ∗ (αi ) i=n+1

 1 =O as n → ∞. v (αn ) + v ∗ (αn )

By combining (6.74), (6.75) and (6.76) with Lemma 1.7.1, we obtain that ∞ D Dαk −M  M|αi |+1 − DM|αi | |αk | f − Sαk f 1 ≥ − v (αk ) + v ∗ (αk ) v (αi ) + v ∗ (αi ) i=k 1 ∞ DM Dαk −M|αk |  |αi |+1 − DM|αi | 1 1 ≥ − v (αk ) + v ∗ (αk ) v (αi ) + v ∗ (αi ) i=k

≥ C −2

∞  i=k

≥C−

1 v (αi ) + v ∗ (αi )

4 . v (αk ) + v ∗ (αk )

6.7 Strong Convergence of Partial Sums in Hp Spaces

317

Hence, lim sup Sαk f − f 1 ≥ C > 0, as k → ∞, k→∞

so (b) is also proved.  Corollary 6.6.10 Let f ∈ H1 and

ωH1

1 ,f Mn

 =o

 1 , as n → ∞. n

Then Sk f − f H1 → 0, as k → ∞. b) There exists a martingale f ∈ H1 for which

ωH1

1 ,f Mn



 1 =O , as n → ∞ n

and Sk f − f 1  0 as k → ∞.

6.7 Strong Convergence of Partial Sums in Hp Spaces In this Section we prove some sharp Hardy type inequalities for partial sums with respect to Vilenkin systems. Theorem 6.7.1 (a) Let 0 < p < 1 and f ∈ Hp . Then there is an absolute constant Cp depending only on p such that ∞ p  Sk f p k=1

k 2−p

p

≤ Cp f Hp .

(b) Let 0 < p < 1 and (n , n ∈ N) be any non-decreasing sequence satisfying the condition lim n = +∞.

n→∞

(6.77)

318

6 Vilenkin-Fourier Coefficients and Partial Sums in Martingale Hardy Spaces

Then there exists a martingale f ∈ Hp such that p ∞  Sk f weak−Lp k

k 2−p

k=1

= ∞.

Proof (a) Analogously to in the proof of the previous theorems without loss the generality we may assume that a be a p-atom with support I = IN . Since Sk a = 0, for k ≤ MN , in view of Theorem 5.3.2 it suffices to prove that ∞ 

IN

|Sk a(x)|p dμ(x) k 2−p

k=MN +1

≤ Cp < ∞

(6.78)

for every such p-atom a with support IN . By applying (1.5) with (6.18) in Theorem 6.4.1, we have that ∞  k=MN +1

=

≤ Cp

|Sk a(x)|p dμ(x)

k 2−p

IN

N−1  1 

∞  k=MN +1



1

k 2−p

k=MN +1 1−p

k 2−p

s=0

∞  k=MN +1

1−p

≤ Cp MN

|Sk a(x)|p dμ(x)

N−1  1 

∞ 

≤ Cp MN

s=0

Is \Is+1

∞  k=MN +1

Is \Is+1

   1/p−1 p Ms  dμ(x) MN

N−1  1 

k 2−p

s=0

N−1 1 

k 2−p

p

Is \Is+1 p−1

Ms

Ms dμ(x)

1−p

+ Cp MN

∞  k=MN +1

s=0

1 ≤ Cp < ∞, k 2−p

so (6.78) holds and the proof of the part (a) is complete. (b) We note that under condition (6.77) there exists an increasing sequence (αk , k ∈ N+ ) of positive integers such that αk ≥ 2 and ∞ 

1

p/4 k=1 M2α

< ∞.

(6.79)

k

Let f = (fn , n ∈ N) be the martingale defined in Example 5.6.10. By using (6.79), we conclude that f ∈ Hp . Let Mαk ≤ j < Mαk +1 . Using (5.54)

6.7 Strong Convergence of Partial Sums in Hp Spaces

319 1/4

and repeating the steps which leads to (5.39) with l = k and λk = 1/M2α , we η obtain that 1/p−1

Sj f =

k−1 M  2αη

1/4

η=0

M2α

DM2α

η +1

− DM2αη +

1/p−1

M2αk

ψM2αk Dj −M2α

k

1/4

M2α

η

k

=: I + I I. We calculate each term separately. By using Corollary 1.6.7 with condition αn ≥ 2 (n ∈ N), we find that DMαn = 0 for x ∈ Gm \I1 , so that I =0

for x ∈ Gm \I1 .

Denote by N0 the subset of positive integers N+ for which n = 0. Then every n ∈ N0 , Mk < n < Mk+1 (k > 1) can be written as n = n0 M0 +

k 

nj Mj ,

j =1

* + where n0 ∈ {1, . . . , m0 − 1} and nj ∈ 0, . . . , mj − 1 , (j ∈ N+ ). Let j ∈ N0 and x ∈ Gm \I1 = I0 \I1 . According to Lemma 1.6.11 we find that     Dαk −Mαk  ≥ cM0 ≥ c > 0 and 1/p−1

|I I | =

M2αk

1/4

M2α

k

  M 1/p−1   2αk . Dj −M2αk (x) = 1/4 M2α k

It follows that 1/p−1

M   Sj f (x) = |I I | = 2αk , 1/4 M2α k

for x ∈ Gm \I1

320

6 Vilenkin-Fourier Coefficients and Partial Sums in Martingale Hardy Spaces

and ⎛ ⎞ 1/p−1 1/p−1 1/p M2αk   M2αk Sj f ⎠ ≥ μ ⎝x ∈ Gm \I1 : Sj f (x) > weak−Lp 1/4 1/4 2M2α 2M2α k

=

1/p−1 M2αk 1/4 2M2α k

k

|Gm \I1 |

1/p−1

≥

cM2αk

(6.80)

.

1/4

M2α

k

Since 

1 ≥ cMk , {n∈N0 :Mk ≤n≤Mk+1 ,} where c is an absolute constant, by applying (6.80) we obtain that M2αk +1 −1

 j =1

p Sj f weak−Lp j j 2−p

≥

M2αk +1 −1



p Sj f weak−Lp j j 2−p

j =M2αk

≥ M2αk

*

j ∈N0 :M2αk ≤j ≤M2αk +1 1−p

j 2−p

+



M2αk

≥ cM2αk

1

p/4 + j 2−p M2α *j ∈N :M ≤j ≤M 0 2αk 2αk +1 k



3/4

≥ cMα Mα1−p k k

3/4

≥c

Sj f p weak−Lp



M2α

k

M2αk +1 *

*

1

j ∈N0 :M2αk ≤j ≤M2αk

 j ∈N0 :M2αk ≤j ≤M2αk +1

+ M 2−p 2αk +1 +1

1

+

3/4

≥ cM2α → ∞, as k → ∞. k

Hence, part (b) is proved, too. 

6.7 Strong Convergence of Partial Sums in Hp Spaces

321

Theorem 6.7.2 (a) Let f ∈ H1 (Gm ). Then there exists an absolute constant C such that 1  Sk f 1 ≤ C f H1 . log n k n

k=1

Proof By Lemma 5.3.2, we only have to prove that 1  Sk a1 ≤C MN . Let x ∈ IN . In this case DMi (x − t) 1IN (t) = 0 for i ≥ N. Recall that wMj (x − t) = wMj (x) for t ∈ IN and j < N. Consequently, by using Lemma 1.6.5, we obtain that  Sk a (x) = a (t) Dk (x − t) dμ (t) IN

 =

a (t) wk (x − t)

N−1 

kj wMj (x − t) DMj (x − t) dμ (t)

j =0

IN

= wk (x)

N−1 

 ki wMj (x) DMj (x)

i=0

= wk (x)

N−1 

a (t) wk (t) dμ (t)

IN

kj wMj (x) DMj (x) a (k) .

j =0

Let x ∈ Is \Is+1 . By using again Lemma 1.6.5 we get that N−1  j =0

DMj (x) ≤

s−1  j =0

DMj (x) ≤ 2Ms .

322

6 Vilenkin-Fourier Coefficients and Partial Sums in Martingale Hardy Spaces

Therefore, according to (1.5) we can conclude that 

N−1  IN j =0

DMj (x) dμ (x) ≤ 2

N−1  Is \Is+1

s=0

≤2

N−1 

(6.82)

Ms dμ (x)

1 ≤ 2N.

s=0

Since n ≥ MN , by using (6.82), Parseval‘s equality and Hölder’s inequality, we obtain that  N−1 ∞  ∞  |Sk a (x)| a (k)| 1  | 1  dμ (x) ≤ DMj (x) dμ(x) log k k log k k k=MN

k=MN

IN

≤

IN

∞ ∞  | a (k)| a (k)| 2N  | ≤2 log k k k k=MN

⎛ ≤ 2⎝

k=MN

⎞1/2 ⎛

∞ 

| a (k)|2 ⎠

k=MN

⎛ ≤

≤

j =0

C 1/2 MN

C

⎜ ⎝



⎞1/2 ∞  1 ⎝ ⎠ k2 k=MN

⎞1/2 ⎟ |a (t)|2 dμ (t)⎠

IN 1/2

1/2 MN

MN ≤ c < ∞.

(6.83)

Let x ∈ IN . Then, by the definition of a p-atom we find that ⎛

 |Sn a (x)| dμ (x) ≤ IN

1 1/2 MN

⎜ ⎝

1 1/2 MN

⎞1/2 ⎟ |Sn a (x)|2 dμ (x)⎠

IN

⎛ ≤



⎜ ⎝



IN

⎞1/2 ⎟ |a (x)|2 dμ (x)⎠

≤ 1.

6.7 Strong Convergence of Partial Sums in Hp Spaces

323

It follows that 1 1 log n k n

k=1



1 1 ≤ C < ∞. log n k n

|Sk a (x)| dμ (x) ≤

(6.84)

n=1

IN

The proof is complete by just combining (6.81)–(6.84).



Theorem 6.7.3 a) Let f ∈ H1 (Gm ). Then there exists an absolute constant C such that 1  Sk f 1 ≤ C f H1 . n∈N n log n n

sup

k=1

b) Let (ϕn , n ∈ N+ ) be a non-decreasing and non-negative sequence satisfying the condition lim

n→∞

log n = +∞. ϕn

(6.85)

Then there exists a function f ∈ H1 such that 1  Sk f 1 = ∞. n∈N nϕn n

sup

k=1

Proof (a) By using Corollary 1.7.2, we can conclude that n n c f H1  1  Sk f 1 ≤ log k ≤ c f H1 n log n n log n k=1

k=1

which completes the proof of part (a). (b) Under the condition (6.85) there exists an increasing sequence of positive integers (αk , k ∈ N) such that log Mαk +1 = +∞ k→∞ ϕMα +1 k lim

and 1/2

∞ 

ϕ2Mα

k=0

log1/2 Mαk

k

< c < ∞.

(6.86)

324

6 Vilenkin-Fourier Coefficients and Partial Sums in Martingale Hardy Spaces

Let f = (fn , n ∈ N) be the martingale defined in Example 5.6.5, where 1/2

λk =

ϕ2Mα

k

log1/2 Mαk

.

By combining (5.28), (5.29) and (5.30), we can conclude that f(j ) =

⎧ ⎨ λk , ⎩0 ,

* + j ∈ Mαk , . . . , 2Mαk − 1 , k ∈ N ∞ * + Mαk , . . . , 2Mαk − 1 j∈ / k=1

and Sj f = SMαk f + λk ψMαk Dj −Mαk =: I1 + I2 .

(6.87)

In view of Theorem 6.5.5 for p = 1, we obtain that I1 1 ≤ SMαk f ≤ c f H1 .

(6.88)

1

By combining (6.87) and (6.88) with Lemmas 1.7.1 and 1.7.5, we get that   Sn f 1 ≥ I2 1 − I1 1 ≥ λk v n − Mαk − c f H1 . Hence, 1  Sk f 1 sup n∈N+ nϕn n

k=1

1 ≥ RMαk ϕMαk * 1 ≥ RMαk ϕMαk *

 Mαk ≤l≤Mαk +1

Mαk ≤l≤Mαk +1

cϕMαk Mαk log1/2 Mαk ϕMαk

Sl f 1    1/2 v l − Mαk ϕMαk



1/2

≥

+

log1/2 Mαk

+

Mαk −1

 l=1

1/2

v (l) − c f H1

 − c f H1

6.8 Final Comments and Open Questions

325

1/2

≥ ≥

cϕMαk log Mαk log1/2 Mαk ϕ2Mαk c log1/2 Mαk 1/2

ϕ2Mα

→ ∞, as k → ∞.

k

This finishes the proof of part (b).  Corollary 6.7.4 There exists a function f ∈ H1 such that 1 Sk f 1 = ∞. n∈N n n

sup

k=1

6.8 Final Comments and Open Questions (1) The proof of Theorem 6.2.1 can be found in Tephnadze [343]. (2) Part (a) of Theorem 6.3.1 was first proved in Weisz [400, 427], but a new and shorter proof is given in this book. The proof of part (b) was given in Tephnadze [348]. (3) Part (a) of Theorem 6.3.2 for the Walsh system was first proved in Fine [261], for Vilenkin system by Simon and Weisz [314], but a new and shorter proof is presented in this book. The proof of part (b) is due to Tephnadze [348]. (4) The proof of Theorem 6.4.1 and Corollaries 6.4.2, 6.4.3 and 6.4.4 can be found in Tephnadze [353] (see also [348] and [355]). (5) The proof of Theorem 6.4.5 and Corollaries 6.4.6, 6.4.7 and 6.4.8 are presented in Tephnadze [353] (see also [355]). (6) The proof of Theorem 6.4.9 and Corollaries 6.4.10, 6.4.11, and 6.4.12 can be found in Blahota et al. [45]. (7) Theorem 6.5.1, Remark 6.5.2, Theorem 6.5.3, Remark 6.5.4 and Theorem 6.5.6 are proved in Tephnadze [353]. (8) Theorems 6.5.6, 6.5.7, 6.5.8 and 6.5.9 can be found in [353]. (9) Theorem 6.6.1, Corollaries 6.6.2, 6.6.3, 6.6.4 and 6.6.5, Theorems 6.6.6 and 6.6.7, Corollary 6.6.8, Theorem 6.6.9 and Corollary 6.6.10 are proved in [360] for bounded Vilenkin system and in [357] for the Walsh system. We note that similar problems are open for unbounded Vilenkin systems. (10) Part (a) of Theorem 6.7.1 for unbounded Vilenkin systems was first proved by Simon [311]. In this book we use some new estimations and present a simpler proof, which is due to Tephnadze [353]. We also prove sharpness of

326

6 Vilenkin-Fourier Coefficients and Partial Sums in Martingale Hardy Spaces

this result in a special sense in part (b) of Theorem 6.7.1, which can be found in Tephnadze [343]. (11) Theorem 6.7.2 even for unbounded Vilenkin systems was proved by Gát [107]. (12) Theorem 6.7.3 and Corollary 6.7.4 can be found in Tutberidze [381]. (13) Similar problems for the partial sums with respect to the two-dimensional Walsh-Fourier and Vilenkin-Fourier series in the martingale Hardy spaces were investigated by Tephnadze [344, 349, 361] and [362]. Open Problem Let f ∈ H1 (Gm ) . Does there exist an absolute constant C such that 1 Sk f weak−L1 ≤ C f H1 , n ∈ N+ , n n

k=1

where Sk f denotes the k-th partial sum of the one-dimensional Vilenkin-Fourier series of f ? Open Problem Let f ∈ H1 . Is it possible to characterize the subspace S1 := {αk , k ∈ N} ⊂ N, such that the maximal operator . S ∗,,∗ defined by   . S ∗,,∗ f := sup Sαk f  k∈N

is bounded from the Hardy space H1 to the space L1 ? Moreover, for any subspace {nk , k ∈ N} ⊂ N for which S1 ∩ {nk , k ∈ N} is not a finite set, does there exists a martingale f ∈ H1 such that sup Snk f 1 = ∞?

k∈N

Remark 6.8.1 Since SM +1 f ≤ C f H for all f ∈ H1 n 1 1 and sup ρ (Mn + 1) = ∞, n∈N

6.8 Final Comments and Open Questions

327

we obtain that  9 S0 := nk , k ∈ N, sup ρ (αk ) < c < ∞ k∈N

is a proper subspace of the set S1 := {αk , k ∈ N} . Open Problem Let f ∈ H1 . Is it possible to characterize the subspace S2 := {αk , k ∈ N} ⊂ N such that the maximal operator . S ∗,,∗ defined by   . S ∗,,∗ f := sup Sαk f  αk ∈N

is bounded from the Hardy space H1 to the space weak − L1 ? Moreover, for any subspace {nk , k ∈ N} ⊂ N for which S2 ∩ {nk , k ∈ N} is not a finite set, does there exists a martingale f ∈ H1 such that sup Snk f weak−L1 = ∞?

k∈N

Remark 6.8.2 According to Remark 6.8.1, we obtain that S0 ⊂ S1 ⊆ S2 . Open Problem Let f ∈ H1 (G). Does there exist an absolute constant C such that n 1  Skκ f 1 ≤ C f H1 , log n k k=1

where Skκ denotes the k-th partial sum with respect to the Kaczmarz system. Remark 6.8.3 The similar problem for 0 < p < 1 and partial sums with respect to the Kaczmarz system was solved by Simon [310].

328

6 Vilenkin-Fourier Coefficients and Partial Sums in Martingale Hardy Spaces

Open Problem Let f ∈ H1 . Does there exist an absolute constant c such that 1  S κ f ≤ C f H , k 1 1 n∈N n log n n

sup

k=1

where Skκ denotes the k-th partial sum with respect to the Kaczmarz system? (b) Let (ϕn , n ∈ N+ ) be a non-negative, non-decreasing sequence satisfying the condition lim

n→∞

log n = +∞. ϕn

Does there exist a function f ∈ H1 such that 1  S κ f = ∞, k 1 n∈N nϕn n

sup

k=1

where Skκ denotes the k-th partial sum with respect to the Kaczmarz system? Remark 6.8.4 The similar remark for 0 < p < 1 and partial sums with respect to unbounded Vilenkin systems was solved by Tutberidze [381]. Open Problem (a) Let f ∈ H1 and

ωH1

1 ,f Mn

 =o

 1 , as n → ∞. n

Is it true or not that κ S f − f → 0, as k → ∞, k H 1

where Skκ denotes the k-th partial sum with respect to the Kaczmarz system? (b) Does there exist a martingale f ∈ H1 for which

ωH1

1 ,f Mn

 =O

 1 , as n → ∞ n

and κ S f − f  0, as k → ∞, k 1 where Skκ denotes the k-th partial sum with respect to the Kaczmarz system.

6.8 Final Comments and Open Questions

329

Open Problem (a) Let 0 < p < 1, f ∈ Hp and

ωHp

1 ,f Mn



 =o



1 1/p−1

Mn

, as n → ∞.

Is is true or not that κ S f − f → 0, as k → ∞, k Hp where Skκ denotes the k-th partial sum with respect to the Kaczmarz system. (b) Let 0 < p < 1. Does there exist a martingale f ∈ Hp for which

ωHp

1 ,f Mn



 =O

1 1/p−1

Mn

 , as n → ∞

and κ S f − f  0, as n → ∞, n weak−Lp where Snκ denotes n-th partial sum with respect to Kaczmarz system. Remark 6.8.5 The similar result for 0 < p ≤ 1 and partial sums with respect to bounded Vilenkin systems was proved in Theorems 6.5.8 and 6.5.9 (see Tephnadze [353]), but this is an open problem for unbounded Vilenkin systems. The analogous problems for Fejér means of the Kaczmarz system and for 0 < p ≤ 1/2 is considered in Tephnadze [350] and the methods of proofs can be applicable to solve similar problems concerning partial sums with respect to the Kaczmarz system. Remark 6.8.6 Concerning convergence and approximation of Fourier series in the classical cases and real Hardy spaces a few results are known, only. We refer to the papers by Oswald [259], Kryakin and Trebels [198] and Storoženko [330], who proved one-to-one analogues of part (a) of Theorem 6.5.8 and Theorem 6.5.9 for the partial sums of the trigonometric series in the classical Hardy spaces, but part (b) which provides sharpness of the above mentioned results in this case seems still to be an open problem. Remark 6.8.7 As mentioned before, several results in this type of harmonic analysis are inspired by the corresponding results in the classical case. Remark 6.8.6 may be seen as a typical example of the opposite, namely that the new results presented in this book can also lead to new questions in the classical case.

Chapter 7

Vilenkin-Fejér Means in Martingale Hardy Spaces

7.1 Introduction Weisz [403] considered the norm convergence of Fejér means of Vilenkin-Fourier series and proved that σk f p ≤ Cp f Hp , p > 1/2, k ∈ N and f ∈ Hp .

(7.1)

This result implies that 1 n2p−1

n p  σk f p k=1

k 2−2p

p

≤ Cp f Hp ,

(1/2 < p < ∞) .

If (7.1) holds for 0 < p ≤ 1/2, then we would have that 1 log[1/2+p] n

n p  σk f p k=1

k 2−2p

p

≤ Cp f Hp ,

(0 < p ≤ 1/2) .

(7.2)

However, in Tephnadze [342] it was proved that the assumption p > 1/2 in (7.1) is essential. In particular, is was proved that there exists a martingale f ∈ H1/2 such that sup σn f 1/2 = +∞.

n∈N

In [354] it was proved that (7.2) holds, though inequality (7.1) is not true for 0 < p ≤ 1/2.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 L.-E. Persson et al., Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series, https://doi.org/10.1007/978-3-031-14459-2_7

331

332

7 Vilenkin-Fejér Means in Martingale Hardy Spaces

In the one-dimensional case Fujii [103] and Simon [303] verified that σ ∗ is bounded from H1 to L1 . Weisz [403] generalized this result and proved the boundedness of σ ∗ from the martingale space Hp to the Lebesgue space Lp for p > 1/2. Simon [307] gave a counterexample, which shows that boundedness does not hold for 0 < p < 1/2. The corresponding counterexample for p = 1/2 is due to Goginava [127], (see also [40] and [41]). Weisz [426] proved that σ ∗ is bounded from the Hardy space H1/2 to the space weak − L1/2 . In [345] and [346] (for Walsh system see [132]) it was proved that the maximal operator . σp∗ with respect to Vilenkin systems defined by . σp∗ f =: sup

n∈N

|σn f | (n + 1)

1/p−2

log2[1/2+p] (n + 1)

,

where 0 < p ≤ 1/2 and [1/2 + p] denotes integer part of 1/2 + p, is bounded from the Hardy space Hp to the Lebesgue space Lp . Moreover, the order of deviant behavior of the n-th Fejér mean was given exactly. As a corollary we get that σn f p ≤ Cp (n + 1)1/p−2 log2[1/2+p] (n + 1) f Hp . By using this estimate it was proved that for any f ∈ Hp (0 < p ≤ 1/2) satisfying the condition

ωHp

1 ,f Mn



 =o



1 1/p−2

Mn

log[1/2+p] n

, as n → ∞,

we have that σk f − f Hp → 0, as k → ∞. Moreover, for every 0 < p ≤ 1/2 there exists a martingale f ∈ Hp for which

ωHp

1 ,f Mn



 =O

1 1/p−2

Mn

log[1/2+p] n

 , as n → ∞

and σk f − f weak−Lp  0, as k → ∞. In this proof it was applied that if f ∈ Hp where 0 < p ≤ 1, then σM f − f → 0, as k → ∞. k Hp

7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces

333

For Vilenkin-Fourier series Weisz [403] proved that the maximal operator σ # defined by   σ # f := sup σMn f  , n∈N

is bounded from the martingale Hardy space Hp to the Lebesgue space Lp for p > 0. For Walsh-Fourier series Goginava [124] proved that the operator |σ2n f | is not bounded from the space Hp to the space Hp , for 0 < p ≤ 1.

7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces First we state the following sharp result: Theorem 7.2.1 (a) Let p > 1/2 and f ∈ Hp . Then the maximal operator σ ∗ of the Vilenkin-Fejér means is bounded from the Hardy space Hp to the Lebesgue space Lp . (b) There exist a martingale f ∈ H1/2 such that sup σn f 1/2 = +∞.

(7.3)

n∈N

Proof (a) First, we note that σn is bounded from L∞ to L∞ (see (3.17) in Corollary 3.2.9). Hence, by Theorem 5.5.1 the proof will be complete if we prove that 

 ∗ p σ a  dμ ≤ c < ∞

(7.4)

I

for every p-atom a, where I denotes the support of the atom a. Let a be an arbitrary p-atom with support I and μ (I ) = MN−1 . We may assume that I = IN . It is easy to see that σn (a) = 0 when n ≤ MN . Therefore 1/p we can suppose that n > MN . Since a∞ ≤ MN it follows that  |a (t)| |Kn (x − t)| dμ (t)

|σn a| ≤ IN



≤ a∞ ≤

1/p MN

|Kn (x − t)| dμ (t) 

IN

|Kn (x − t)| dμ (t) . IN

334

7 Vilenkin-Fejér Means in Martingale Hardy Spaces

Let x ∈ INk,l , 0 ≤ k < l ≤ N. From Corollary 3.2.15, we can deduce that 1/p Ml Mk MN2

|σn a| ≤ Cp MN

1/p−2

= Cp MN

Ml Mk .

(7.5)

The expression on the right-hand side of (7.5) does not depend on n. Thus,  ∗  σ a (x) ≤ Cp M 1/p−2Ml Mk , for x ∈ I k,l , 0 ≤ k < l ≤ N. N N

(7.6)

By using (7.6) combined with the identity (1.5), we obtain that  IN



mj −1

N−2  N−1   ∗  σ a (x)p dμ (x) =



k=0 l=k+1 xj =0,j ∈{l+1,...,N−1}

+

N−1  k=0

≤ cp

INk,N

N−2  N−1 

N−1  k=0

(7.7)

 ∗  σ a (x)p dμ (x)

k=0 l=k+1

+ cp

 ∗  σ a (x)p dμ (x)

INk,l

ml+1 . . . mN−1 1−2p p p MN Ml Mk MN

1 1−p p M Mk MN N

1−2p

≤ cp MN

N−2  N−1 

M Ml Mk k + cp p Ml MN p

k=0 l=k+1

p

N−1

p

k=0

=: I + I I.

We estimate each term separately. For I we have that I =

1−2p cp MN

≤

1−2p cp MN

N−2  N−1 

(7.8)

1−p k=0 l=k+1 Ml N−2 

2p−1 Mk

k=0

≤ cp

p

Mk

N−2 

Mk

2p−1 N−1 

k=0

MN

2p−1

N−1 

1−p

Mk

1−p l=k+1 Ml 1−p

Mk

1−p l=k+1 Ml

7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces

≤ cp

N−2  k=0

≤ cp

k=0

N−1 

1 2(N−k)(2p−1)

N−2 

335

[N − k] 2(N−k)(2p−1)

1

l=k+1

2(p−1)(l−k)

≤ Cp < ∞.

It is obvious that I I ≤ cp

N−1 

Mk

k=0

MN

p

p

≤ Cp < ∞.

(7.9)

We combine (7.7)–(7.9) to see that (7.4) holds which completes the proof of part (a). (b) Let f = (fn , n ∈ N) be the martingale defined in Example 5.6.7 in the case when p = q = 1/2. We have that M

σqαk f =

2α 1 k 1 Sj f + qαk qαk

j =1

q

αk 

(7.10)

Sj f

j =M2αk +1

=: I I I + I V . According to (5.43) and (5.46) for p = q = 1/2, we can conclude that M

2α 1 k |I I I | ≤ |Si f | qαk

(7.11)

j =1

≤

2 2RM2α k−1 M2αk 1/2 qαk α k−1

≤

2 2RM2α k−1 1/2

αk−1

≤

Mα2k 3/2

.

16αk

By applying (5.45) obtained by letting l = k, we can rewrite I V as  IV =

 qαk − M2αk SM2αk f qαk

=: I V1 + I V2 .

+

M2αk ψM2αk 1/2

αk qαk

q

αk 

j =M2αk +1

Dj −M2αk

336

7 Vilenkin-Fejér Means in Martingale Hardy Spaces

According to (5.46) for j = M2αk and invoke estimate (5.43) for p = q = 1/2, we find that   2RM 2 Mα2k 2αk−1   |I V1 | ≤ SM2αk f  ≤ ≤ . 1/2 3/2 αk−1 16αk

(7.12)

Let @

  x ∈ I2η+1 e2η−1 + e2η ,

η=

A 2αk + 1, . . . , αk − 3. 3

By applying Corollary 3.2.13 we have that     2 144qαk −1 Kqαk −1 (x) ≥ M2η . Hence, for I V2 we readily get that   qαk −1     M2αk  |I V2 | = 1/2 Dj  ψM2αk  αk qαk  j =1 =

≥ ≥

(7.13)

 M2αk qαk −1   K  q −1 α 1/2 k qαk α k     qαk −1 Kqαk −1  1/2

2αk 2 M2η 1/2

.

288αk

Therefore, by using (7.10), (7.11), (7.12) and (7.13), we find that     σqαk f (x) ≥ |I V2 | − (|I I I | + |I V1 |)   144Mα2k 1 2 M2η − . ≥ 1/2 αk 288α

(7.14)

k

Thus for sufficiently large k, we have that M2η ≥ 144Mαk , αk > 1 and it follows that for some c > 0,  cM 2    2η   σqαk f (x) ≥ 1/2 , x ∈ I2η+1 e2η−1 + e2η , αk

@ η=

A 2αk + 1, . . . , αk − 3. 3

7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces

337

Hence,  Gm

1/2    σqαk f (x) dμ (x) ≥

≥

≥



α k −3

 



σqα f k η=[2αk /3]+1 I2η+1 e2η−1 +el2η

c

α k −3

1/4

αk

c

1/2  (x) dμ (x)

   I2η+1 e2η−1 + el  M2η 2η

η=[2αk /3]+1 α k −3

1/4 αk η=[2αk /3]+1

3/4

1 ≥ αk

→ ∞, as k → ∞. 

Thus (7.3) holds. Our next sharp result reads: Theorem 7.2.2

(a) Let f ∈ H1/2 . Then the maximal operator σ ∗ of Vilenkin-Fejér means is bounded from H1/2 to the space weak − L1/2 . (b) Let 0 < p < 1/2. Then there exists a martingale f ∈ Hp such that sup σn f weak−Lp = +∞. n∈N

Proof (a) By Theorem 5.5.2, we have to prove that ' ( tμ x ∈ IN : σ ∗ a ≥ t 2 ≤ c < ∞,

t ≥0

(7.15)

for every 1/2-atom a. We may assume that a is an arbitrary 1/2-atom with support I, μ (I ) = MN−1 and I = IN . It is easy to see that σn a (x) = 0,

when n ≤ MN .

Therefore, we can suppose that n > MN . Let t ∈ IN . Since σn is bounded from L∞ to L∞ (see (3.17) in Corollary 3.2.9) and a∞ ≤ MN2 , by combining (7.5) and (7.6) for p = 1/2 we get that |σn a (x)| ≤ CMl Mk for x ∈ INk,l , 0 ≤ k < l ≤ N and   ∗ σ a (x) ≤ CMl Mk for x ∈ I k,l , 0 ≤ k < l ≤ N. N

(7.16)

338

7 Vilenkin-Fejér Means in Martingale Hardy Spaces

It follows that for such k < l ≤ N we have the following estimate |σn a (x)| ≤ CMN2 for x ∈ INk,l and also that ( ' μ x ∈ INk,l : σ ∗ f > CMs2 = 0, s = N + 1, N + 2, . . .

(7.17)

Let θ ∈ N such that 2θ−1 ≤ R < 2θ , where R = sup mn . n

Suppose that Ml Mk > Ms2 for some s ≤ k < l ≤ N

(7.18)

It is evident that inequality (7.18) holds for all l > k ≥ s, that is, Ml Mk > Ms2 , where l > k ≥ s.

(7.19)

If l > s > k, from (7.18) we can conclude that Ml > Ms (mk mk+1 ms−1 ) ≥ 2s−k Ms ≥ R (s−k)/θ Ms , ms ms+1 . . . ml−1 > R (s−k)/θ , R l−s > R (s−k)/θ , l − s = ln(M l−s ) > ln(R (k−s)/θ ) = (s − k)/θ , l > (s − k)/θ + s. Hence, Ml Mk ≥ Ms2 , where s > k, l > [(s − k)/θ] + s.

(7.20)

By combining (7.16), (7.19) and (7.20) we get that N ' ' ( N−1 (   x ∈ IN : σ ∗ f ≥ CMs2 ⊂ x ∈ INk,l : σ ∗ f ≥ CMs2 k=s l=k+1 s 

N 

k=0 l=(s−k)/θ+s

' ( x ∈ INk,l : σ ∗ f ≥ CMs2

7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces

339

and N ' ( N−1   μ x ∈ IN : σ ∗ f ≥ CMs2 ≤

   k,l  IN 

mj−1



(7.21)

k=s l=k+1 xj =0, j ∈{l+1,...,N−1}

+

s 

   k,l  IN 

mj−1

N 



k=0 l=[(s−k)/θ]+s xj =0, j ∈{l+1,...,N−1}

≤

N−1 

s N   1 + Ml

k=s l=k+1

≤

N−1  k=s

N 

k=0 l=[(s−k)/θ]+s

1 Ml

 1 1 C + ≤ . Ms M[(s−k)/θ]+s Ms s

k=0

In view of (7.17) and (7.21) we can conclude that ( ' M s μ x ∈ IN : σ ∗ f ≥ Ms2 < C < ∞, which shows (7.15) as well as part (a). (b) Let 0 < p < 1/2 and f = (fn , n ∈ N) be the martingale defined in Example 5.6.7 in the case when 0 < p < q = 1/2. We have that σM2αk +1 f =

M2αk



1 M2αk + 1

Sj f +

j =0

SM2αk +1 f M2αk + 1

=: I I I + I V . We combine (5.46) and (5.47) and invoke (5.43) in the case when p < q = 1/2 to obtain the following estimates: 1/p

M2αk 2RM2αk−1 |I I I | ≤ M2αk + 1 α 1/2 k−1

≤

1/p 2RM2αk−1 1/2 αk−1 1/p−2

≤

Mαk

3/2

16αk

340

7 Vilenkin-Fejér Means in Martingale Hardy Spaces

and |I V | ≥

    SM2αk +1 f  M2αk + 1

1/p−2

≥

M2αk

.

2αk

Let x ∈ Gm . We conclude that     σM2αk +1 f (x) ≥ |I V | − |I I I | 1/p−2

≥

1/p−2

M2αk

−

1/2

2αk

Mαk

3/2

16αk

1/p−2

≥

M2αk

1/2

.

4αk

It follows that 1/p−2

M2αk

1/2

4αk

 )

  M 1/p−2   2αk μ x ∈ Gm : σM2αk +1 f (x) ≥ 1/2 4αk

:1/p

1/p−2

≥

M2αk

1/2

4αk

→ ∞, as k → ∞.

Hence, also part (b) is proved so the proof is complete. We note the following consequence of Theorems 7.2.1 and 7.2.2 : Corollary 7.2.3 (a) There exists a martingale f ∈ H1/2 such that ∗ σ f = +∞. 1/2 (b) Let 0 < p < 1/2. Then there exists a martingale f ∈ Hp such that ∗ σ f = +∞. weak−Lp Also our next result is sharp in the similar sense.



7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces

341

Theorem 7.2.4 (a) Let 0 < p < 1/2. Then the maximal operator . σp∗ defined by . σp∗ f := sup

n∈N

|σn f | (n + 1)1/p−2

is bounded from the Hardy martingale space Hp to the Lebesgue space Lp . (b) Let (n , n ∈ N) be any non-decreasing sequence satisfying the condition (n + 1)1/p−2 = +∞. n→∞ n lim

(7.22)

Then

sup

σM2n +1 fk k M2n +1 k

weak−Lp

fk Hp

k∈N

= ∞.

(7.23)

Proof (a) First we note that σn is bounded from L∞ to L∞ (see (3.17) in Corollary 3.2.9). Hence, by Theorem 5.5.1 the proof will be complete if we show that     ∗ p σp (x) dμ(x) ≤ Cp < ∞ .

(7.24)

I

for every p-atom a, where I denotes the support of the atom. Let a be an arbitrary p-atom with support I and μ (I ) = MN−1 . We may assume that I = IN . It is easy to see that σn a = 0 when n ≤ MN . Therefore 1/p we can suppose that n > MN . Since a∞ ≤ MN , it follows that |σn a| (n + 1)1/p−2

≤ ≤

1 (n + 1)1/p−2 a∞

|a (t)| |Kn (x − t)| dμ (t) 

cMN

(n + 1)1/p−2

IN

|Kn (x − t)| dμ (t)

(n + 1)1/p−2 1/p

≤



IN

 |Kn (x − t)| dμ (t) . IN

342

7 Vilenkin-Fejér Means in Martingale Hardy Spaces

Let x ∈ INk,l , 0 ≤ k < l ≤ N. From Corollary 3.2.15 we can deduce that 1/p

|σn a|

≤

(n + 1)

1/p−2

Cp MN

1/p−2 MN

Ml Mk = Cp Ml Mk . MN2

(7.25)

The expression on the right-hand side of (7.25) does not depend on n. Thus,     ∗ σp a (x) ≤ Cp Ml Mk , for x ∈ INk,l , 0 ≤ k < l ≤ N. .

(7.26)

By using (7.26) together with the identity (1.5) we obtain that 

p    ∗ σp a (x) dμ (x) .

IN

=

+

(7.27) 

mj−1

N−2  N−1 



k,l k=0 l=k+1 xj =0,j ∈{l+1,...,N−1} IN

N−1  k=0

≤ Cp

INk,N

p    ∗ σp a (x) dμ (x) .

N−2  N−1 

 1 ml+1 . . . mN−1 p p p p Ml Mk + Cp MN Mk MN MN N−1

k=0 l=k+1

≤ Cp

p    ∗ σp a (x) dμ (x) .

N−2  N−1 

k=0

 M Ml Mk k + Cp 1−p Ml M p

k=0 l=k+1

p

p

N−1 k=0

N

=: I + I I. We estimate each term separately: I = Cp

≤ Cp

≤ Cp

N−2  N−1 

N−2  N−1 

p

p

Ml Mk

1−2p k=0 l=k+1 Ml

2p

Ml

1

1−2p k=0 l=k+1 Ml N−2  N−1  k=0 l=k+1

≤ Cp

1

N−2  k=0

1 2l(1−2p)

1 2k(1−2p)

< Cp < ∞.

(7.28)

7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces

343

It is obvious that N−1 

Cp

II ≤

p

Mk

p 1−2p MN MN k=0

≤

Cp 1−2p

MN

< c < ∞.

(7.29)

By combining (7.27) with (7.28)–(7.29), we obtain (7.24) and part (a). (b) Under condition (7.22), there exists an increasing sequence of positive integers (λk , k ∈ N) such that 1/p−2

λk k→∞ λk lim

= ∞.

It is evident that for every λk there exists a positive integer m,k such that qmk < λk < 2qm,k . Since (n , n ∈ N) is a non-decreasing function we have that 1/p−2

lim

M2m,

k

k→∞ M2m, +1 k



1/p−2 M2m,k + 1 1 ≥ lim 2 k→∞ M2m, +1 k

≥ c lim

k→∞

1/p−2 λk

λ k

= ∞.

  Let (nk , k ∈ N) ⊂ m,k , k ∈ N be a sequence of positive numbers such that 1/p−2

lim

k→∞

M2nk M2n

k

+1

=∞

and fk is the atom defined in Example 5.6.2. By combining (5.23) and (5.24) in Example 5.6.2, we find that     σM2nk +1 fk  M2nk +1

  M  2n +1  k  1    = Sj fk  M2nk +1 M2nk + 1  j =0  =

  1  

SM2nk +1 fk  M2nk +1 M2nk + 1

M2n

  1  

DM2nk +1 − DM2n  k +1 M2nk + 1

M2n

  1     ψM2n  k +1 M2nk + 1

= k

=

k

344

7 Vilenkin-Fejér Means in Martingale Hardy Spaces

=

M2n

k

≥

1   +1 M2nk + 1

c . M2nk M2nk +1

Hence, ⎧ ⎨

μ x ∈ Gm : ⎩

  σM2n

k

+1 fk

  (x)

M2nk +1

≥

⎫ ⎬

c M2nk +1 ≥ μ (Gm ) = 1. ⎭ M2nk

Therefore, by using (5.25) in Example 5.6.2, we get that  ) c M2nk M2n

k

+1

μ x ∈ Gm :

    σM2n +1 fk (x) k M2n +1 k

:1/p ≥

c M2nk M2n

k

+1

fk Hp ≥ =

c 1/p−1 M2nk M2nk M2nk +1 1/p−2 cM2n k

M2nk +1

→ ∞, as k → ∞

and also part (b) is proved.



We also point out the following consequence of Theorem 7.2.4, which we need later on. Corollary 7.2.5 Let 0 < p < 1/2 and f ∈ Hp . Then there exists an absolute constant Cp depending only on p, such that σn f p ≤ Cp (n + 1)1/p−2 f Hp , n ∈ N+ .

(7.30)

Proof According to part (a) of Theorem 7.2.4 we conclude that |σn f | σn f ≤ sup ≤ Cp f H , n ∈ N+ , p (n + 1)1/p−2 1/p−2 n∈N (n + 1) p p which implies that (7.30) holds.



7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces

345

We also mention the following corollaries: Corollary 7.2.6 Let (n , n ∈ N) be any non-decreasing sequence satisfying condition (7.22). Then there exists a martingale f ∈ Hp such that σn f sup = ∞. n∈N n weak−Lp Corollary 7.2.7 Let (n , n ∈ N) be any non-decreasing sequence satisfying condition (7.22). Then the following maximal operator |σn f | n∈N n sup

is not bounded from the Hardy space Hp to the space weak − Lp . Theorem 7.2.8 ∼∗

(a) The maximal operator σ defined by |σn f |

∼∗

σ f := sup

2 n∈N log (n + 1)

is bounded from the Hardy space H1/2 to the Lebesgue space L1/2 . (b) Let (n , n ∈ N) be any non-decreasing sequence satisfying the condition log2 (n + 1) = ∞. n→∞ n lim

(7.31)

Then σ f qnk k q sup k∈N

nk

1/2

fk H1/ 2

= ∞,

(7.32)

where fk are 1/2-atoms. Proof (a) It is enough to prove that   1/2  ∼∗ σ a(x) dμ(x) ≤ c < ∞

(7.33)

I

for every 1/2-atom a, where I denotes the support of the atom. Let a be again an arbitrary 1/2-atom with support I = IN and μ (I ) = MN−1 . Since σn (a) = 0

346

7 Vilenkin-Fejér Means in Martingale Hardy Spaces

when n ≤ MN , we can suppose that n > MN . Since a∞ ≤ MN2 we obtain that  |σn a(x)| 1 |a (t)| |Kn (x − t)| dμ (t) ≤ log2 (n + 1) log2 (n + 1) IN  a∞ |Kn (x − t)| dμ (t) ≤ log2 (n + 1) IN  MN2 |Kn (x − t)| dμ (t) . ≤ log2 (n + 1) IN Let x ∈ INk,l , 0 ≤ k < l ≤ N. Then, from Corollary 3.2.15 it follows that |σn a(x)|

≤

log2 (n + 1)

cMN2 Ml Mk cMl Mk = . N 2 MN2 N2

(7.34)

Since the expression on the right-hand side of (7.34) does not depend on n,  ∗  cMl Mk . for x ∈ INk,l , 0 ≤ k < l ≤ N. σ a (x) ≤ N2 By applying (7.35) combined with the identity (1.5) we obtain that 

1/2  ∗ . σ a (x) dμ (x) IN

=

mj −1

N−2  N−1 





k,l k=0 l=k+1 xj =0,j ∈{l+1,...,N−1} IN

+

N−1  k=0

≤c

INk,N

N−2  N−1  k=0 l=k+1

+c

N−1  k=0

1/2  ∗  ∼ σ a (x) dμ (x)

1/2  ∗ . σ a (x) dμ (x) 1/2

1/2

ml+1 . . . mN−1 Ml Mk MN N 1/2

1/2

1 MN Mk MN N

=: I + I I.

For the first term we get that I ≤

1/2 N−2 N−1 N−2 c   Mk c  ≤ 1 ≤ c < ∞. 1/2 N N k=0 l=k+1 Ml k=0

(7.35)

7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces

347

Moreover, II ≤

N−1 

1

1/2 MN N k=0

1/2

Mk

≤

c < c < ∞. N

We conclude that (7.33) follows by just combining the estimates above and the proof of part (a) is complete. (b) Under condition (7.31) there exists an increasing sequence (λk , k ∈ N) of positive integers such that log2 (λk + 1) = ∞. k→∞ λ k lim

It is evident that for every λk there exists a positive integer m,k such that qm ≤ k λk < qmk +1 < c qm . Since n is a non-decreasing function, we have that k

lim

 2 mk

k→∞

q m,

log2 (λk + 1) = ∞. k→∞ λ k

≥ c lim

k

  Let (nk , k ∈ N) ⊂ mk , k ∈ N be a subsequence of positive numbers N+ such that n2k =∞ k→∞ qn k lim

and let fk be the atom defined in Example 5.6.2. We combine (5.23) and (5.24) in Example 5.6.2 and invoke (1.40) in Lemma 1.6.4 to obtain that     σqnk fk  qnk

=

1 qnk qnk

=

1 qnk qnk

=

1

=

qnk qnk 1 qnk qnk

  qn    k   Sj fk    j =1     qn  k      Sj fk     j =M2nk +1     qnk

    Dj − DM2nk   j =M2n +1  k   qnk −1

   Dj +M2nk − DM2n   k   j =1

348

7 Vilenkin-Fejér Means in Martingale Hardy Spaces

= =

1 qnk qnk

  q  n −1  k    D j   j =1 

 1 qnk −1   Kqnk −1  . qnk qnk

Let x ∈ I2s+1 (e2s−1 + e2s ). Then, in view of Corollary 3.2.13, we can conclude that     2 σqnk fk (x) cM2s ≥ . qnk M2nk qnk Hence,     n k −3   σq fk 1/2  σq fk 1/2  nk   nk     dμ ≥  dμ Gm  qnk  I2s+1 (e2s−1 +e2s )  qnk 



s=0

≥c

≥c

n k −3

M2s μ (I2s+1 (e2s−1 + e2s ))

s=0

qnk M2nk

1/2

n k −3

≥

1 1/2

s=0

1/2

1/2

M2nk qnk

cnk . 1/2 1/2 M2nk qnk

From (5.25) in Example 5.6.2 we have that

 σ f 1/2 2  qnk k  dμ Gm  qn  k

fk H1/2

≥

cn2k M2nk M2nk qnk

≥

cn2k → ∞, qnk

Thus, (7.32) holds so part b) is proved.

as

k → ∞. 

We also point out the following consequence of Theorem 7.2.8 which will be used later.

7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces

349

Corollary 7.2.9 Let f ∈ H1/2. Then there exists an absolute constant c such that σn f 1/2 ≤ c log2 (n + 1) f H1/2 , n ∈ N+ .

(7.36)

Proof According to part (a) of Theorem 7.2.8, we readily conclude that σn f log2 (n + 1)

1/2

≤ sup

n∈N

log2 (n + 1) |σn f |

≤ c f H1/2 ,

n ∈ N+ ,

1/2



which implies that (7.36) holds and the proof is complete.

Corollary 7.2.10 Let (n , n ∈ N) be any non-decreasing sequence satisfying condition (7.31). Then there exists a martingale f ∈ H1/2 such that σn f = ∞. sup n∈N n 1/2 Corollary 7.2.11 Let (n , n ∈ N) be any non-decreasing sequence satisfying condition (7.31). Then the following maximal operator |σn f | n∈N n sup

is not bounded from the Hardy space H1/2 to the Lebesgue space L1/2 . Theorem 7.2.12 (a) Let 0 < p ≤ 1/2 and (αk , k ∈ N) be a subsequence of positive numbers such that sup ρ (αk ) =  < c < ∞,

(7.37)

k∈N

where ρ (n) is defined in (1.7). Then the maximal operator . σ ∗, defined by   . σ ∗, f := sup σαk f  k∈N

is bounded from the Hardy space Hp to the Lebesgue space Lp . (b) Let 0 < p < 1/2 and (αk , k ∈ N) be a subsequence of positive numbers satisfying the condition sup ρ (αk ) = ∞. k∈N

(7.38)

350

7 Vilenkin-Fejér Means in Martingale Hardy Spaces

Then there exists a martingale f ∈ Hp such that sup σαk f weak−Lp = ∞.

(7.39)

k∈N

Proof (a) Since . σ ∗, is bounded from L∞ to L∞ , the proof of part (a) is complete if 

 ∗,  . σ a (x) < c < ∞

(7.40)

IN

holds for every p-atom a with support IN and μ (IN ) = MN−1 . Analogously to 1/p the proof of Theorem 7.2.8 we may assume that nk > MN . Since a∞ ≤ MN we find that      σn a  ≤ |a (t)| Knk (x − t) dμ (t) (7.41) k IN



≤ a∞ 1/p

≤ MN



IN

IN

  Kn (x − t) dμ (t) k   Kn (x − t) dμ (t) . k

i,j

i,j

Let x ∈ IN and i < j < nk  . Then x − t ∈ IN for t ∈ IN and, according to Lemma 3.2.1, we obtain that   KM (x − t) = 0 for all nk  ≤ l ≤ |nk | . l i,j

Let x ∈ IN , 0 ≤ i < j < nk  ≤ l ≤ |nk | . By applying (7.41) and (3.12) in Lemma 3.2.7, we get that |nk |     σn a (x) ≤ M 1/p k N

l=nk  IN

  KM (x − t) dμ (t) = 0. l

i,j

Let x ∈ IN , where nk  ≤ j ≤ N. Then, in view of Corollary 3.2.15, we can conclude that    Kn (x − t) dμ (t) ≤ cMi Mj . k MN2 IN By using again (7.41) we obtain that   σn a (x) ≤ Cp M 1/p−2 Mi Mj . k N

7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces

351

If we define := mink∈N αk  , then   ∗, i,j . σ a (x) = 0, for x ∈ IN , 0 ≤ i < j ≤

(7.42)

and   ∗, 1/p−2 i,j . Mi Mj , for x ∈ IN , i < ≤ j ≤ N − 1. (7.43) σ a (x) ≤ Cp MN Analogously to (6.34) we can conclude that N − ≤  and 1−2p

MN

1−2p M

≤ R (N− )(1−2p) ≤ R (1−2p) ≤ Cp < ∞,

(7.44)

where R = supk mk . Let p = 1/2. By combining the estimates (7.42)–(7.44) with identity (1.5), we can conclude that   ∗, 1/2 . σ a  dμ IN

=



m k −1

N−2  N−1 

i,j IN

i=0 j =i+1 xk =0,k∈{j +1,...,N−1}

≤



m k −1

−1N−1  

N−1   ∗, 1/2 . σ a  dμ + i=0

 INi,N

 ∗, 1/2 . σ a  dμ

 ∗, 1/2 . σ a  dμ

i,j

i=0 j = xk =0,k∈{j +1,...,N−1} IN



m k −1

N−2  N−1 

+

i= j =i+1 xk =0,k∈{j +1,...,N−1}

≤c

N−1   1/2 Mi i=0

≤ cM 1/2

1

1/2 j = +1 Mj

1 1/2 M

+c

N−2 

+c

1/2

≤ N − + c ≤ C < ∞.

1 1/2

Mi

i=0 N−1 

1/2

Mi

i=

Mi

i=

N−2 

i,j IN

N−1   ∗, 1/2 . σ a  dμ+

1

1/2 j =i+1 Mj

+c

+c

 INi,N

N−1 

Mi

i=0

MN

1/2 1/2

 ∗, 1/2 . σ a  dμ

352

7 Vilenkin-Fejér Means in Martingale Hardy Spaces

Let 0 < p < 1/2. By combining (7.42), (7.43) and (7.44) with (1.5), we have that   ∗, p . σ a  dμ IN

=



m k −1

N−2  N−1 

i,j

N−1   ∗, p .  σ a dμ +

i=0 j =i+1 xk =0,k∈{j +1,...,N−1} IN

≤



m k −1

−1N−1  

+



m k −1

N−2  N−1 

i,j IN

i= j =i+1 xk =0,k∈{j +1,...,N−1} 1−2p

≤ Cp MN



p

Mi

i=0

N−1 

i=0

1

N−1   ∗, p . σ a  dμ + i=0 1−2p

1−p j = +1 Mj

INk,N

 ∗, p . σ a  dμ

 ∗, p . σ a  dμ

i,j IN

i=0 j = xk =0,k∈{j +1,...,N−1}



+ MN

N−2 

p

Mi

i=

N−1 

 INi,N

1

1−p j =i+1 Mj

 ∗, p . σ a  dμ

+ Cp

N−1 

Mi

i=0

MN

p

p

1−2p

≤

Cp MN

1−2p

M

+ Cp

≤ Cp R (|nk |−nk )(1−2p) + Cp < ∞. We conclude that (7.40) holds for 0 < p ≤ 1/2 so the proof of part (a) is complete. (b) Let (nk , k ∈ N) be a sequence of positive numbers satisfying condition (7.38). Then M|nk | = ∞. k∈N Mnk  sup

(7.45)

Under condition (7.45) there exists a sequence (αk , k ∈ N) ⊂ (nk , k ∈ N) such that (1−2p)/2 ∞  Mα  k

(1−2p)/2 k=0 M|αk |

≤ Cp < ∞.

Let f = (fn , n ∈ N) be the martingale defined in Example 5.6.6, where (1/p−2)/2

λk =

RMαk 

(1/p−2)/2

M|αk |

,

R = sup mn . n∈N

(7.46)

7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces

353

By applying Theorem 5.3.2 we can conclude that f ∈ Hp . Using (5.37) with λk defined by (7.46) we obtain that ⎧ * + 1/2p (1/p−2)/2 ⎪ , j ∈ M|αk | , . . . , M|αk |+1 − 1 , k ∈ N+ , ⎨ M|αk | Mαk  ∞ * + f(j ) = ⎪ M|αk | , . . . , M|αk |+1 − 1 . j∈ / ⎩ 0, k=0

(7.47) Therefore, M

|α | 1 k 1 σαk f = Sj f + αk αk j =1

αk 

Sj f =: I + I I.

j =M|α | +1 k

Let M|αk | < j ≤ αk . Then, according to (7.47) we have that 1/2p

(1/p−2)/2

Sj f = SM|α | f + M|αk | Mαk  k

(7.48)

Dj −M|α | . k

By applying (7.48) we can rewrite I I as 1/2p

(1/p−2)/2

M|αk | Mαk  αk − M|αk | II = SM|α | f + k αk αk

ψM|α

k|

αk  j =M|αk | +1

Dj −M|α

k|

=: I I1 + I I2 . In view of Corollary 6.4.10, we find that

p

I I1 weak−Lp ≤

αk − M|αk | αk

p p SM|αk | f

p

weak−Lp

≤ Cp f Hp < ∞. (7.49)

Moreover, by using part (a) of Theorem 7.2.12 for I we obtain that

p

I weak−Lp =

M|αk | αk

p p σM|αk | f

weak−Lp

p

≤ Cp f Hp < ∞.

Under condition (7.38) we can conclude that αk  = |αk |

and

> ? αk − M|αk | = αk  .

(7.50)

354

7 Vilenkin-Fejér Means in Martingale Hardy Spaces α −1,αk 

Let x ∈ Iαkk+1

. According to Lemma 3.2.11 we find that 1/2p

|I I2 | =

αk 1/2p

= ≥

      j =1

(1/p−2)/2 αk −M|αk |

M|αk | Mαk 

(1/p−2)/2

M|αk | Mαk  αk

     αk − M|αk | Kαk −M|α |  k

1/2p−1 (1/p−2)/2  cM|αk | Mαk  αk 1/2p−1

≥ cM|αk |

(1/p+2)/2

Mαk 

   Dj  

    − M|αk | Kαk −M|α |  k

.

It follows that p

I I2 weak−Lp (7.51)

p ' ( (1/p−2)/2 (1/p+2)/2 (1/p−2)/2 (1/p+2)/2 − Cp ≥ Cp M|αk | Mαk  μ x ∈ Gm : |I I2 | ≥ Cp M|αk | Mαk  ( ' 1/2−p 1/2+p k −1,αk  ≥ Cp M|αk | Mαk  μ Iα α +1 k

≥

1/2−p Cp M|αk | 1/2−p Mαk 

→ ∞, as k → ∞.

By now combining (7.49)–(7.51) above we can conclude that p p p σα f p ≥ I I2 weak−Lp − I I1 weak−Lp − I weak−Lp k weak−Lp ≥

1 p I I2 weak−Lp 2 1/2−p

≥

Cp M|αk |

1/2−p

Mαk 

→ ∞, as k → ∞, 

which shows part (b).

The following well-known result (see Weisz [403]) follows immediately from part (a) of Theorem 7.2.12. Corollary 7.2.13 Let 0 < p ≤ 1. Then the maximal operator σ # defined by   σ # f := sup σMn f  n∈N

is bounded from the Hardy space Hp to the Lebesgue space Lp .

7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces

355

Proof It is obvious that |Mn | = [Mn ] = n

sup ρ (Mn ) = 0 ≤ C < ∞.

and

n∈N

Since condition (7.37) is satisfied, the proof follows from part (a) of Theorem 7.2.12 for the case 0 < p ≤ 1/2. For 1/2 < p ≤ 1, it follows from Theorem 7.2.1.    Theorem 7.2.14 Let 0 < p ≤ 1. Then the operator σMn f  is not bounded from the martingale Hardy space Hp to Hp . Proof Let fk be the p-atom from Example 5.6.2. By combining (1.40) in Lemma 1.6.4 and (5.23), (5.24) in Example 5.6.2, we find that σM2nk +1 fk =

=

=

=

M2nk +1



1 M2nk +1

Sj fk

j =1 M2nk +1



1 M2nk +1

Sj fk

j =M2nk +1 M2nk +1



1 M2nk +1

j =M2nk +1



Dj − DM2nk

 Dj +M2nk − DM2nk

M2nk

1 M2nk +1

j =1

M

=

2nk ψM2nk 

M2nk +1

Dj =

j =1

ψM2nk m2nk

KM2nk .

It is evident that SMN

     σM2nk +1 fk (x) =

    σM2nk fk (t) DMN (x − t) dμ (t)



    σM2nk fk (t) DMN (x − t) dμ (t)

Gm

≥ I2nk



≥c I2nk

≥ cM2nk

    KM2nk +1 (t) DMN (x − t) dμ (t)  I2nk

DMN (x − t) dμ (t) .

356

7 Vilenkin-Fejér Means in Martingale Hardy Spaces

Since ψj (t) = 1,

for t ∈ I2nk , j = 0, 1, . . . , M2nk − 1,

we obtain that     SMN σM2nk +1 fk (x) ≥ cM2nk

1 DMN (x) , N = 0, 1, . . . , MN − 1, M2nk

so that     SMN σM2nk +1 fk (x) ≥ c

sup 1≤N 0) .

(8.1)

Then there exists a martingale f ∈ Hp , such that sup tn f weak−Lp = ∞. n∈N

(8.2)

400

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Proof (a) Let the sequence (qk , k ∈ N) be non-decreasing. By combining (4.23) and (4.24) and using Abel transformation, we get that     n  1    |tn f | ≤  qn−j Sj f  Q  n j =1  ⎛ ⎞ n−1      1 ⎝ qn−j − qn−j −1  j σj f  + q0 n |σn f |⎠ ≤ Qn j =1

⎞ n−1  c ⎝  qn−j − qn−j −1 j + q0 n⎠ σ ∗ f ≤ Qn ⎛

j =1

≤ σ ∗ f, so that t ∗ f ≤ σ ∗ f.

(8.3)

In view of (8.3) and Theorem 7.2.2 we can conclude that the maximal operators t ∗ is bounded from the Hardy space H1/2 to the space weak − L1/2 . The proof of part (a) is complete. Let f = (fn , n ∈ N) be the martingale defined in Example 5.6.7 in the case when 0 < p < q = 1/2. We have that tM2αk +1 f (x) =

M2αk



1 QM2αk +1

j =0

qM2αk +1−j Sj f (x) +

q0 QM2αk +1

SM2αk +1 f (x)

=: I + I I. According to (5.46) in Example 5.6.7, we can conclude that |I | ≤

M2αk



1 QM2αk +1 1/p

≤

2RM2αk−1

1

1/2 αk−1

QM2αk +1

1/p

≤

2RM2αk−1 1/2

αk−1 1/p−2

≤

j =0

    qj SM2αk +1−j f (x)

Mαk

3/2

16αk

.

M2αk

 j =0

qM2αk +1−j

(8.4)

8.2 Maximal Operators of Nörlund Means on Hp Spaces

401

Let x ∈ Gm . If we now apply (5.47) for I I we find that |I I | = ≥

q0 QM2αk +1 q0

    SM2αk +1 f (x)

(8.5)

1/p−1

M2αk

QM2αk +1 4α 1/2 k

.

Without loss the generality we may assume that c = 1 in (8.1). By combining (8.4) and (8.5) for any x ∈ Gm , we obtain that     tM2αk +1 f (x) ≥ |I I | − |I | ≥

1/p−1

M2αk

q0

QM2αk +1 4α 1/2 k 1/p−2

≥

M2αk

1/2

4αk

1/p−2

−

4RMαk

3/2

αk

1/p−2

−

4RMαk

3/2

αk

1/p−2

≥

M2αk

1/2

.

8αk

Moreover, )

  M 1/p−2   2αk μ x ∈ Gm : tM2αk +1 f (x) ≥ 1/2 8αk

: = μ (Gm ) = 1.

Therefore, 1/p−2

M2αk

1/2

8αk

)

 M 1/p−2    2αk μ x ∈ Gm : tM2αk +1 f (x) ≥ 1/2 8αk

: (8.6)

1/p−2

=

M2αk

1/2

8αk

μ (Gm )

1/p−2

=

M2αk

1/2

8αk

→ ∞, as k → ∞.

In view of (8.6), we conclude that (8.2) holds.



402

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

One immediate consequence of Theorem 8.2.1 is the following: Corollary 8.2.2 (a) Let f ∈ H1/2 . Then there exists an absolute constant c such that ∗ σ f ≤ c f H1/2 , weak−L1/2 ∗ B f

weak−L1/2

≤ c f H1/2

and ∗ β f ≤ c f H1/2 . weak−L1/2 (b) Let 0 < p < 1/2. Then there exists a martingale f ∈ Hp such that ∗ σ f =∞ weak−Lp ∗ B f

weak−Lp

=∞

and ∗ β f = ∞. weak−Lp Next, we consider weighted maximal operators of Nörlund means with nondecreasing sequences (qk , k ∈ N). Theorem 8.2.3 Let 0 < p < 1/2 and the sequence (qk , k ∈ N) be non-decreasing. Then the maximal operator . tp∗ defined by . tp∗ f := sup

n∈N

|tn f | (n + 1)1/p−2

is bounded from the Hardy martingale space Hp to the Lebesgue space Lp . Proof The idea of the proof is similar to that of part (a) of Theorem 7.2.4, but the situation is more general here, so we give the details. First we note that tn is bounded from L∞ to L∞ (see Lemma 4.4.6). Let a be an arbitrary p-atom, with support I and μ (I ) = MN−1 . We may assume that I = IN . It is easy to see that Sn a (x) = tn a (x) = 0,

when n ≤ MN .

8.2 Maximal Operators of Nörlund Means on Hp Spaces

403

Therefore, we can suppose that n > MN . Hence, 1  qn−k Sk a(x) Qn n

tn a(x) =

(8.7)

k=1

=

n 1  qn−k Sk a(x) Qn k=MN

=

 n 1  qn−k a (x) Dk (x − t) dμ (t) . Qn IN k=MN

1/p

Since a∞ ≤ MN , it follows that |tn a(x)| (n + 1)1/p−2

≤



a∞ (n + 1)1/p−2



1/p

≤

IN

MN

(n + 1)1/p−2

IN

    n  1    qn−k Dk (x − t) dμ (t) Q  n k=MN      n  1     dμ (t) . q D − t) (x n−k k Q   n  k=MN

Let x ∈ INk,l , 0 ≤ k < l ≤ N. According to Lemma 4.4.5 we can deduce that 1/p

|tn a(x)| (n + 1)1/p−2

≤

cp MN

1/p−2 MN

Ml Mk = cp Ml Mk . MN2

Therefore,    .∗ tp a (x) ≤ cp Ml Mk

for x ∈ INk,l , 0 ≤ k < l ≤ N.

By combining (1.5) with (8.8) we obtain that  IN

=

+

p   .∗ tp a (x) dμ (x)

N−2  N−1 

mj−1





k,l k=0 l=k+1 xj =0,j ∈{l+1,...,N−1} IN

N−1  k=0

INk,N

p   .∗ tp a (x) dμ (x)

p   .∗ tp a (x) dμ (x)

(8.8)

404

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

≤ cp

N−2  N−1  k=0 l=k+1

≤ cp

N−2  N−1  k=0 l=k+1

 1 ml+1 . . . mN−1 p p p p Ml Mk + cp MN Mk MN MN N−1 k=0

 M Ml Mk k + cp ≤ cp < ∞. 1−p Ml M p

p

p

N−1 k=0

N



The proof is complete.

Remark 8.2.4 Since the Fejér means are examples of Nörlund means with nondecreasing sequence (qk , k ∈ N), we immediately obtain from part (b) of Theorem 7.2.4 that the asymptotic behaviour of the sequence of weights

1/ (k + 1)1/p−2 , k ∈ N can not be improved in Theorem 8.2.3. .p∗ and β .p∗ Corollary 8.2.5 Let 0 < p < 1/2. Then the maximal operators . σp∗ , B defined by . σp∗ f := sup

n∈N

|σn f | (n + 1)

1/p−2

.p∗ f := sup B

,

n∈N

|Bn f | (n + 1)1/p−2

and .p∗ f := sup β

n∈N

|βn f | (n + 1)1/p−2

,

respectively, are bounded from Hp to Lp . Next, we state the following result for a weighted maximal operator of T means. Theorem 8.2.6 Let the sequence (qk , k ∈ N) be non-decreasing. Then the maximal ∼∗

operator t 1 defined by ∼∗ t 1f

:= sup n∈N

|tn f | log (n + 1) 2

is bounded from the Hardy space H1/2 to the Lebesgue space L1/2 .

8.2 Maximal Operators of Nörlund Means on Hp Spaces

405

Proof Analogously to the proof of Theorem 8.2.3, we may assume that n > MN and a is a p-atom with support I = IN . Since a∞ ≤ MN2 if we apply (8.7), then we obtain that      n   1  a∞ |tn a(x)|  dμ (t)  ≤ q D − t) (x n−k k   2 2 log (n + 1) log (n + 1) IN  Qn k=M  N      n  1   MN2  ≤ qn−k Dk (x − t) dμ (t) .  2 log (n + 1) IN  Qn k=M  N Let x ∈ INk,l , 0 ≤ k < l ≤ N. Then, from Lemma 4.4.5, it follows that |tn a (x)| log2

(n + 1)

≤

cMN2 log2

Ml Mk cMl Mk = 2 log2 (n + 1) (n + 1) MN

and  cMl Mk  ∗ . . t1 a (x) ≤ N2

(8.9)

By combining (1.5) with (8.9), we find that 

1/2  ∗ . t1 a (x) dμ (x) IN

=



k,l k=0 l=k+1 xj =0,j ∈{l+1,...,N−1} IN

+

N−1  k=0

≤c

INk,N

N−2  N−1  k=0 l=k+1

≤



mj −1

N−2  N−1 

N−2  N−1 

1/2  ∗ . t1 a (x) dμ (x) 1/2

1/2

ml+1 . . . mN−1 Ml Mk MN N

1/2 Mk c 1/2 k=0 l=k+1 NMl

The proof is complete.

1/2   ∼∗ t1 a (x) dμ (x)  

+c

N−1 

1 1/2

k=0 MN

+c

1/2 Mk

N

N−1  k=0

1/2

1/2

1 MN Mk MN N

≤ c < ∞. 

406

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Remark 8.2.7 As in Remark 8.2.4, we obtain from part (b) of Theorem 7.2.8 that the asymptotic behaviour of the sequence of weights

1/ log2 (n + 1) , n ∈ N can not be improved in Theorem 8.2.6. As a consequence we immediately get the following result. ∼∗ ∼ ∗

∼∗

Corollary 8.2.8 The maximal operators σ 1 , B 1 and β 1 defined by |σn f |

∼∗

σ 1 f := sup

n∈N

log (n + 1) 2

∼∗

B 1 f := sup

,

n∈N

|Bn f | log2 (n + 1)

and ∼∗

β 1 f := sup

n∈N

|βn f | log (n + 1) 2

,

respectively, are bounded from H1/2 to L1/2 . The next results deal with Nörlund means with non-increasing sequences (qk , k ∈ N). We begin by stating a divergence result for all such summability methods when 0 < p < 1/2. Theorem 8.2.9 Let 0 < p < 1/2. Then, for all Nörlund means with non-increasing sequence (qk , k ∈ N) there exists a martingale f ∈ Hp such that sup tn f weak−Lp = ∞. n∈N

Proof We use the martingale defined in Example 5.6.7 (see also the proof of part (b) of Theorem 8.2.1). It is obvious that for every non-increasing sequence (qk , k ∈ N), q0 QM2αk +1

≥

1 M2αk + 1

.

Since tM2αk +1 f =

1 QM2αk +1

=: I + I I.

M2αk

 j =0

qM2αk +1−j Sj f +

q0 SM +1 f QM2αk +1 2αk

8.2 Maximal Operators of Nörlund Means on Hp Spaces

407

By combining (8.4) and (8.5), we see that 1/p−2   M2αk   . tM2αk +1 f  ≥ |I I | − |I | ≥ 1/2 8αk

Analogously to (8.6) we get that sup tM2αk +1 f

weak−Lp

k∈N

= ∞. 

The proof is complete.

Corollary 8.2.10 Let 0 < p < 1/2 and tn be Nörlund means with non-increasing sequence (qk , k ∈ N). Then the maximal operator t ∗ is not bounded from the martingale Hardy space Hp to the space weak − Lp , that is, there exists a martingale f ∈ Hp such that sup t ∗ f weak−Lp = ∞.

n∈N

Next, we state the following result for the Nörlund means with non-increasing sequence (qk , k ∈ N). Theorem 8.2.11 (a) The maximal operator t ∗ of the Nörlund summability method with nonincreasing sequence (qk , k ∈ N) satisfying the condition (4.5) and (4.6), is bounded from the Hardy space H1/(1+α) to the space weak − L1/(1+α) for 0 < α ≤ 1. (b) Let 0 < α ≤ 1 and (qk , k ∈ N) be a non-increasing sequence satisfying the conditions nα ≥ Cα > 0 n→∞ Qn lim

(8.10)

and |qn − qn+1 | ≥ Cα nα−2 ,

n ∈ N.

Then there exists a martingale f ∈ H1/(1+α) such that sup tn f 1/(1+α) = ∞.

n∈N

(8.11)

408

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Proof (a) By Theorem 5.5.2 the proof of part (a) is complete if we prove that ( ' tμ x ∈ I : t ∗ f ≥ t 1+α ≤ c < ∞,

t ≥ 0,

(8.12)

for every 1/ (1 + α)-atom a. We may assume that a is an arbitrary 1/ (1 + α)atom with support I, μ (I ) = MN−1 and I = IN . It is easy to see that tm a (x) = 0,

when m ≤ MN .

(8.13)

Therefore, we can suppose that m > MN . Let t ∈ IN . Since tm is bounded from L∞ to L∞ (the boundedness follows from Corollary 4.5.16) and 1/(1+α)

a∞ ≤ MN

,

we obtain that  |a (t)| |Fm (x − t)| dμ (t)

|tm a (x)| ≤ IN

≤ a∞

 |Fm (x − t)| dμ (t) IN

 ≤ MN1+α

|Fm (x − t)| dμ (t) . IN

Let x ∈ INk,l , 0 ≤ k < l ≤ N and m > MN . From Corollary 4.4.16 we get that |tm a (x)| ≤ Cα Mk Mlα . Hence,  ∗ t a (x) ≤ Cα Mk M α . l

(8.14)

Let n ≥ N. According to (8.14) we can conclude that ( ' μ x ∈ IN : t ∗ f ≥ Cα Ms1+α = 0, s = N + 1, N + 2, . . .

(8.15)

Thus, we can suppose that 0 < s ≤ N. Let R = supn mn . It is obvious that for fixed R there exists θ ∈ N so that 2θ−1 ≤ R < 2θ .

8.2 Maximal Operators of Nörlund Means on Hp Spaces

409

Suppose that Mk Mlα > Ms1+α for some s ≤ k < l ≤ N.

(8.16)

If l > s > k, we can conclude from (8.16) that Mlα > Msα (mk mk+1 ms−1 ) ≥ 2s−k Msα ≥ R (s−k)/θ Msα , (ms ms+1 . . . ml−1 )α > R (s−k)/θ , R (l−s)α > R (s−k)/θ , R l−s > R (s−k)/θ , l − s = ln(M l−s ) > ln(R (k−s)/θ ) = (s − k)/θ , l > (s − k)/θ + s. Hence, Ml Mk ≥ Ms1+α , where s > k, l > (s − k)/θ + s.

(8.17)

It is evident that (7.18) holds true for all l > k ≥ s, that is, Ml Mk > Ms1+α , where l > k ≥ s.

(8.18)

By applying (1.5) and (7.16), combining with (8.17) and (8.18) we get that N ' ' ( N−1 (   x ∈ IN : t ∗ f ≥ Cα Ms1+α ⊂ x ∈ INk,l : t ∗ f ≥ Cα Ms1+α k=s l=k+1 s 

N 

' ( x ∈ INk,l : t ∗ f ≥ Cα Ms1+α

k=0 l=(s−k)/θ+s

and N ( N−1 '   ∗ 1+α ≤ μ x ∈ IN : t f ≥ Cα Ms

   k,l  IN 

mj−1



k=s l=k+1 xj =0, j ∈{l+1,...,N−1}

+

s 

   k,l  IN 

mj−1

N 



k=0 l=[(s−k)/θ]+s xj =0, j ∈{l+1,...,N−1}

≤

N−1 

n N   1 + Ml

k=s l=k+1

N 

k=0 l=[(s−k)/θ]+s

1 Ml

410

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces N−1 

≤

k=s

 1 1 + Mk M[(s−k)/θ]+s s

k=0

Cα . Ms

≤

(8.19)

In view of (8.15) and (8.19), we can conclude that ( ' sup Mn μ x ∈ IN : t ∗ f ≥ Cα Mn1+α ≤ cα < ∞

n∈N

and it follows that (8.12) holds so part (a) is proved. (b) Under condition (8.10) there exists an increasing sequence (qk , k ∈ N) of positive integers such that α M2α k +1

QM2αk +1

> Cα > 0, k ∈ N.

(8.20)

To prove part (b), we use the martingale defined in Example 5.6.7 in the case when p=q =

1 . 1+α

In particular, as we proved there, the martingale belongs to the space H1/(1+α) . Moreover, ⎧ α * + M2α ⎪ ⎪ ⎨ α 1/2k , j ∈ M2αk , . . . , M2αk +1 − 1 , k ∈ N, k f(j ) = ∞ * + ⎪ ⎪ M2αk , . . . , M2αk +1 − 1 . j∈ / ⎩ 0, k=1

We have that tM2αk +M2s f =

+

1 QM2αk +M2s 1 QM2αk +M2s

=: I + I I.

M2αk

 j =0

qM2αk +M2s −j Sj f

M2αk +M2s



j =M2αk +1

qM2αk +M2s −j Sj f

8.2 Maximal Operators of Nörlund Means on Hp Spaces

411

According to (5.46) we can conclude that |I | ≤

≤

≤

M2αk



1 QM2αk +M2s

j =0

α+1 2RM2α k−1 1/2

  qM2αk +M2s −j Sj f  M2αk



1 QM2αk +M2s

αk−1 α+1 2RM2α k−1 1/2

αk−1

≤

(8.21)

Mααk 3/2

j =0

qM2αk +M2s −j

.

16αk

Let x ∈ Is /Is+1 and M2αk + 1 ≤ j ≤ M2αk + M2s . In view of the second inequality of (5.45) in the case when l = k and p = 1/ (1 + α), we find that Sj f = SM2α f +

α ψ M2α M2α Dj −M2α k k

k

1/2

k

.

αk

Hence, II =

+

M2αk +M2s



1 QM2αk +M2s

j =M2αk +1

qM2αk +M2s −j

α ψ M2α M2αk Dj −M2αk k 1/2

αk

M2αk +M2s



1 QM2αk +M2s

j =M2αk +1

qM2αk +M2s −j SM2α f k

=: I I1 + I I2 . By using again (5.46) we get that |I I2 | ≤

≤

α+1 2RM2α k−1 1/2

αk−1 α+1 2RM2α k−1 1/2

αk−1

M2αk +M2s



1 QM2αk +M2s ≤

Mαα+1 k 1/2

16αk

.

j =M2αk +1

qM2αk +M2s −j

412

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Let x ∈ Is /Is+1

for

[αk /2] < s ≤ αk .

Then, according to (8.20) we find that |I I1 | =

=

≥

≥

    M2s α   ψM2αk M2α  k  qM2s −j Dj  1/2 QM2αk +M2s   αk j =1     α M2α  M2s  k   q j M2s −j   1/2  αk QM2αk +M2s j =1     α M2α   M2s k  qM2s −j j   1/2  αk QM2αk +1 j =1      M2s  c  qM2s −j j  . 1/2  α  R αk j =1 1

We invoke Abel transformation and apply (8.11) to get that      M2s    q j M2s −j  =  j =1  ≥

     M2s   j (j + 1)    q − q M2s −j M2s −j −1   2  j =1 2 cα M2s

M2s 

1/2 αk

j =[M2s /2]

2 ≥ cα M2s

 q M

 − qM2s −j −1  j 2

 q M

 − qM2s −j −1 

2s −j

M2s 

2s −j

j =[M2s /2] 2 ≥ cα M2s

[M 2s /2]

  qj − qj +1 

j =1 2 ≥ cα M2s

[M 2s /2] j =1

1 j α−2

α−1 2 α+1 ≥ cα M2s M2s ≥ cα M2s .

8.2 Maximal Operators of Nörlund Means on Hp Spaces

413

Hence, |I I1 | ≥

Cα 1/2

αk

    α+1  M2s  Cα M2s  qM2s −j j  ≥ .  1/2 j =1  αk

By now using the estimates above we obtain that  Gm

 1/(1+α)   dμ ≥ |I I1 | − |I I2 | − |I | tM2αk +M2s f  ≥

1+α Cα M2s

−

1/2

αk

4RMαα+1 k 3/2

αk

(8.22) ≥

1+α Cα M2s 1/2

.

αk

By combining (8.21) and (8.22), we find that 

α k −1

 ∗ 1/(1+α) t f  dμ ≥

Gm



s=[αk /2] Is /Is+1

≥ Cα

α k −1

M2s 1/2(1+α)

s=[αk /2]

≥ Cα

≥ ≥

1/(1+α)    dμ tM2αk +M2s f 

α k −1

M2s αk 1

1/2(1+α) s=[αk /2] αk

Cα

α k −1

1

1/2(1+α)

αk

s=[αk /2]

Cα αk 1/2(1+α) αk

≥ Cα αk

1/2

→ ∞, as k → ∞,

which proves part (b).  Corollary 8.2.12 (a) Let 0 < α ≤ 1. Then the maximal operators σ α,∗ and V α,∗ are bounded from H1/(1+α) to weak − L1/(1+α) . (b) There exists a martingale f ∈ H1/(1+α) such that sup σnα f 1/(1+α) = ∞

n∈N

414

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

and sup Vnα f 1/(1+α) = ∞

n∈N

Our next result reads as follows. Theorem 8.2.13 (a) Let 0 < p < 1/ (1 + α) , 0 < α ≤ 1 and (qk , k ∈ N) be a non-increasing sequence satisfying the condition nα = c > 0, 0 < α ≤ 1. n→∞ Qn lim

(8.23)

Then there exists a martingale f ∈ Hp such that sup tn f weak−Lp = ∞.

(8.24)

n∈N

(b) Let (qk , k ∈ N) be a non-increasing sequence satisfying the condition nα = ∞, n→∞ Qn lim

(0 < α ≤ 1) .

(8.25)

Then there exists a martingale f ∈ H1/(1+α) , such that sup tn f weak−L1/(1+α) = ∞.

(8.26)

n∈N

Proof (a) Under condition (8.23), there exists an increasing sequence (αk , k ∈ N) of positive integers such that α M2α k

QM2αk

≥ c, k ∈ N

and in addition, all conditions in Example 5.6.7 are fulfilled. To prove part (a) we use the martingale defined in Example 5.6.7 in the case when 0 < p < q = 1/ (1 + α) . Since tM2αk +1 f =

1 QM2αk +1

=: I + I I.

M2αk

 j =0

qM2αk +1−j Sj f +

1 SM +1 f QM2αk +1 2αk

8.2 Maximal Operators of Nörlund Means on Hp Spaces

415

By combining (8.4) and (8.5), we get that 1/p−1 1/p   M2αk 2RMαk−1 1   − . tM2αk +1 f  ≥ |I I | − |I | = 1/2 Q 1/2 M2αk +1 4αk αk−1

Without loss of generality, we may assume that c = 1 in (7.79). By using (5.43) we find that 1/p  M 1/p−1−α  2RMαk−1   2αk − tM2αk +1 f  ≥ 1/2 1/2 4αk αk−1 1/p−1−α

≥

M2αk

1/p−1−α

−

1/2

4αk

RMαk

3/2

16αk

1/p−1−α

≥

M2αk

1/2

8αk

so that 1/p−1−α

M2αk

8αk

)

  M 1/p−1−α   2αk · μ x ∈ Gm : tM2αk +1 f  ≥ 8αk

:

1/p−1−α

=

M2αk

8αk

→ ∞,

as k → ∞. This means that (8.24) holds and part (a) is proved. (b) Under the condition (8.25) there exists an increasing sequence (αk , k ∈ N), which satisfies the conditions 1/2(1+α) +1 k

∞ Q  M2α

α/2(1+α)

k=0 k−1 

M2αk

α/2+1

QM2αη +1 M2αη

≤ c < ∞,

(8.27)

α/2+1

≤ QM2αk +1 M2αk

(8.28)

η=0

and α/2

α/2+1

32RQM2αk−1 +1 M2αk−1
MN . Let x ∈ IN . Since a∞ ≤ MN , we obtain that  |tn a (x)| ≤ |a (t)| |Fn (x − t)| dμ (t) IN



≤ a∞ 1/p

≤ MN

|Fn (x − t)| dμ (t) 

IN

|Fn (x − t)| dμ (t) . IN

420

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Let x ∈ INk,l , 0 ≤ k < l < N. Then, from Lemma 4.4.15 we get that 1/p−1

|tn a (x)| ≤

cα,p MN

Mlα Mk

(8.37)

nα

and |tn a (x)| ≤ cα,p Mlα Mk . n1/p−1−α

(8.38)

Hence,    ∼∗  t a (x) ≤ cα,p M α Mk . l   p,α

(8.39)

For x ∈ INk,N , 0 ≤ k < N., by Lemma 4.4.15, we have that 1/p−1

|tn a (x)| ≤ cα,p MN

(8.40)

Mk .

By combining (1.5) and (8.39) we obtain that  IN

  N−2  N−1  ∼∗ p  t a  dμ =  p,α 

mj −1





k,l k=0 l=k+1 xj =0,j ∈{l+1,...,N−1} IN

+

N−1  k=0

≤ Cα,p

INk,N

  ∼∗ p  t a  dμ  p,α 

N−2  N−1  k=0 l=k+1

≤ Cα,p

N−2  k=0

  ∼∗ p  t a  dμ  p,α 

p

Mk

 1 1 αp p 1−p p Ml Mk + Cα,p M Mk Ml MN N

N−1 

N−1 k=0

1

1−αp l=k+1 Ml

+ Cα,p

N−1 

Mk

k=0

MN

p

p

≤ Cα,p < ∞.

Thus (8.36) as well as part (a) hold. (b) Let 0 < p < 1/ (1 + α) . Under condition (8.34), there exist positive integers nk such that  lim

k→∞

1/p−1−α M2nk + 1 = ∞, 0 < p < 1/ (1 + α) . M2nk +1

8.2 Maximal Operators of Nörlund Means on Hp Spaces

421

To prove part (b), we apply the p-atoms defined in Example 5.6.2. Under conditions (4.5) and (4.6), if we invoke (5.23) and (5.24) we find that         tM2nk +1 fk  SM2nk +1  = M2nk +1 QM2nk +1 M2nk +1     q0 DM2nk +1 − DM2nk  = QM2nk +1 M2nk +1     q0 ψM2nk  = QM2nk +1 M2nk +1 =

Cα . α  M2n M2nk +1 k

Hence, ⎧ ⎨

μ x ∈ Gm : ⎩

    tM2nk +1 fk (x) M2nk +1

≥

⎫ ⎬

Cα = 1. α  M2n M2nk +1 ⎭ k

(8.41)

By combining (5.25) and (8.41) we have that  ) α M2n

k

Cα M2n

k

+1

μ x ∈ Gm :

    tM2n +1 fk (x) k M2n +1 k

:1/p ≥

α M2n

k

q0 M2n

k

+1

fk Hp 1/p−1−α

≥

≥

Cα M2nk

M2nk +1

1/p−1−α Cα M2nk + 1 M2nk +1

→ ∞, as k → ∞.

This means that (8.35) holds and part (b) is also proved.  Theorem 8.2.19 (a) Let f ∈ Hp , where 0 < p < 1/ (1 + α) for some 0 < α ≤ 1 and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying conditions (4.8) and (4.9).

422

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces ∼∗

Then the maximal operator t p,α defined by ∼∗ t p,α f

:= sup n∈N

|tn f | (n + 1)1/p−1−α

is bounded from the martingale Hardy space Hp to the Lebesgue space Lp . (b) Let 0 < p < 1/ (1 + α) for some 0 < α ≤ 1 and (n , n ∈ N+ ) be any nondecreasing sequence satisfying the condition (n + 1)1/p−1−α = ∞. n→∞ n lim

(8.42)

Then there exists a Nörlund mean with non-increasing sequence (qk , k ∈ N) satisfying the conditions (4.8) and (4.9) and a martingale f ∈ Hp such that

sup

tM2n +1 fk k M2n +1

weak−Lp

k

fk Hp

k∈N

= ∞.

(8.43)

Proof (a) According to Lemma 4.4.18, we can conclude that 

|Fn (t)| dt Gm

(n + 1)1/p−1−α

⎧ ⎫  |n|  ⎬  cα ⎨ α KM (t) dt ≤ Mj ϕMj j ⎭ Qn ⎩ Gm

(8.44)

j =0

≤

α cα M|n| ϕM|n|

Qn (n + 1)1/p−1−α cα ϕM|n| ≤ < Cα < ∞, (n + 1)1/p−1−α ∼∗,p

which means that the operator t is bounded from L∞ to L∞ . Hence, according to Theorem 5.5.1 the proof of part (a) will be complete, if we prove that p   ∗,p ∼   t a(x) dμ (x) < ∞, (8.45)   IN

for every p-atom a. We may assume that a is an arbitrary p-atom with support I, μ (I ) = MN−1 and I = IN . Since tn a = 0, when n ≤ MN , we can suppose

8.2 Maximal Operators of Nörlund Means on Hp Spaces

423

1/p

that n > MN . Let x ∈ IN . Since a∞ ≤ MN

we obtain that

 |a (t)| |Fn (x − t)| dμ (t)

|tn a (x)| ≤ IN



≤ a∞

1/p MN

|Fn (x − t)| dμ (t) ≤ IN

 |Fn (x − t)| dμ (t) . IN

Let x ∈ Il+1 (sk ek + sl el ) , 0 ≤ k < l < N. From Lemma 4.4.19 it follows that 1/p

|tn a (x)| ≤

Cα Mlα ϕMl Mk MN nα MN

≤

Cα ϕMl Mlα Mk 1−1/p

nα MN

≤

Cα ϕMl Mlα Mk 1+α−1/p

.

(8.46)

MN

Let x ∈ IN (sk ek ) , 0 ≤ k < N. According to Lemma 4.4.19 we have that 1/p

|tn a (x)| ≤

Cα MN Mk 1/p−1 ≤ Cα MN Mk . MN

(8.47)

By combining (1.5) with (8.46)–(8.47) we obtain that  IN

=

 ∼ p  ∗,p  t a(x) dμ(x)  

k=0 sk =1 l=k+1 sl =1

+

   tn a(x) p  sup  1/p−1−α  dμ(x) n

N−2 k −1 N−1 l −1   m  m

   tn a p  sup  1/p−1−α  dμ n

N−1 k −1   m k=0 sk =1

≤ cα

IN (sk ek ) n>MN

N−2  N−1  k=0 l=k+1

≤ cα

Il+1 (sk ek +sl el ) n>MN

N−2  N−1  k=0 l=k+1

 (mk − 1) αp p p (mk − 1) (ml − 1)  MN Mk ϕMl Mlα Mk + cα Ml+1 MN N−1 k=0



ϕMl Mlα Mk Ml+1

p + cα

N−1 

p

Mk

(1−αp)

k=0

MN

and  IN

 ∼ p N−2  p N−1   ∗,p  t a(x) dμ(x) ≤ cα Mk  

1−αp l=k+1 Ml

k=0

≤ cα

N−2  k=0

p

ϕMl

p

p

Mk

ϕMk 1−αp

Mk

+ cα

+ cα

424

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

≤ cα

Mk

+ cα

p

N−2 

ϕMk p(1/p−(1+α))

k=0

≤ cα

ϕMk 1−p(α+1)

k=0

≤ cα

p

N−2 

N−2 

Mk

+ cα

1

p(1/p−(1+α))/2 k=0 Mk

+ cα ≤ Cα < ∞.

Thus (8.45) holds and the proof of part (a) is complete. (b) Under condition (8.42) there exists positive integers (nk , k ∈ N) such that  lim

k→∞

1/p−1−α M2nk + 1 = ∞, 0 < p < 1/ (1 + α) . M2nk +1

To prove part (b), we apply the p-atoms defined in Example 5.6.2. Under conditions (4.8) and (4.9), if we invoke (5.23) and (5.24) we find that         q0 SM2nk +1  tM2nk +1 fk  = M2nk +1 QM2nk +1 M2nk +1     q0 DM2nk +1 − DM2nk  = QM2nk +1 M2nk +1     q0 ψM2nk  = QM2nk +1 M2nk +1 ≥

Cα . α  M2n M2nk +1 ϕM2nk +1 k

Hence, ⎧ ⎨

μ x ∈ Gm : ⎩

    tM2nk +1 fk (x) M2nk +1

≥

⎫ ⎬

Cα = 1. α  M2n M2nk +1 ϕM2nk +1 ⎭ k

(8.48)

8.2 Maximal Operators of Nörlund Means on Hp Spaces

425

By applying (8.48) we have that  ) Cα

α  M2n M2n k

k

+1 ϕM2n +1 k

μ x ∈ Gm :

    tM2n +1 fk (x) k M2n +1

:1/p ≥

q0

α  M2n M2n k

k

k

+1 ϕM2n +1 k

fk Hp 1/p−1−α

≥

Cα M2nk

M2nk +1 ϕM2nk +1  1/p−1−α Cα M2nk + 1 ≥ → ∞, as k → ∞. M2nk +1 ϕM2nk +1 This proves (8.43) as well as part (b).  We also state the following consequence of Theorem 8.2.19: Corollary 8.2.20 Let f ∈ Hp , where 0 < p < 1/ (1 + α) for some 0 < α ≤ 1. ∼ α,∗

∼ α,∗

Then the maximal operators σ p,α and V p,α defined by ∼ α,∗ σ p,α f

:=

 α  σ f  n

(n + 1)1/p−1−α

and  α  V f 

∼ α,∗

V p,α f :=

n

(n + 1)1/p−1−α

,

respectively, are bounded from Hp to Lp . (b) Let 0 < p < 1/ (1 + α) for some 0 < α ≤ 1 and (n , n ∈ N) be any non-negative, non-decreasing sequence satisfying condition (8.34). Then

sup k∈N

σα M2n +1 fk k M2nk +1

weak−Lp

fk Hp

=∞

and

sup k∈N

Vα M2n +1 fk k M2n +1 k

weak−Lp

fk Hp

= ∞.

We now formulate our next result in this Section.

426

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Theorem 8.2.21 (a) Let f ∈ H1/(1+α), where 0 < α ≤ 1 and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying conditions (4.5) and (4.6). Then the maximal ∼∗

operator t α defined by ∼∗ tα

:=

|tn f | 1+α

log

(n + 1)

is bounded from H1/(1+α) to L1/(1+α). (b) Let (n , n ∈ N+ ) be any non-decreasing sequence satisfying the condition log1+α (n + 1) = +∞. n→∞ n lim

(8.49)

Then there exists a Nörlund mean with non-increasing sequences (qk , k ∈ N) satisfying conditions (8.10) and (8.11) such that    t n f k  supn    n 1/(1+α) sup = ∞. (8.50) f H1/(1+α) k∈N Proof (a) According to Theorem 5.5.1 the proof of part (a) will be complete if we prove that  IN

  ∼∗ 1/(1+α)  t a dμ < ∞  α 

(8.51)

for every 1/ (1 + α)-atom a. We may assume that a is an arbitrary 1/ (1 + α)atom with support I, μ (I ) = MN−1 and I = IN . It is easy to see that tm a (x) = 0, when m ≤ MN . Therefore we can suppose that m > MN . Let x ∈ IN . Since tm is bounded from L∞ to L∞ (the boundedness follows 1/(1+α) from Corollary 4.5.16) and a∞ ≤ MN , we obtain that  |a (t)| |Fm (x − t)| dμ (t)

|tm a (x)| ≤ IN

≤ a∞

 |Fm (x − t)| dμ (t) IN

 ≤ MN1+α

|Fm (x − t)| dμ (t) . IN

8.2 Maximal Operators of Nörlund Means on Hp Spaces

427

Let x ∈ INk,l , 0 ≤ k < l < N. From Lemma 4.4.15 it follows that Cα Mk Mlα MNα mα

|tm a (x)| ≤

(8.52)

and so, for n > MN , that |tn a (x)| log1+α n

≤

Cα Mk Mlα . N 1+α

Hence,   α  ∼∗ tα a (x) ≤ Cα Mk MN .   N 1+α

(8.53)

Let x ∈ INk,N , 0 ≤ k < N. In view of Lemma 4.4.15 we have that |tm a (x)| ≤ Cα Mk MNα .

(8.54)

According to (1.5) and (8.53) we obtain that  IN

=

   ∼ ∗ 1/(1+α) tα a  dμ   mj −1

N−2  N−1 





   ∼ ∗ 1/(1+α) tα a  dμ  k,l 

k=0 l=k+1 xj =0,j ∈{l+1,...,N−1} IN

+

N−1  k=0

INk,N

   ∼ ∗ 1/(1+α) tα a  dμ  

N−2 N−1 N−1 cα   1 α/(1+α) 1/(1+α) cα  1 α/(1+α) 1/(1+α) M Mk + M Mk ≤ N Ml l N MN N k=0 l=k+1

≤

cα N

N−2  N−1  k=0 l=k+1

k=0

α/(1+α) 1/(1+α) Ml Mk

Ml

+

cα N

N−1 

1/(1+α)

Mk

1/(1+α)

k=0 MN

≤ Cα < ∞,

so (8.51) holds and the proof of part (a) is complete. (b) Under condition (8.49), there exists a sequence of positive integers m,k such that M2m,k +1 ≤ λk < 2M2m,k +1 .

428

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Since (n , n ∈ N) be a non-negative, non-decreasing sequence we have that

lim

k→∞

 1+α mk M2m, +1

log1+α (λk ) = ∞. k→∞ M2λ +1 k

≥ c lim

k

  Let (nk , k ∈ N) ⊂ mk , k ∈ N be a sequence of positive numbers such that n1+α k = ∞. k→∞ M2n +1 k lim

To prove part (b) we use the 1/ (1 + α)-atoms defined in Example 5.6.2. If we apply (1.40) in Lemma 1.6.4 with (5.23) and (5.24) and invoke Abel transformation, we get that     tM2nk +M2s fk  M2nk +M2s =

1 M2nk +M2s QM2nk +M2s

=

1 M2nk +M2s QM2nk +M2s

=

1 M2nk +M2s QM2nk +M2s

=

1 M2nk +M2s QM2nk +M2s

  +M2s  M2n

  k  qM2nk +M2s −j Dj − DM2nk   j =M2n +1  k    M2s

   qM2s −j Dj +M2nk − DM2nk   j =1      M2s    ψM  q D M2s −j j   2nk   j =1    M2s     qM2s −j j  .  j =1 

Let x ∈ I2s /I2s+1 , s = [nk /2] , . . . , nk . If we again use Abel transformation, then, under the conditions (8.10) and (8.11), we find that        M2s  tM2nk +M2s fk     2 c  qM2s −j − qM2s −j −1 j  ≥ α  M2nk +M2s M2nk +M2s M2n   k j =1 ≥

c

M2s 

α M2nk +M2s M2n k j =[M2s ]/2

 q M

2s −j

 − qM2s −j −1  j 2

8.2 Maximal Operators of Nörlund Means on Hp Spaces

≥

≥

≥

429

M2s 

2 cM2s

α M2nk +M2s M2n k j =[M2s ]/2 2 cM2s

[M 2s ]/2

α M2nk +M2s M2n k

j =1

2 cM2s

[M 2s ]/2

α M2nk +M2s M2n k

j =1

≥

2 M α−1 cM2s 2s α M2nk +M2s M2n k

≥

α+1 cM2s α M2nk +M2s M2n k

≥

α+1 cM2s α . M2nk +1 M2n k

 q M

2s −j

 − qM2s −j −1 

  qj − qj +1 

j α−2

Hence,  1/(1+α)  tn fk   dμ ≥ sup   n∈N+ n



 Gm

≥

≥

≥ ≥

n k −1 s=[nk /2]

c



   tM +M fk 1/(1+α)  2nk 2s  dμ   I2s /I2s+1  M2nk +M2s  n k −1



1/(1+α) M2n +1 s=[nk /2] I2s /I2s+1 k

c

n k −1

α+1 M2s

1/(1+α) α/(1+α) M2n +1 s=[nk /2] M2nk k

c

n k −1

1

1/(1+α) α/(1+α) M2n +1 s=[nk /2] M2nk k

cnk . α/(1+α) 1/(1+α) M2nk M2n +1 k

M2s dμ α/(1+α) M2nk 1 α+1 M2s

430

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Therefore, by using (5.25) for p = 1/ (1 + α) we have that

1+α    tn fk  1/(1+α) dμ Gm supn∈N  n  fk H1/(1+α)

≥

cn1+α cn1+α α k k M ≥ → ∞, as k → ∞, 2nk α  M2n M2nk +1 M2nk +1 k

which implies (8.50), so also part (b) is proved.  Corollary 8.2.22 Let f ∈ H1/(1+α), where 0 < α ≤ 1. Then the maximal ∼ α,∗

∼ α,∗

operators σ α and V α defined by ∼ α,∗ σα f

:=

 α  σ f 

log

n 1+α

 α  V f 

∼ α,∗

(n + 1)

and V α f :=

n

log1+α (n + 1)

,

are bounded from H1/(1+α) to L1/(1+α). (b) Let 0 < α ≤ 1 and (n , n ∈ N+ ) be any non-decreasing sequence satisfying condition (8.49). Then  α   σ n f k  supn    n 1/(1+α) sup =∞ f  k∈N H1/(1+α) and

sup k∈N

 α   V n f k  supn    n

1/(1+α)

f H1/(1+α)

= ∞.

8.3 Maximal Operators of T Means on Hp Spaces In this Section we investigate maximal operators and weighted maximal operators of T means. First, we consider T means with non-increasing sequence (qk , k ∈ N). Our first result of this Section reads: Theorem 8.3.1 (a) The maximal operator T ∗ of T means with non-increasing sequence (qk , k ∈ N) is bounded from H1/2 to weak − L1/2 .

8.3 Maximal Operators of T Means on Hp Spaces

431

(b) Let 0 < p < 1/2 and (qk , k ∈ N) be a non-increasing sequence satisfying condition c qn+1 ≥ , Qn+2 n

(c ≥ 1) .

(8.55)

Then there exists a martingale f ∈ Hp such that sup Tn f weak−Lp = ∞.

(8.56)

n∈N

Proof (a) Let the sequence (qk , k ∈ N) be non-increasing. By applying (4.88) and (4.91) we get that     n−1  1    |Tn f | ≤  qj Sj f   Qn j =1  ⎛ ⎞ n−2    1 ⎝  ≤ qj − qj +1  j σj f  + qn−1 (n − 1) |σn f |⎠ Qn j =1

⎛ ⎞ n−2    1 ⎝ qj − qj +1 j + qn−1 (n − 1)⎠ σ ∗ f ≤ σ ∗ f ≤ Qn j =1

so that T ∗ f ≤ σ ∗ f.

(8.57)

If we apply (8.57) and Theorem 7.2.2, we can conclude that T ∗ is bounded from H1/2 to weak − L1/2 . (b) To prove part (b) we apply the martingale f from Example 5.6.7 for 0 < p < q = 1/2. Then, f ∈ Hp for 0 < p < 1/2. It is evident that TM2αk +2 f =

1 QM2αk +2

M2αk

 j =0

qj Sj f +

qM2αk +1 QM2αk +2

SM2αk +1 f =: I + I I.

(8.58)

432

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

According to (5.46) in Example 5.6.7, we can conclude that |I | ≤

Mαk



1 QMαk +2

(8.59)

j =0

1/p

≤

  qj Sj f  Mαk



2RMαk−1

1

1/2

QMαk +2

αk−1

1/p

qj ≤

2RMαk−1 1/2

αk−1

j =0

1/p−2

≤

Mαk

3/2

.

16αk

If we now apply (5.47) for I I we find that  1/p−1  1/p−1  qM2αk +1  M2αk qM2αk +1 M2αk  |I I | = SM2αk +1 f  ≥ .   QM2α +2 4α 1/2 QM2αk +2  α 1/2 k k k

(8.60)

Without loss the generality we may assume that c = 1 in (8.55). By combining (8.59) and (8.60), we get that 1/p−1 1/p   qM2αk +1 M2αk 4RMαk   − TMαk +2 f  ≥ |I I | − |I | ≥ QM2αk +2 4αk αk−1 1/p−2

≥

M2αk

4αk

Moreover, )

  M 1/p−2 αk   μ x ∈ Gm : TM2αk +2 f (x) ≥ 8αk

1/p−2

1/p

4RMαk − αk−1

(8.61)

≥

M2αk

1/2

.

8αk

: = μ (Gm ) = 1.

(8.62)

Let 0 < p < 1/2. Then 1/p−2

M2αk

8αk

 )

  M 1/p−2 2αk   · μ x ∈ Gm : TM2αk +2 f (x) ≥ 8αk

:1/p (8.63)

1/p−2

=

M2αk

8αk

→ ∞, as k → ∞,

and we conclude that (8.56) holds. 

8.3 Maximal Operators of T Means on Hp Spaces

433

In particular, we have the following special cases of Theorem 8.3.1: Corollary 8.3.2 (a) The maximal operators σ ∗ , B −1,∗ , σ α,−1,∗ and V1α,−1,∗ defined by σ ∗ f := sup |σn f |, n∈N

    B −1,∗ f := sup Bn−1 f  n∈N

    σ α,−1,∗ f := sup σnα,−1 f ,

    V1α,−1,∗ f := sup Bnα,−1 f ,

n∈N

n∈N

respectively, are bounded from H1/2 to weak-L1/2 . Theorem 8.3.3 Let 0 < p < 1/2, f ∈ Hp and (qk , k ∈ N) be a sequence of non-increasing numbers. Then the maximal operator T.p∗ defined by T.p∗ f := sup

n∈N+

|Tn f | (n + 1)1/p−2

(8.64)

is bounded from Hp to Lp . Proof By using (4.88) and (4.91), we get that       1 n−1    ≤ q S f j j   1/p−2 1/p−2 (n + 1) (n + 1)  Qn j =1  ⎞ ⎛ n−2    1 ⎝   1 qj − qj +1  j σj f  + qn−1 (n − 1) |σn f |⎠ ≤ (n + 1)1/p−2 Qn j =1 |Tn f |

1

⎛ ⎞    n−2  1 ⎝  qj − qj +1  j σj f  qn−1 (n − 1) |σn f | ⎠ + ≤ Qn (j + 1)1/p−2 (n + 1)1/p−2 j =1

⎞ ⎛ n−2  |σn f | 1 ⎝  ≤ qj − qj +1 j + qn−1 (n − 1)⎠ sup 1/p−2 Qn n∈N+ (n + 1) j =1 ≤ sup

|σn f |

1/p−2 n∈N+ (n + 1)

=. σp∗ f,

so that σp∗ f. T.p∗ f ≤ .

(8.65)

Hence, if we apply (8.65) and Theorem 7.2.4, then we can conclude that T.p∗ is bounded from Hp to Lp for 0 < p < 1/2. The proof is complete. 

434

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Remark 8.3.4 Since the Fejér means are examples of T means with a nonincreasing sequence (qk , k ∈ N), we obtain from part  (b) of Theorem 7.2.4 that the asymptotic behaviour of the sequence of weights 1/ (n + 1)1/p−2 , n ∈ N in Theorem 8.3.3 can not be improved. As a consequence of Theorem 8.3.3, we conclude that the following result holds. ∼∗

∼ −1,∗ ∼ α,−1,∗ , σp

Corollary 8.3.5 Let 0 < p < 1/2. Then the maximal operators σ p , B p ∼ α,−1,∗

and V p

defined by

∼∗ σ pf

∼ α,−1,∗

σp

:= sup n∈N

|σn f | (n + 1)1/p−2

f := sup n∈N

∼ −1,∗

,

   α,−1  σn f  (n + 1)1/p−2

Bp

f := sup n∈N

∼ α,−1,∗

,

Vp

 −1  B f  n

(n + 1)1/p−2

f := sup n∈N

,

   α,−1  Vn f  (n + 1)1/p−2

,

respectively, are bounded from Hp to Lp . Theorem 8.3.6 Let f ∈ H1/2 and (qk , k ∈ N) be a sequence of non-increasing numbers. Then the maximal operator T.∗ defined by .∗ f := sup T n∈N+

|Tn f | log2 (n + 1)

is bounded from H1/2 to L1/2 . Proof By using (4.88) and (4.91) we get that     n−1  1     ≤ q S f j j   2 2 log (n + 1) log (n + 1)  Qn j =1  ⎛ ⎞ n−2    1   1 ⎝ qj − qj +1  j σj f  + qn−1 (n − 1) |σn f |⎠ ≤ log2 (n + 1) Qn |Tn f |

1

j =1

⎛ ⎞    n−2  1  qj − qj +1  j σj f  qn−1 (n − 1) |σn f | ⎝ ⎠ + ≤ 2 2 Qn log log + 1) + 1) (j (n j =1 ⎞ ⎛ n−2    |σn f | 1 ⎝ ≤ qj − qj +1 j + qn−1 (n − 1)⎠ sup 2 Qn n∈N+ log (n + 1) j =1

≤ sup n∈N+

|σn f | log2 (n + 1)

=. σ ∗ f,

8.3 Maximal Operators of T Means on Hp Spaces

435

so that T.∗ f ≤ . σ ∗ f.

(8.66)

Hence, if we apply (8.66) and Theorem 7.2.8, we can conclude that T.∗ is bounded from H1/2 to L1/2 . The proof is complete.  Remark 8.3.7 Part (b) of Theorem 7.2.4 implies that   the asymptotic behaviour of the sequence of weights 1/ log2 (n + 1) , n ∈ N in Theorem 8.3.6 can not be improved. As a consequence of Theorem 8.3.6 we immediately get the following result. ∼ ∗ ∼ −1,∗ ∼ α,−1,∗ , σ1

Corollary 8.3.8 (a) The maximal operators σ 1 , B 1 by ∼∗ σ 1f

∼ α,−1,∗

σ1

:= sup n∈N

|σn f | log2 (n + 1)

f := sup n∈N

∼ −1,∗

,

   α,−1  σn f  log2 (n + 1)

B1

,

f := sup n∈N

∼ α,−1,∗

V1

and Bnα,−1 defined

 −1  B f  n

log2 (n + 1)

f := sup n∈N

,

   α,−1  Bn f  log2 (n + 1)

,

respectively, are bounded from H1/2 to L1/2 . Next we present the following result concerning maximal operators and weighted maximal operators of T means with non-decreasing sequence (qk , k ∈ N). Theorem 8.3.9 (a) The maximal operator T ∗ of T means with non-decreasing sequence (qk , k ∈ N) satisfying the condition qn−1 =O Qn

 1 n

(8.67)

is bounded from H1/2 to weak − L1/2 . (b) Let 0 < p < 1/2. For any non-decreasing sequence (qk , k ∈ N), there exists a martingale f ∈ Hp such that sup Tn f weak−Lp = ∞. n∈N

(8.68)

436

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Proof (a) Let the sequence (qk , k ∈ N) be non-decreasing. By applying (4.88) and (4.91), we get that     n−1  1   |Tn f | ≤  qj Sj f   Qn j =1  ⎛ ⎞ n−2    1 ⎝  ≤ qj − qj +1  j σj f  + qn−1 (n − 1) |σn f |⎠ Qn j =1

⎞ n−2  1 ⎝  − qj − qj +1 j − qn−1 (n − 1) + 2qn−1 (n − 1)⎠ σ ∗ f ≤ Qn ⎛

j =1

≤

1 (2qn−1 (n − 1) − Qn ) σ ∗ f ≤ cσ ∗ f, Qn

so that T ∗ f ≤ cσ ∗ f.

(8.69)

We use (8.69) and Theorem 7.2.2 to see that T ∗ is bounded from H1/2 to weak− L1/2 . (b) To prove part (b) we apply the martingale f from Theorem 8.3.1 (see also Example 5.6.7 for 0 < p < q = 1/2). It is easy to prove that for every nonincreasing sequence (qk , k ∈ N), it automatically holds that qMαk +1 /QMαk +2 ≥ c/Mαk . Hence, according to (8.58)–(8.61), we can conclude that 1/p−2   Mαk   . TMαk +2 f  ≥ |I I | − |I | ≥ 8αk

Analogously to the proofs of (8.62) and (8.63), we then get that sup TMαk +2 f

k∈N

weak−Lp

= ∞,

which means that (8.68) holds, so part (b) is proved. 

8.3 Maximal Operators of T Means on Hp Spaces

437

As a consequence of Theorem 8.3.9, we get that the following result holds for concrete summability methods: Corollary 8.3.10 Let 0 < p < 1/2. Then the maximal operators B −1,∗ and β −1,∗ defined by         B −1,∗ f := sup Bn−1 f , β −1,∗ f := sup βn−1 f , n∈N

n∈N

respectively, are bounded from Hp to Lp . Moreover, we state the following theorem concerning weighted maximal operator of T means generated by non-decreasing sequence (qk , k ∈ N): Theorem 8.3.11 Let 0 < p < 1/2, f ∈ Hp and (qk , k ∈ N) be a sequence of non-decreasing numbers satisfying condition (4.10). Then the maximal operator T.p∗ defined by T.p∗ f := sup

n∈N+

|Tn f | (n + 1)1/p−2

is bounded from Hp to Lp . Proof Since the sequence (qk , k ∈ N) is non-decreasing and satisfying condition (4.10), by applying (4.88) and (4.91) we find that |Tn f | (n + 1)1/p−2

    n−1  1    ≤ qj Sj f  1/p−2  Q (n + 1)  n j =1  ⎛ ⎞ n−2    1   1 ⎝ qj −qj +1  j σj f  +qn−1 (n − 1) |σn f |⎠ ≤ (n + 1)1/p−2 Qn 1

j =1

⎛ ⎞    n−2      qj − qj +1 j σj f 1 qn−1 (n − 1) |σn f | ⎝ ⎠ ≤ + 1/p−2 1/p−2 Qn + 1) + 1) (j (n j =1 ⎛ ⎞ n−2  |σn f | 1  ⎝ qj +1 − qj j + qn−1 (n − 1)⎠ sup ≤ 1/p−2 Qn n∈N+ (n + 1) j =1

≤

|σn f | 2qn−1 (n − 1) − Qn + q0 sup 1/p−2 Qn n∈N+ (n + 1)

≤ c sup n∈N+

|σn f | (n + 1)1/p−2 log2[1/2+p] (n + 1)

= c. σp∗ f,

438

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

thus .p∗ f ≤ c. T σp∗ f

(8.70) 

Now (8.70) and Theorem 7.2.4 show the theorem.

Remark 8.3.12 Part (b) of Theorem 7.2.4 yields that  the asymptotic behaviour of the sequence of weights 1/ (n + 1)1/p−2 , n ∈ N in Theorem 8.3.11 can not be improved. In particular, we get the following result: .p−1,∗ and Corollary 8.3.13 (a) Let 0 < p < 1/2. Then the maximal operators B −1,∗ .p β defined by .p−1,∗ f := sup B

n∈N

 −1  B f  n

(n + 1)1/p−2

and .p−1,∗ f β

:= sup

 −1  β f  n

1/p−2 n∈N (n + 1)

,

respectively, are bounded from Hp to Lp . Theorem 8.3.14 Let f ∈ H1/2 and (qk , k ∈ N) be a sequence of non-decreasing numbers satisfying condition (4.10). Then the maximal operator T.p∗ defined by .∗ f := sup T n∈N+

|Tn f | log (n + 1) 2

is bounded from the martingale Hardy space H1/2 to the space L1/2 . Proof Let the sequence (qk , k ∈ N) be non-decreasing satisfying condition (4.10). By applying (4.88) and (4.91), we find that     n−1    |Tn f | 1 1   ≤ q S f j j   log2 (n + 1) (n + 1)1/p−2 log2  Qn j =1  ⎛ ⎞ n−2      1 1 ⎝ qj − qj +1  j σj f  + qn−1 (n − 1) |σn f |⎠ ≤ log2 (n + 1) Qn j =1

8.4 Strong Convergence of Nörlund Means in Hp Spaces

439

⎛ ⎞    n−2  1 ⎝ qj − qj +1  j σj f  qn−1 (n − 1) |σn f | ⎠ + ≤ Qn log2 (j + 1) log2 (n + 1) j =1 ⎞ ⎛ n−2  |σn f | 1 ⎝  ≤ qj +1 − qj j + qn−1 (n − 1)⎠ sup 2 Qn n∈N+ log (n + 1) j =1

≤

|σn f | 2qn−1 (n − 1) − Qn + q0 sup 2 Qn n∈N+ log (n + 1)

≤ c sup n∈N+

|σn f | log (n + 1) 2

= c. σ ∗ f.

Hence .∗ f ≤ c. σ ∗f T

(8.71) 

and (8.71) and Theorem 7.2.8 prove the theorem.

Remark 8.3.15 By part (b) of Theorem 7.2.4, the asymptotic behaviour of the sequence of weights 1/ log2 (n + 1) , n ∈ N in Theorem 8.3.14 can not be improved. Moreover, Theorem 8.3.14 implies the following result: .−1,∗ and β .−1,∗ Corollary 8.3.16 (a) Let f ∈ H1/2 . Then the maximal operators B 1 1 defined by .−1,∗ f := sup B 1

n∈N

 −1  B f  n

log2 (n + 1)

and

.−1,∗ f β 1

:= sup n∈N

 −1  β f  n

log2 (n + 1)

,

respectively, are bounded from H1/2 to L1/2 .

8.4 Strong Convergence of Nörlund Means in Hp Spaces In this Section we investigate strong convergence of Nörlund means in Hp spaces with monotone coefficients. First we consider Nörlund means with non-decreasing sequence (qk , k ∈ N). Theorem 8.4.1 Let 0 < p < 1/2, f ∈ Hp and the sequence (qk , k ∈ N) be nondecreasing. Then there exists an absolute constant Cp depending only on p such

440

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

that ∞ p  tk f p k=1

k 2−2p

p

≤ Cp f Hp .

Proof The proof is similar to that of part (a) of Theorem 7.5.1, but since this case is more general we give the details. By Theorem 5.3.2 the proof is complete if we prove that ∞ p  tm ap m=1

m2−2p

≤ Cp

(8.72)

for every p-atom a with support I , μ (I ) = MN−1 . We may assume that I = IN . It is easy to see that Sn a (x) = tn a (x) = 0,

when n ≤ MN .

Therefore, we can suppose that n > MN . Let x ∈ IN . Since the Nörlund means tn with non-decreasing sequence (qk , k ∈ N) are bounded from L∞ to L∞ (the boundedness follows from Lemma 4.4.6) and 1/p a∞ ≤ MN we obtain that 

p

|tm a|p dμ ≤ IN

a∞ ≤ 1, 0 < p ≤ 1/2. MN

Hence, ∞  m=1

IN

|tm a|p dμ m2−2p

≤

∞  k=1

1 ≤ c < ∞. m2−2p

It is easy to see that     n    1  |tn a (x)| =  a (t) qn−k Dk (x − t) dμ (t) Qn  IN  k=MN      n  1   |a (t)|  qn−k Dk (x − t) dμ (t) ≤ IN  Qn  k=MN

(8.73)

8.4 Strong Convergence of Nörlund Means in Hp Spaces

441

    n  1     dμ (t) ≤ a∞ q D − t) (x n−k k Q  IN  n k=M  N      n  1   1/p  ≤ MN qn−k Dk (x − t) dμ (t) . Q IN  n k=M  N 

Let x ∈ INk,l , 0 ≤ k < l ≤ N. Then, in view of Lemma 4.4.5, we get that 1/p−2

|tm a (x)| ≤ Cp Ml Mk MN

for 0 < p < 1/2.

(8.74)

By combining (1.5) with (8.74), we find that  |tm a|p dμ

(8.75)

IN

=



mj−1

N−2  N−1 



k=0 l=k+1 xj =0, j ∈{l+1,...,N−1}

≤ Cp

N−2  N−1  k=0 l=k+1 1−2p

≤ Cp MN

INk,l

|tm a|p dμ +

N−1 

|tm a|p dμ

INk,N

k=0

 1 ml+1 · · · mN−1 p p 1−2p p 1−p Ml Mk MN + Cp M M MN MN k N N−1 k=0

N−2  N−1 

p p Ml Mk

k=0 l=k+1

Ml

N−1 

p Mk + Cp p MN k=0

1−2p

≤ Cp MN

.

Moreover, according to (8.75), we get that ∞  m=MN +1

IN

|tm a|p dμ m2−2p

≤ Cp

∞  m=MN +1

1−2p

MN ≤ Cp < ∞, (0 < p < 1/2) . m2−2p

Finally, by combining this estimate with (8.73) we obtain (8.72) so the proof is complete.  Remark 8.4.2 Since the Fejér means are examples of Nörlund means with nondecreasing sequence (qk , k ∈ N), we can conclude from part (b)  of Theorem 7.5.1  that the asymptotic behaviour of the sequence of weights 1/k 2−2p , k ∈ N in Theorem 8.4.1 can not be improved.

442

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

In particular, we point out the following concrete cases: Corollary 8.4.3 Let 0 < p < 1/2 and f ∈ Hp . Then there exists an absolute constant cp depending only on p such that the following inequalities hold: ∞ p  σk f p

p

≤ cp f Hp ,

k 2−2p

k=1

∞ p  Bk f p

k 2−2p

k=1

p

≤ cp f Hp

and ∞ p  βk f p

k 2−2p

k=1

p

≤ cp f Hp .

Next, we investigate the case p = 1/2 : Theorem 8.4.4 Let f ∈ H1/2 and the sequence (qk , k ∈ N) be non-decreasing satisfying condition (4.10). Then there exists an absolute constant c, such that 1/2

1  tk f 1/2 1/2 ≤ c f H1/2 . log n k n

k=1

Proof The proof is similar to that of part (a) of Theorem 7.5.4, but since this case is more general we present the details. According to Theorem 5.3.2, it suffices to prove that 1/2

1  tm a1/2 ≤c MN . Let x ∈ IN . Since tn is bounded from L∞ to L∞ (the boundedness follows from Lemma 4.31) and a∞ ≤ MN2 , we obtain that 

1/2

|tm a|1/2 dμ ≤ IN

a∞ ≤ c < ∞. MN

8.4 Strong Convergence of Nörlund Means in Hp Spaces

443

Hence, 1  log n n

IN

|tm a|1/2 dμ m

m=1

1 1 ≤ c < ∞, n ∈ N. log n m n

≤

(8.77)

m=1

It is easy to see that  |a (t)| |Fm (x − t)| dμ (t)

|tm a (x)| ≤ IN



≤ a∞

|Fm (x − t)| dμ (t) IN

 ≤

|Fm (x − t)| dμ (t) .

MN2

IN

Let x ∈ INk,l , 0 ≤ k < l < N. Then, in view of Lemma 4.4.3, we get that cMl Mk MN . m

|tm a (x)| ≤

(8.78)

Let x ∈ INk,N . Then, according to Lemma 4.4.3, we find that |tm a (x)| ≤ cMk MN .

(8.79)

By combining (8.78) and (8.79) with (1.5), we can conclude that  |tm a (x)|1/2 dμ (x) IN

≤c

N−2  N−1  k=0 l=k+1 1/2

≤ cMN

1/2

ml+1 · · · mN−1 Ml MN

N−2  N−1  k=0 l=k+1

1/2

1/2

Ml Mk m1/2 Ml

+c

1/2

1/2

Mk MN m1/2

N−1 

Mk

k=0

MN

+c

k=0

1/2 1/2

N−1 

1 1/2 1/2 M MN MN k

1/2

≤

cMN N + c. m1/2

It follows that 1 log n 1 ≤ log n

n  m=MN +1 n  m=MN +1

IN

|tm a (x)|1/2 dμ (x) m



1/2

cMN N c + 3/2 m m

 ≤ C < ∞.

(8.80)

444

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

By combining (8.77) and (8.80), we find that (8.76) holds so the proof is complete.  We also state the following special cases: Corollary 8.4.5 Let f ∈ H1/2. Then there exists an absolute constant C such that 1/2

n 1  σm f 1/2 1/2 ≤ C f H1/2 , log n m m=1

1/2

1  Bm f 1/2 1/2 ≤ C f H1/2 log n m n

m=1

and α 1/2 n 1  βm f 1/2 1/2 ≤ C f H1/2 . log n m m=1

Next, we investigate Nörlund means generated by non-increasing sequences (qk , k ∈ N). First we consider the case 0 < p < 1/ (1 + α) where 0 < α ≤ 1. Theorem 8.4.6 Let f ∈ Hp , where 0 < p < 1/ (1 + α) , 0 < α ≤ 1 and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying conditions (4.5) and (4.6). Then there exists an absolute constant Cα,p depending only on α and p such that p

∞ t f   k Hp k=1

k 2−(1+α)p

p

≤ Cα,p f Hp .

Proof Analogously to the proofs of the previous theorems we may assume that a is an arbitrary p-atom with support I, μ (I ) = MN−1 and I = IN and m > MN . In view of Theorem 5.3.2 it suffices to prove that ∞ p  tm ap ≤ Cα,p < ∞ m2−(1+α)p

(8.81)

m=1

for every p-atom a. Let x ∈ IN . Since tm is bounded from L∞ to L∞ (the 1/p boundedness follows from Corollary 4.5.16) and a∞ ≤ MN we obtain that  p |tn a|p dμ ≤ a∞ MN−1 ≤ 1. IN

8.4 Strong Convergence of Nörlund Means in Hp Spaces

445

Hence, ∞  n=MN

p IN |tn a| dμ n2−(1+α)p

≤

∞  n=1

1 ≤ Cα,p < ∞. n2−(1+α)p

Therefore, according to (1.5) and (8.37)–(8.40), we can conclude that ∞  n=MN +1

=

p IN |tn a| dμ n2−(1+α)p

(8.82)

mj −1

N−2  N−1 



k=0 l=k+1 xj =0,j ∈{l+1,...,N−1}

≤ Cα,p



∞  m=MN +1

|t a|p dμ INk,l n n2−(1+α)p

n 

+

N−1 

n=MN +1 k=0

|t a|p dμ INk,N n n2−(1+α)p

1−p N−2 N−1 pα p 1−p N−1  Mp MN MN   Ml Mk k + m2−p Ml m2−(1+α)p MN k=0 l=k+1

 .

k=0

Moreover, since N−2  N−1 

pα p N−2  p N−1  Ml Mk 1 ≤ Mk 1−pα Ml k=0 l=k+1 k=0 l=k+1 Ml

≤

≤

N−2 

1

p

1−pα k=0 Mk N−2 

Mk

1

1−p(α+1) k=0 Mk

≤ Cα,p < ∞

and N−1  k=0

p

Mk p−1 ≤ MN < ∞, MN

in view of (8.82), we obtain that ∞  n=MN +1

p IN |tn a| dμ n2−(1+α)p


MN and a is an arbitrary p-atom, with support I, μ (I ) = MN and I = IN . 1/p Let x ∈ IN . Since a∞ ≤ MN if we apply Lemma 4.4.18, then we obtain that    p p p p p |tn a|p dμ ≤ Fn 1 a∞ dμ ≤ Cα,p ϕn a∞ dμ ≤ Cα,p ϕn . IN

IN

IN

Hence, ∞  n=MN +1

p IN |tn a| dμ n2−(1+α)p

∞ 

≤ Cα,p

n=MN +1

p

ϕn

n2−(1+α)p

p

≤

Cα,p ϕMN (1−(1+α)p)

MN

p

≤

Cα,p ϕMN p(1/p−(1+α))

MN

≤ Cα,p < ∞.

By combining (1.5), (8.46) and (8.47), we can conclude that ∞  n=MN +1

=

∞  n=MN +1

p IN |tn a| dμ n2−(1+α)p

⎛ ⎝

N−2  N−1 

mj −1



k=0 l=k+1 xj =0,j ∈{l+1,...,N−1}

|t a|p dμ INk,l n n2−(1+α)p

+

N−1  k=0

⎞

|t a|p dμ INk,N n ⎠ n2−(1+α)p

8.4 Strong Convergence of Nörlund Means in Hp Spaces



∞ 

≤

n=MN +1

1−p

< Cα,p MN

1−p N−2  N−1 

Cα,p MN n2−p ∞ 

n=MN +1

k=0 l=k+1

447

pα p p 1−p N−1 p Cα,p MN  Mk Ml ϕl Mk + 2−(1+α)p Ml MN n

1 + Cα,p n2−p



k=0

∞  n=MN +1

1 ≤ Cα,p < ∞, n2−(1+α)p 

so that (8.83) holds and the proof is complete. As a consequence we get the following corollary:

Corollary 8.4.8 Let 0 < α ≤ 1, 0 < p < 1/ (1 + α) and f ∈ Hp . Then there exists an absolute constant Cα,p depending only on α and p such that α p ∞ σ f  m p m(1+α)(1−p)

m=1

p

≤ Cα,p f Hp

and ∞ V α f p  m p m=1

p

m(1+α)(1−p)

≤ Cα,p f Hp .

Finally, we consider strong convergence of Nörlund means generated by a nonincreasing sequence (qk , k ∈ N) for the endpoint case p = 1/(1 + α). Theorem 8.4.9 Let f ∈ H1/(1+α) where 0 < α ≤ 1 and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying conditions (4.5) and (4.6). Then there exists an absolute constant Cα depending only on α such that 1/(1+α)

1  tm f H1/(1+α) 1/(1+α) ≤ Cα f H1/(1+α) . log n m n

m=1

Proof By Theorem 5.3.2, we have to prove that 1/(1+α)

1  tm a1/(1+α) ≤ Cα < ∞ log n m n

(8.84)

m=1

for every 1/ (1 + α)-atom a. Analogously to the proofs of the previous theorems, we may assume that a is an arbitrary 1/ (1 + α)-atom with support I, μ (I ) = MN−1 and I = IN and m > MN .

448

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Let x ∈ IN . Since tm is bounded from L∞ to L∞ (the boundedness follows from Corollary 4.5.16) and a∞ ≤ MN1+α , we obtain that 

1/(1+α)

|tm a|1/(1+α) dμ ≤ a∞

IN

/MN ≤ 1.

Hence, n 1  log n

IN

|tm a (x)|1/(1+α) dμ m

m=MN

1 1 ≤ Cα < ∞. log n m n

≤

m=1

According to (1.5) combined with (8.52) and (8.54), we can conclude that 1 log n 1 = log n 1 + log n ≤

+

Cα log n 1 log n

n 

IN

|tm a|1/(1+α) dμ

n 

(8.85)

m

m=MN +1

mj−1

N−2  N−1 



INk,l

|tm a|1/(1+α) dμ m

m=MN +1 k=0 l=k+1 xj =0, j ∈{l+1,...,N−1} n 

N−1 

INk,N

|tm a|1/(1+α) dμ m

m=MN +1 k=0 n  m=MN +1 n  m=MN +1

α/(1+α) N−2  N−1  α/(1+α) 1/(1+α) ml+1 · · · mN−1 MN Ml Mk α/(1+α)+1 MN m k=0 l=k+1

N−1 1  1/(1+α) α/(1+α) Mk MN . mMN k=0

Moreover, since N−2  N−1 

α/(1+α)

Ml

1/(1+α) ml+1 · · · mN−1

Mk

MN

k=0 l=k+1

=

N−2  N−1 

α/(1+α)

Ml

1/(1+α)

Mk

k=0 l=k+1

=

N−2 

1/(1+α)

Mk

k=0

≤

N−2  k=0

N−1 

1

1/(1+α) l=k+1 Ml

1

1/(1+α)

Mk

1 Ml

1/(1+α)

Mk

≤

N−2  k=0

1≤N

8.5 Strong Convergence of T Means in Hp Spaces

449

and N−1 

1/(1+α)

Mk

α/(1+α)

MN

1/(1+α)

≤ MN

α/(1+α)

MN

≤ MN ,

k=0

according to (8.85) we obtain that 1 log n

n 

IN

m

m=MN +1

⎛ n Cα ⎝  ≤ log n

m=MN +1

|tm a|1/(1+α) dμ

α/(1+α)

NMN + mα/(1+α)+1

n  m=MN +1

⎞ 1⎠ ≤ Cα < ∞. m 

Thus (8.84) holds so the proof is complete. We finish this Section by stating the following

Corollary 8.4.10 Let 0 < α ≤ 1 and f ∈ H1/(1+α). Then there exists an absolute constant Cα depending only on α such that α 1/(1+α) n 1  σm f 1/(1+α) 1/(1+α) ≤ Cα f H1/(1+α) log n m m=1

and n α 1/(1+α) 1  Um f 1/(1+α) 1/(1+α) ≤ Cα f H1/(1+α) . log n m m=1

8.5 Strong Convergence of T Means in Hp Spaces In this Section we investigate strong convergence of T means in Hp spaces with monotone coefficients. First we consider T means with non-increasing sequence (qk , k ∈ N) : Theorem 8.5.1 Let 0 < p < 1/2, f ∈ Hp and (qk , k ∈ N) be a sequence of non-increasing numbers. Then there exists an absolute constant Cp depending only on p such that ∞ p  Tk f p k=1

k 2−2p

p

≤ Cp f Hp .

450

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Proof By Theorem 5.3.2, the proof will be complete, if we prove that p

∞ T a  k Hp

k 2−2p

k=1

≤ Cp

(8.86)

for every p-atom a with support I , μ (I ) = MN−1 . We may assume that I = IN . It is easy to see that Sk a (x) = Tk a (x) = 0,

when k ≤ MN .

Therefore, we can suppose that k > MN . 1/p Let x ∈ IN . Since a∞ ≤ MN we obtain that 

p

|Tk a|p dμ ≤ IN

a∞ ≤ c < ∞. MN

Hence, ∞ 

IN

|Tk a|p dμ k 2−2p

k=1

≤

∞  k=1

c k 2−2p

≤ C < ∞, 0 < p < 1/2.

(8.87)

It is easy to see that      |Tk a (x)| =  a (t) Fk (x − t) dμ (t) IN     k    1  a (t) ql Dl (x − t) dμ (t) = Qk   IN l=MN      k  1    ≤ a∞ ql Dl (x − t) dμ (t) Q IN  k l=M  N      k    1/p  1  dμ (t) . q D − t) ≤ cMN (x l l Q  IN  k 

(8.88)

l=MN

i,j

Let x ∈ IN , 0 ≤ i < j ≤ N. Then, in view of Lemma 3.2.15, we get that 1/p−2

|Tk a (x)| ≤ cMi Mj MN

for 0 < p < 1/2.

(8.89)

8.5 Strong Convergence of T Means in Hp Spaces

451

Hence, by using (1.5), (8.88) and (8.89) we find that  |Tk a|p dμ

(8.90)

IN

=



mj−1

N−2  N−1 



i,j

|Tk a|p dμ +

i=0 j =i+1 xj =0, j ∈{i+1,...,N−1} IN

≤c

N−2  N−1  i=0 j =i+1

≤

1−2p cMN

N−1  i=0

INi,N

|Tk a|p dμ

N−1  1 p 1−2p mj +1 · · · mN−1  p 1−p Mi Mj MN +c M M MN MN i N i=0



N−2  N−1  i=0 j =i+1

Mj Mk Mj

p +c

N−1 

p

Mi

p MN i=0

1−2p

≤ cMN

.

Moreover, according to (8.90), we get that ∞  k=MN +1

IN

|Tk a|p dμ k 2−2p

∞ 

≤

k=MN +1

1−2p

cMN < c, (0 < p < 1/2) . k 2−2p

(8.91)

By just combining (8.87) and (8.91), we find that (8.86) holds so the proof is complete.  We also point out the following special cases: Corollary 8.5.2 Let 0 < p < 1/2 and f ∈ Hp . Then there exists an absolute constant Cp depending only on p such that the following inequalities hold: ∞ p  σk f p k=1

k 2−2p

∞ p  Bk f p k=1

k 2−2p

p

≤ Cp f Hp ,

p

≤ Cp f Hp

and ∞ p  Rk f p k=1

k 2−2p

p

≤ Cp f Hp .

For the limit case p = 1/2 we have the following result: Theorem 8.5.3 Let f ∈ H1/2 and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying condition (4.11). Then there exists an absolute constant C such

452

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

that 1/2

1  Tk f 1/2 1/2 ≤ C f H1/2 . log n k n

(8.92)

k=1

Proof By Theorem 5.3.2, the proof will be complete, if we prove that 1/2

1  Tk aH1/2 ≤C log n k n

(8.93)

k=1

for every 1/2-atom a with support I , μ (I ) = MN−1 . We may assume that I = IN . Since Sk a (x) = Tk a (x) = 0,

when k ≤ MN ,

we can suppose that k > MN . Let x ∈ IN . Since Tn is bounded from L∞ to L∞ (boundedness follows from Lemma 8.65) and a∞ ≤ MN2 , we obtain that 

1/2

|Tk a|1/2 dμ ≤ IN

a∞ ≤ c < ∞. MN

Hence, 1  log n n

k=1

IN

|Tk a|1/2 dμ k

c 1 ≤ C < ∞. log n k n

≤

(8.94)

k=1

Analogously to (8.88) we find that     k    1 |Tk a (x)| =  a (t) ql Dl (x − t) dμ (t) Qk  IN  l=MN      k  1    ql Dl (x − t) dμ (t) ≤ a∞ Q IN  k l=M  N      k  1   2  ≤ MN ql Dl (x − t) dμ (t) . Q IN  k  l=MN

(8.95)

8.5 Strong Convergence of T Means in Hp Spaces

453

i,j

Let x ∈ IN , 0 ≤ i < j < N. Then, in view of Lemma 4.5.2, we get that cMi Mj MN . k

|Tk a (x)| ≤

(8.96)

Let x ∈ INi,N . Then, according to Lemma 4.5.2, we obtain that |Tk a (x)| ≤ cMi MN .

(8.97)

By combining (1.5), (8.95), (8.96) and (8.97) we obtain that  |Tk a (x)|1/2 dμ (x) IN

≤c

N−2  N−1  i=0 j =i+1 1/2

≤ cMN

 1/2 1/2 N−1  1 MN mj +1 · · · mN−1 Mi Mj 1/2 1/2 + c M MN MN MN i k 1/2 i=0



N−2  N−1  i=0 j =i+1

1/2

Mi Mj k 1/2 Mj

+c

N−1 

Mi

i=0

MN

1/2 1/2

1/2

≤

cMN N + C. k 1/2

It follows that 1 log n 1 ≤ log n

n  k=MN +1 n  k=MN +1

IN

|Tk a (x)|1/2 dμ (x) k



1/2

cMN N c + k 3/2 k

(8.98)

 ≤ C < ∞.

By just combining (8.94) and (8.98), we see that (8.93) holds so the proof is complete.  As a consequence we get that the following: Corollary 8.5.4 Let f ∈ H1/2. Then there exists an absolute constant C such that 1/2

n 1  σm f 1/2 1/2 ≤ C f H1/2 , log n m m=1

1/2

1  Bm f 1/2 1/2 ≤ C f H1/2 log n m n

m=1

454

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

and 1/2

1  Rm f 1/2 1/2 ≤ C f H1/2 . log n m n

m=1

Now, we investigate T means generated by non-decreasing sequences (qk , k ∈ N). First we consider the case 0 < p < 1/2. Theorem 8.5.5 Let 0 < p < 1/2, f ∈ Hp and (qk , k ∈ N) be a sequence of non-decreasing numbers. Then there exists an absolute constant Cp depending only on p such that the following inequality holds: ∞ p  Tk f p k=1

k 2−2p

p

≤ Cp f Hp .

Proof If we use Lemma 4.5.9, Lemma 4.5.10 and follow the analogical steps as in the proof of Theorem 8.5.1, we can readily prove Theorem 8.5.5. We omit the details.  In particular, we have the following: Corollary 8.5.6 Let 0 < p ≤ 1/2 and f ∈ Hp . Then there exists an absolute constant Cp depending only on p such that the following inequalities hold: ∞ p  Bk f p k=1

k 2−2p

p

≤ Cp f Hp

and ∞ β α f p  k p k 2−2p

k=1

p

≤ Cp f Hp .

Next, we consider the case p = 1/2 : Theorem 8.5.7 Let f ∈ H1/2 and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying condition (4.10). Then there exists an absolute constant C such that the following inequality holds: 1/2

1  Tk f 1/2 1/2 ≤ C f H1/2 . log n k n

(8.99)

k=1

Proof Using Lemmas 4.5.9 and 4.5.10, the inequality can be proved analogously to Theorem 8.5.3. 

8.6 Maximal Operators of Riesz and Nörlund Logarithmic Means on Hp Spaces

455

We also state the following special cases: Corollary 8.5.8 Let f ∈ H1/2. Then there exists an absolute constant C such that 1/2

1  Bm f 1/2 1/2 ≤ C f H1/2 log n m n

m=1

and n α 1/2 1  βm f 1/2 1/2 ≤ C f H1/2 . log n m m=1

8.6 Maximal Operators of Riesz and Nörlund Logarithmic Means on Hp Spaces In our previous Sections we investigated Nörlund means with non-increasing sequences (qk , k ∈ N), but the case when qk = 1/k was excluded, since this sequence does not satisfies the condition (4.5) for any 0 < α ≤ 1. On the other hand, Riesz logarithmic means are not examples of Nörlund means. In this Section we fill this gap simultaneously for both cases. Theorem 8.6.1 (a) The maximal operator R ∗ of the Riesz logarithmic means is bounded from the Hardy space H1/2 to the space weak − L1/2 . (b) Let 0 < p ≤ 1/2. Then there exists a martingale f ∈ Hp such that ∗ R f = +∞. p Proof (a) By using Abel transformation, we obtain that σn f 1  σj f + . ln j +1 ln n−1

Rn f =

j =1

Consequently, R ∗ f ≤ cσ ∗ f.

(8.100)

456

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Since σ ∗ is bounded from H1/2 to weak − L1/2 (see Theorem 7.2.2), by using (8.100) we can conclude that ∗ R f

weak−L1/2

≤ c f H1/2

and the proof of part (a) is complete. (b) Let f = (fn , n ∈ N) be martingale defined in Example 5.6.7. Set qns = M2n + M2s − 1, n > s. Then, we have that qs

k

qs

M

2α αk Sj f 1 k Sj f 1 1  = + Rqαs f = k lqαs j lqαs j lqαs

j =1

j =1

k

k

αk 

j =M2αk +1

Sj f =: I + I I. j

According to (5.46) we have that |I | ≤

M2αk −1 

  Sj f (x) j

1 lqαs

j =1

k

M2αk −1

1/p

 1 j

1 2RM2αk−1 ≤ αk α 1/2 k−1

j =1

1/p

≤

2RM2αk−1 1/2

αk−1

1/p

2RMαk

≤

.

3/2

αk

Let M2αk ≤ j ≤ qαs k . By using the second inequality of (5.45) in the case when l = k, we deduce that 1/p−1

Sj f = SM2αk f +

M2αk

ψM2α Dj −M2α k

k

1/2

(8.101)

.

αk

Hence, we can rewrite I I as qs

II =

αk 

1 lqαs

k

j =M2αk

=: I I1 + I I2 .

SM2α f k

j

1/p−1

+

1 M2αk lqαs

k

ψM2αk

1/2

αk

qs

αk Dj −M2α 

k

j =M2αk

j

(8.102)

8.6 Maximal Operators of Riesz and Nörlund Logarithmic Means on Hp Spaces

457

In view of (5.46) we find that qs

|I I1 | ≤

αk 

1 lqαs

j =M2αk

k

 1   SM2αk f  j

(8.103)

  2RM 1/p 2αk−1   ≤ SM2αk f  ≤ . 1/2 αk−1 Let x ∈ I2s \I2s+1 for s = [2αk /3] , . . . , αk . Since M 2s −1

Dj (x)

j =0

j +M2αk

≥

M 2s −1

j

j =0

j +M2αk

≥

M 2s −1

j

j =0

2M2αk

≥

2 cM2s , M2αk

we obtain that 1/p−1

|I I2 | =

1 M2αk

1/2

lqαs

αk

k

1/p−1

≥

c M2αk αk α 1/2 k 1/p−2

≥

cM2αk

  M 2s −1  Dj ψM  2αk j +M2αk  j =0

     

2 M2s M2αk

2 M2s

3/2

.

αk

By combining (5.41)–(5.43) with (8.102)–(8.104), we get that    s  Rqαk f  = |I I2 + I + I I1 | ≥ |I I2 | − |I | − |I I1 | 1/p

≥ |I I2 | −

2RMαk

1/p−2

≥

cM2αk

2 M2s

3/2

αk

1/p−2

≥

3/2

αk

cM2αk

3/2

αk

2 M2s

1/p

−

.

cMαk

3/2

αk

(8.104)

458

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Consequently, 

αk 

  ∗ R f (x)p dμ (x) ≥ Cp

Gm



p    s Rqαk f (x) dμ (x)

(8.105)

s=[2αk /3] I \I 2s 2s+1 αk 

≥ Cp



1−2p

3p/2

dμ (x)

αk

s=[2αk /3]I \I 2s 2s+1

≥ Cp

2p

cM2αk M2s

1−2p

2p−1

αk 

M2αk M2s

s=[2αk /3]

αk

3p/2

⎧ ⎨ Cp 2αk (1−2p)/3 , when 0 < p < 1/2, 3p/2 αk ≥ ⎩ 1/4 cαk , when p = 1/2. Hence, 

 ∗ p R f  dμ → ∞ as k → ∞, Gm

which finishes the proof.  Remark 8.6.2 If we follow the steps in the proof of the estimate (8.105) for Rqα0 , we k can prove a stronger result for part b) when 0 < p < 1/2, namely, if 0 < p < 1/2, then there exists a martingale f ∈ Hp such that sup Rn f p = +∞. n∈N+

It is well-known that the maximal operator R ∗ is not bounded form the Hardy space Hp to the space Lp for 0 < p ≤ 1/2 (see e.g. [352]). However, by using a suitable weight such boundedness indeed holds for the maximal operator. Our result for the case 0 < p < 1/2 reads as follows. Theorem 8.6.3 ∼∗

(a) Let 0 < p < 1/2. Then the maximal operator R p defined by ∼∗

R p f := sup

log n |Rn f |

1/p−2 n∈N (n + 1)

is bounded from Hp to Lp .

8.6 Maximal Operators of Riesz and Nörlund Logarithmic Means on Hp Spaces

459

(b) Let 0 < p < 1/2 and (ϕn , n ∈ N) be a non-negative, non-decreasing sequence satisfying the condition (n + 1)1/p−2 = ∞. log (n + 1) ϕn

(8.106)

Then the maximal operator defined by |Rn f | n∈N ϕn sup

is not bounded from Hp to weak − Lp . Proof (a) By Theorem 5.3.2 it suffices to prove that   ∼ ∗ p   R a  dμ ≤ Cp < ∞  p 

(8.107)

I

holds for every p-atom a, where I denotes the support of the atom. Let a be an arbitrary p-atom with support I and μ (I ) = MN−1 . We may assume that I = IN . Then Kn a (x) = Rn a (x) = 0,

when n ≤ MN . 1/p

Therefore we can suppose that n > MN . Since a∞ ≤ MN , by using (4.108) we have that log (n + 1) (n + 1)1/p−2

1/p 

|Rn a (x)| ≤

n−1 

log (n + 1) MN

(n + 1)1/p−2 ln

IN j =MN +1

1/p

+

  Kj (x − t)

log (n + 1) MN

(n + 1)1/p−2 ln

j +1

dμ (t)

 |Kn (x − t)| dμ (t) . IN

Let x ∈ IN (xk ek + xl el ) , where 1 ≤ xk ≤ mk − 1, 1 ≤ xl ≤ ml − 1, k = 0, . . . , N − 2, l = k + 1, . . . , N − 1. From Lemmas 3.2.14 and 4.9.3, it follows that log (n + 1) (n + 1)1/p−2

|Rn a (x)| ≤ Cp Ml Mk .

(8.108)

460

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Let x ∈ IN (xk ek ) , 1 ≤ xk ≤ mk − 1, k = 0, . . . , N − 1. By applying again Lemmas 3.2.14 and 4.9.3, we have that log (n + 1) (n + 1)1/p−2

|Rn a (x)| ≤ cNMN Mk .

(8.109)

By combining (1.5), (8.108), and (8.109), we get that  IN

p  ∗ N−2 k −1 N−1 l −1   m  m  ∼ R a (x) dμ (x) =   p k=0 xk =1 l=k+1 xl =1

+

N−1 k −1   m k=0 xk =1

≤ Cp

IN (xk ek )

N−2  N−1  k=0 l=k+1

+Cp

N−1  k=0

IN (xk ek +xl el )

p  ∗  ∼ R a (x) dμ (x)   p

 ∗ p ∼  R a (x) dμ (x)  p 

1 (Ml Mk )p Ml

1 (NMN Mk )p ≤ Cp < ∞, MN

which means that (8.107) holds so part (a) is proved. (b) Let (λk , k ∈ N) be an increasing sequence of positive integers, which satisfies condition (8.106). Then there exists (nk , k ∈ N) ⊂ (λk , k ∈ N) such that 

1/p−2 M2nk + 1 = ∞. lim k→∞ ϕM2n +1 log(M2nk + 1) k Let fk (x) be the atom defined in Example 5.6.2. By combining (5.23) and (5.24) in Example 5.6.2, we find that     RM2nk +1 fk (x) ϕM2nk +1

=

    Rqn0 f (x) k

ϕqn0

k

≥

1

  ϕM2nk +1 lM2nk +1 M2nk + 1

8.6 Maximal Operators of Riesz and Nörlund Logarithmic Means on Hp Spaces

≥

c ϕM2nk +1 log (M2nk )M2nk

≥

c nk M2nk ϕM2nk +1

461

for x ∈ I0 \I1 = Gm \I1 . Therefore, in view of (5.25) we get that c nk M2nk ϕM2n

k

 μ x ∈ Gm +1

  ∗  ∼ : R p fk (x) ≥

91/p c nk M2nk ϕM2n

k

+1

fk (x)Hp 1/p−2

≥

μ(Gm \I1 )   ϕM2nk +1 log M2nk + 1 cM2nk

1/p−2

cM2nk   ϕM2nk +1 log M2nk + 1  1/p−2 c M2nk + 1   → ∞, as k → ∞. ≥ ϕM2nk +1 log M2nk + 1 ≥

This completes the proof of the theorem.  Next we state the corresponding result for the case p = 1/2 : Theorem 8.6.4 .∗ defined by (a) The maximal operator R |Rn f | n∈N log (n + 1)

.∗ f := sup R

is bounded from H1/2 to L1/2 . (b) Let (ϕn , n ∈ N) be a non-negative, non-decreasing sequence satisfying the condition lim

n→∞

log (n + 1) = +∞. ϕn

Then the maximal operator defined by |Rn f | n∈N ϕn sup

is not bounded from H1/2 to L1/2 .

(8.110)

462

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Proof By Theorem 5.3.2 it is enough to show that   ∼ ∗ 1/2   R a  dμ ≤ c < ∞  

(8.111)

I

for every 1/2-atom a, where I denotes the support of the atom. Let a be an arbitrary 1/2-atom with support I and μ (I ) = MN−1 . We may assume that I = IN . It is easy to see that Rn a (x) = σn a (x) = 0,

when n ≤ MN .

Therefore we can suppose that n > MN . Since a∞ ≤ MN2 , if we apply (4.108), we find that  |Rn a (x)| 1 |a (t)| |Ln (x − t)| dμ (t) (8.112) = log (n + 1) log (n + 1) IN  a∞ |Ln (x − t)| dμ (t) ≤ log (n + 1) IN    n−1  Kj (x − t) cMN2 ≤ dμ (t) log (n + 1) ln j +1 IN j =MN +1

+

cMN2 log (n + 1) ln

 |Kn (x − t)| dμ (t) . IN

Let x ∈ IN (xk ek + xl el ) , where 1 ≤ xk ≤ mk − 1, 1 ≤ xl ≤ ml − 1, k = 0, . . . , N − 2, l = k + 1, . . . , N − 1. From Lemmas 3.2.14 and 4.9.3 it follows that |Rn (a)| cMl Mk ≤ . log (n + 1) N2

(8.113)

Let x ∈ IN (xk ek ) , 1 ≤ xk ≤ mk − 1, k = 0, . . . , N − 1. Applying again Lemmas 3.2.14 and 4.9.3, we have that |Rn a (x)| MN Mk ≤ ≤ cMN Mk . log (n + 1) N

(8.114)

8.6 Maximal Operators of Riesz and Nörlund Logarithmic Means on Hp Spaces

463

By combining (1.5), (8.113) and (8.114), we get that  IN

=

1/2  ∗  ∼ R a (x) dμ (x)  

N−2 k −1 N−1 l −1   m  m Il+1 (xk ek +xl el )

k=0 xk =1 l=k+1 xl =1

+

N−1 k −1   m IN (xk ek )

k=0 xk =1

≤c

N−2  N−1  k=0 l=k+1

1 Ml

1/2  ∗  ∼ R a (x) dμ (x)  

1/2  ∗  ∼ R a (x) dμ (x)  

√ N−1  1 3 Ml Mk +c MN Mk ≤ C < ∞, N MN k=0

which shows part (a). (b) Let(λk , k ∈ N) be an increasing sequence of positive integers, which satisfies condition (8.110). For every λk there exists a positive integers (nk , k ∈ N) ⊂ (λk , k ∈ N) such that lim

k→∞

nk ϕM2nk +1

= ∞.

Let fk be the atom defined in Example 5.6.2. By combining (5.23) and (5.24) in Example 5.6.2, we find that    s  Rqnk fk  ϕqns

k

 qs    nk   Sj fk  1  =  s  ϕ qnk lqns j =M +1 j  k 2nk

  qs   nk D − D  j M2nk   1   =  ϕqns lqns  j  k k j =M2n +1 k

   M2s D   j +M2nk − DM2nk  1  . =  ϕqns lqns  j + M2nk  k k j =1

Thus, by applying (1.40) in Lemma 1.6.4, we find that    s  Rqnk fk  ϕqns

k

  M2s Dj  1  = . ϕqns lqns j + M2nk k

k

j =1

464

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Let x ∈ I2s \I2s+1 . Then    s  Rqnk fk  ϕqns

≥

k

M2s j 1  ϕqns lqns j + M2nk k

k

j =0

2s 1  1 j ϕqns lqns 2M2nk

M

≥

k

≥

j =0

k

2 cM2s . ϕqns lqns M2nk k

k

Hence,  Gm

 ∗ 1/2 n k −1  ∼  R fk  dμ ≥   I2s \I2s+1 s=1

≥c

n k −1 s=1

≥c

n k −1 s=1

   Rq s fk 1/2  nk    dμ  ϕqns  k

M2s 1 B M ϕqns lqns M2nk 2s k

k

1 B ϕM2nk +1 lM2n

k +1

M2nk

cnk ≥ B . ϕM2nk +1 lM2nk +1 M2nk Therefore, by using (5.25) we have that 

 ∗ 1/2 2 ∼    dμ Gm R fk  fk H1/2

≥

cnk → ∞, as k → ∞, ϕM2nk +1

so also part (b) is proved and the proof is complete. Next we state the following result concerning Nörlund logarithmic means. Theorem 8.6.5 Let 0 < p ≤ 1. Then there exists a martingale f ∈ Hp such that ∗ L f = +∞. p



8.6 Maximal Operators of Riesz and Nörlund Logarithmic Means on Hp Spaces

465

Proof We note that qs

αk Sj f 1  Lqαs f = s k lqαk ,s qαk − j

(8.115)

j =1

M2αk −1



1

=

l

qαs k

Sj f −j

qαs k

j =1 qs

αk Sj f 1  + s s qαk qαk − j

j =M2αk

=: I + I I. In view of (5.46), we get the following estimate: M

|I | ≤

1/p 2αk−1 M2αk−1 1  1 αk qαs k − j α 1/2 j =0 k−1 1/p

≤

M2αk−1 1/2

αk−1

(8.116)

1/p

≤

Mαk

3/2

.

αk

Moreover, according to the second inequality of (5.45) in the case when l = k (see also (8.101)), we can rewrite I I as qs

II =

αk 

1 lqαs

k

j =M2αk

SM2αk f qαk ,s − j

1/p−1

+

1 M2αk

1/2

lqαs

αk

k

qs

αk  Dj −M2αk

ψM2αk

j =M2αk

qαs k − j

(8.117)

=: I I1 + I I2 . Applying again (5.46), we have that 1/p

|I I1 | ≤

2RM2αk−1 1/2

αk−1

1/p

≤

2RMαk 3/2

αk

.

(8.118)

466

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Let x ∈ I2s \I2s+1 , s = [2αk /3] , . . . , αk . Since M 2s −1 j =0

M 2s −1 Dj j = −j M2s M2s − j j =0

=

M 2s −1 j =0

M 2s −1 M2s M2s − j − − j M2s M2s − j j =0

= Cp sM2s − M2s ≥ Cp sM2s , we obtain that |I I2 | =

    

1/p−1 M2s −1 

1 M2αk

1/2

lqαk ,s

αk

j =0

  Dj   M2s − j 

(8.119)

1/p−1

≥

Cp M2αk

x ∈ I2s /I2s+1 .

sM2s ,

3/2

αk

By combining (5.43) with (8.115)–(8.119) for x ∈ I2s \I2s+1 , s = [2αk /3] , . . . , αk and 0 < p ≤ 1, we get that    s  Lqαk f (x) = |I I I + I V1 + I V2 | ≥ |I V2 | − |I I I | − |I V1 | 1/p−1

≥

Cp M2αk 3/2

αk

sM2s −

2RMαk 3/2

αk

1/p−1

≥

Cp M2αk 3/2

αk

sM2s .

Consequently, 

  ∗ L f (x)p dμ (x) ≥ Gm

≥

αk 



s=[2αk /3] I2s \I2s+1 αk 



s=[2αk /3] I2s \I2s+1

≥ Cp

αk 

  ∗ L f (x)p dμ (x) p    s Lqαk f (x) dμ (x)



s=[2αk /3] I2s \I2s+1

1−p

M2αk−1 3p/2 αk

p

s p M2s dμ

8.6 Maximal Operators of Riesz and Nörlund Logarithmic Means on Hp Spaces

≥ Cp

αk 

467

1−p

M2αk−1 p/2

αk

s=[2αk /3]

p−1

M2s

⎧ ⎨ Cp 2αk (1−p)/3 , when 0 < p < 1, p/2 αk ≥ ⎩ 1/2 cαk , when p = 1. Hence, 

  ∗ L f (x)p dμ(x) → ∞, as k → ∞. Gm



The proof is complete.

Finally, in this Section we state and prove some results concerning the weighted maximal operator of Nörlund logarithmic means. First we consider the case 0 < p < 1. Theorem 8.6.6 (a) Let 0 < p < 1. Then the maximal operator . L∗p defined by .∗p f := sup L

n∈N

|Ln f | (n + 1)1/p−1

is bounded from Hp (Gm ) to Lp (Gm ) . (b) Let 0 < p < 1 and (ϕn , n ∈ N) be a non-negative, non-decreasing sequence satisfying the condition n1/p−1 = +∞. n→∞ ϕn log n lim

(8.120)

Then there exists a martingale f ∈ Hp (Gm ) such that the maximal operator |Ln f | n∈N ϕn sup

is not bounded from Hp (Gm ) to Lp (Gm ) .

468

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Proof (a) Since |Ln f |

≤

(n + 1)1/p−1

1 sup |Sk f | (n + 1)1/p−1 1≤k≤n |Sk f |

≤ sup

1/p−1 1≤k≤n (k + 1)

|Sn f |

≤ sup

1/p−1 n∈N (n + 1)

,

we have that sup

|Ln f |

n∈N (n +

1/p−1

1)

≤ sup

|Sn f |

n∈N (n + 1)

1/p−1

.

Therefore, by using Theorem 6.4.1, we obtain that |Ln f | |Sn f | sup ≤ sup ≤ Cp f H p (n + 1)1/p−1 (n + 1)1/p−1 n∈N

p

n∈N

p

and part (a) is proved. (b) Let (λk , k ∈ N) be an increasing sequence of positive integers such that 1/p−1

λk = ∞. k→∞ log(λk )ϕλk lim

Choose (nk , k ∈ N) ⊂ (λk , k ∈ N) such that

1/p−1 1/p−1 M2nk + 2 λk   lim ≥ c lim = ∞. k→∞ log M2nk + 2 ϕM2n +2 k→∞ log(λk )ϕλk k

(8.121)

Let fk be the atom defined in Example 5.6.2. By combining (5.23) and (5.24) in Example 5.6.2, we find that        LM +2 fk  − D  D M +1 M 2n 2nk k  2nk   =  ϕM2n +2  lM2nk +1 ϕM2nk +1 k     ψM2nk  1 = = . lM2nk +2 ϕM2nk +1 lM2nk +1 ϕM2nk +2

8.6 Maximal Operators of Riesz and Nörlund Logarithmic Means on Hp Spaces

Hence, )

    μ x ∈ Gm : LM2nk +2 fk  ≥

1 lM2nk +2 ϕM2nk +2

469

: = μ (Gm ) = 1.

(8.122)

By combining (5.25), (8.121) and (8.122), we get that

lM2n

k

≥

1  +2 ϕ M2nk +2

1/p−1 M2n k

lM2nk +2 ϕM2nk +2

 μ x ∈ Gm :

    LM2nk +2 fk  ≥

fk p

1/p−1 c M2nk + 2   ≥ → ∞, log M2nk + 2 ϕM2nk +2

91/p lM2n

k

1 +2 ϕM2n

k +2



as

k → ∞,

so part (b) is proved as well.  Our corresponding result for the case p = 1 reads as follows. Theorem 8.6.7 ∼∗

(a) The maximal operator L defined by ∼∗

|Ln f | log (n + 1) n∈N

L f := sup

is bounded from H1 (Gm ) to L1 (Gm ) . (b) Let (ϕn , n ∈ N) be a nonnegative, non-decreasing sequence satisfying the condition lim

n→∞

log (n + 1) = +∞. ϕn

(8.123)

Then there exists a martingale f ∈ H1 (Gm ) such that the maximal operator defined by |Ln f | n∈N ϕn sup

is not bounded from H1 (Gm ) to L1 (Gm ) .

470

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

Proof (a) It is obvious that n |Ln f | |Sk f | |Sn f | 1  1 ≤ sup ≤ sup . ln (n − k) k∈N log (k + 1) n∈N log (n + 1) n∈N log (n + 1)

sup

k=0

Thus, by using Theorem 6.4.5 we can conclude that sup |Ln f | ≤ sup |Sn f | ≤ C f H . 1 n∈N log (n + 1) 1 n∈N log (n + 1) 1 The proof of part (a) is complete. (b) Under condition (8.123), there exist positive integers (nk , k ∈ N) ⊂ (λk , k ∈ N) such that lim

k→∞

nk   = ∞. ϕ qnk

Let fk be the atom defined in Example 5.6.2. By combining (5.23), (5.24) and (5.25), we find that q

nk Sj fk 1 1  = Lqnk fk = lqnk qnk − j lqnk

j =1

=

1 lqnk

q

nk 

Dj − DM2nk

j =M2nk +1

qnk − j

qnk −1 −1

 Dj +M2n − DM2n k k j =1

qnk −1 − j

.

Hence, by applying (1.40) in Lemma 1.6.4 we get that   qnk −1       Dj 1     f = Lqnk k   lqnk  qnk −1 − j  j =1   qnk −1     Dj 1   = ψ M2nk  lqnk  qnk −1 − j  j =1  lqnk −1   Fqnk −1  lqnk     ≥ c Fqnk −1  .

=

8.7 Strong Convergence of Riesz and Nörlund Logarithmic Means in Hp Spaces

471

By now using Lemma 4.9.4, we can conclude that

Gm

    Lqnk fk  dμ ϕqn k

fk H1

    Lqnk fk 

 ≥

ϕqnk c Fqnk −1

dμ

Gm

≥ ≥

1

ϕqnk cnk → ∞, as k → ∞, ϕqnk

which shows part (b). 

8.7 Strong Convergence of Riesz and Nörlund Logarithmic Means in Hp Spaces Our first result in this Section reads: Theorem 8.7.1 Let 0 < p < 1/2 and f ∈ Hp (Gm ). Then there exists an absolute constant Cp , depending only on p such that the inequality p

∞ logp n R f   n Hp n=1

n2−2p

p

≤ Cp f Hp

holds, where Rn f denotes the nth Riesz logarithmic mean of f. Proof (a) By Theorem 5.3.2 it suffices to prove that ∞  n=1

logp n |Rn a|p dμ I

n2−2p

≤ Cp < ∞, for 0 < p < 1/2

(8.124)

for every p-atom a, where I denotes the support of the atom. Let a be an arbitrary p-atom with support I and μ (I ) = MN−1 . We may assume that I = IN . It is easy to see that Rn a(x) = σn a (x) = 0,

when n ≤ MN .

472

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces 1/p

Therefore, we can suppose that n > MN . Since a∞ ≤ MN , if we apply (4.108), then we can conclude that |Rn a (x)| 1/p  n−1  MN ≤ ln

  Kj (x − t) j +1

IN j =MN +1

(8.125) dμ (t) +

1/p MN



ln

|Kn (x − t)| dμ (t) . IN

Let x ∈ IN (xk ek + xl el ) , where 1 ≤ xk ≤ mk − 1, 1 ≤ xl ≤ ml − 1, k = 0, . . . , N − 2, l = k + 1, . . . , N − 1. From Lemmas 3.2.14 and 4.9.3, it follows that 1/p−2

|Rn a (x)| ≤

cMl Mk MN log(n + 1)

(8.126)

.

Let x ∈ IN (xk ek ) , 1 ≤ xk ≤ mk − 1, k = 0, . . . , N − 1. By applying again Lemmas 3.2.14 and 4.9.3, we have that 1/p−1

|Rn a (x)| ≤ MN

(8.127)

Mk .

By combining (1.5) and (8.125)–(8.127), we obtain that  |Rn a| dμ = p

IN

+

mj−1

N−2  N−1 



k,l k=0 l=k+1 xj =0,j ∈{l+1,...,N−1} IN

N−1  k=0

≤c

INk,N

N−2  N−1 

N−1  k=0

≤

|Rn a|p dμ

|Rn a|p dμ

k=0 l=k+1

+



1−2p

ml+1 . . . mN−1 (Ml Mk )p MN MN logp (n + 1)

1 p 1−p M M MN k N

1−2p N−2  N−1  (Ml Mk )p N−1  Mp cMN k + p p log (n + 1) Ml MN k=0 l=k+1

k=0

(8.128)

8.8 Final Comments and Open Questions

473

=

1−2p N−2 N−1  1  M 1−p N−1  Mp cMN k k + p 1−2p 1−p logp (n + 1) M M M N k=0 l=k+1 l k=0 k

≤

cMN + Cp . logp (n + 1)

1−2p

It is easy to see that ∞ 

1

n=MN +1

≤

n2−2p

Cp 1−2p

for 0 < p < 1/2.

(8.129)

MN

By combining (8.128) and (8.129), we get that ∞  n=MN +1

≤

∞ 

logp n 

n=MN +1 1−2p

≤ Cp MN

p IN |Rn a| n2−2p 1−2p

Cp MN n2−p ∞ 

n=MN +1

dμ

Cp + 2−p n 1

n2−2p

+

 + Cp ∞ 

n=MN +1

1 n2−p

It means that (8.124) holds so the proof is complete.

+ Cp ≤ Cp < ∞. 

8.8 Final Comments and Open Questions (1) The proof of Theorem 8.2.1, Corollary 8.2.5, Theorem 8.2.9, Corollary 8.2.10 and Theorem 8.2.11 can be found in [271] (see also [356]). (2) Theorems 8.2.3 and 8.2.6 and Remarks 8.2.4 and 8.2.7 and Corollary 8.2.8, Theorem 8.4.1, Remark 8.4.2, Corollary 8.4.3, Theorem 8.4.4 and Corollary 8.4.5 are given in [272]. (5) The proof of theorem 8.2.13, Corollaries 8.2.14, 8.2.15, 8.2.16 and 8.2.17 can be found in Memic et al. [222]. (6) Theorems 8.2.18, 8.2.19 and Corollary 8.2.20 are due to Blahota and Tephnadze [39] (see also [25]). (7) Theorem 8.2.21 and Corollary 8.2.22 can be found in Bhahota et al. [42]. (8) Theorem 8.3.1, Corollary 8.3.2, Theorem 8.3.9 and Corollary 8.3.10 are due to Tutberidze [382]. Similar problem for the Walsh-Kaczmarz system was studied by Gogolashvili and Tephnadze [142, 143].

474

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

(9) The proof of Theorem 8.3.3, Remark 8.3.4, Corollary 8.3.5, Theorem 8.3.6, Remark 8.3.7, Corollary 8.3.8, Theorem 8.3.11, Remark 8.3.12, Corollary 8.3.13, Theorem 8.3.14, Remark 8.3.15 and Corollary 8.3.16 can be found in Tutberidze [384] (see also Baramidze [21]). (13) The proof of Theorems 8.4.6, 8.4.7 and Corollary 8.4.8 can be found in Blahota and Tephnadze [39] (see also [25]). (14) Theorem 8.4.9 and Corollary 8.4.10 are due to Blahota et al. [42]. (15) Theorem 8.5.1, Corollary 8.5.2, Theorem 8.5.3, Corollary 8.5.4, Theorem 8.5.5, Corollary 8.5.6, Theorem 8.5.7 and Corollary 8.5.8 can be found in Tutberidze [384]. (16) Theorem 8.6.1 is due to Tephnadze [341, 352]. (17) The proof of Theorems 8.6.3, 8.6.4 and Remark 8.6.2 can be found in Tephnadze [352]. (18) The proof of Theorem 8.7.1 is given in Lukkassen et al. [216]. (19) Theorem 8.6.5 is due to Tephnadze [341]. (20) The proof of Theorem 8.6.6 can be found in Tephnadze [341]. (21) Theorem 8.6.7 was proved in [274]. Open Problem Let 0 < α ≤ 1. Does there exist a martingale f ∈ H1/(1+α), such that sup σnα f H1/(1+α) = ∞.

n∈N+

Open Problem Let 0 < α ≤ 1. To investigate T means generated by nondecreasing sequences under some suitable restrictions on the sequences (qk , k ∈ N), which provides boundedness of maximal operators of T means from the martingale Hardy spaces H1/(1+α) to the space weak-L1/(1+α). Open Problem Let f ∈ H1/2 . Does there exist an absolute constant c such that the following inequality 1/2

1  Rk f 1/2 1/2 ≤ c f H1/2 log n k n

k=1

holds? Remark 8.8.1 Stronger results were proved for Riesz means than for Fejér means, when 0 < p < 1/2, but in the case p = 1/2, it is still not known if it is possible to prove an analogical result for Riesz means as has been proved for Fejér means.

8.8 Final Comments and Open Questions

475

Open Problem (a) Let f ∈ Hp , where 0 < p < 1. Does there exist an absolute constant Cp depending only on p such that the following inequality ∞ p  logp k Lk f p

k 2−p

k=1

p

≤ Cp f Hp

holds? (b) For 0 < p < 1 and any non-negative, non-decreasing sequence (n , n ∈ N) satisfying the condition lim n = ∞,

n→∞

is it possible to find a martingale f ∈ Hp (Gm ) such that ∞ p  logp n Ln f p n

n2−p

n=1

= ∞?

Open Problem (a) Let f ∈ Hp , where 0 < p ≤ 1, and

ωHp

1 ,f Mn



 =o

log n 1/p−1

Mn

log2[p] n

 ,

as n → ∞.

Does the following convergence result hold: Lk f − f Hp → 0, as k → ∞?

(8.130)

Open Problem For any 0 < p < 1, is it possible to find non-negative, nondecreasing sequence (n , n ∈ N) such that the maximal operator . L∗p defined by |Ln f | n∈N n+1

. L∗p f := sup

is bounded from the Hardy space Hp (Gm ) to the Lebesgue space Lp (Gm )? Moreover, is it true that the rate of (n , n ∈ N) is sharp, that is, for any nonnegative, non-decreasing sequence (ϕn , n ∈ N) satisfying the condition lim

n→∞

n = ∞, ϕn

476

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

there exists a martingale f ∈ Hp (Gm ) such that the maximal operator |Ln f | n∈N ϕ (n + 1) sup

is not bounded from the Hardy space Hp (Gm ) to the space Lp (Gm )? Remark 8.8.2 According to Theorem 8.6.6, we can conclude that there exist absolute constants C1 and C2 such that C1 n1/p−1 ≤ n ≤ C2 n1/p−1 . log(n + 1) Open Problem (a) Let f ∈ Hp , where 0 < p < 1. Does there exist an absolute constant Cp depending only on p such that the following inequality holds: ∞ p  logp k Lk f p k=1

k 2−p

p

≤ Cp f Hp ,

(0 < p < 1)?

(b) For 0 < p < 1 and any nonnegative, non-decreasing sequence (n , n ∈ N) satisfying the condition lim n = ∞,

n→∞

is it possible to find a martingale f ∈ Hp (Gm ) such that ∞ p  logp n Ln f p n

n2−p

n=1

= ∞?

Open Problem (a) Let f ∈ Hp , where 0 < p ≤ 1, and

ωHp

1 ,f Mn



 =o

log n 1/p−1

Mn

log2[p] n

 ,

as n → ∞.

Does the following convergence result hold: Lk f − f Hp → 0, as k → ∞?

8.8 Final Comments and Open Questions

477

(b) Let 0 < p ≤ 1. Does there exist a martingale f ∈ Hp for which

ωHp

1 ,f Mn



 =O

log n 1/p−1

Mn

log2[p] n

 , as n → ∞

and Lk f − f weak−Lp  0, as k → ∞? Open Problem Does there exist a martingale f ∈ H1/2 (Gm ) such that sup Rn f 1/2 = ∞? n∈N

It is also interesting to investigate boundedness of RMn f from the martingale Hardy space Hp (Gm ) to the space Lp (Gm ) when 0 < p ≤ 1/2 : Open Problem Does there exist a martingale f ∈ Hp (Gm ), where 0 < p < 1/2, such that sup RMn f p = ∞?

n∈N

If we prove this result, we can conclude that the maximal operator defined by   sup RMn f  n∈N

is not bounded from the martingale Hardy space Hp (Gm ) to the space Lp (Gm ) when 0 < p < 1/2. Open Problem (a) Let f ∈ Hp (Gm ), where 1/2 < p ≤ 1. Is the maximal operator t ∗ of the Nörlund summability method with non-increasing sequence (qk , k ∈ N), satisfying the condition sup n∈N

1−p n Mn  p p−1 Q M < ∞, p QMn j =1 Mj j

(8.131)

bounded from the Hardy space Hp (Gm ) to the space Lp (Gm )? (b) For 1/2 < p ≤ 1 and non-increasing sequence (qk , k ∈ N) satisfying the condition 1−p n Mn  p p−1 sup p QMj Mj = ∞, Q n∈N Mn j =1

478

8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces

does there exist a martingale f ∈ Hp (Gm ) such that sup tn f p = ∞? n∈N

Open Problem Let f ∈ Hp (Gm ), where 0 < p < 1/ (1 + α) , 0 < α ≤ 1 and (qk , k ∈ N) is a sequence of non-increasing numbers satisfying the condition (8.131) for p = 1/(1 + α). Does there exist an absolute constant cα,p depending only on α and p such that p

∞ t f   k Hp k=1

k 2−(1+α)p

p

≤ cα,p f Hp ?

Open Problem Let f ∈ H1/(1+α)(Gm ), where 0 < α ≤ 1 and (qk , k ∈ N) be a sequence of non-increasing numbers satisfying condition (8.131) for p = 1/(1+α). Does there exist an absolute constant cα depending only on α such that 1/(1+α)

n 1  tm f H1/(1+α) 1/(1+α) ≤ cα f H1/(1+α) ? log n m m=1

Open Problem Let f ∈ Hp , where 1/2 < p ≤ 1. (a) Is the maximal operator T ∗ of the T summability method with non-decreasing sequence (qk , k ∈ N), satisfying the condition sup n∈N

1−p n Mn  p p−1 Q M < ∞, p QMn j =1 Mj j

(8.132)

bounded from the Hardy space Hp to the space Lp ? (b) For 1/2 < p ≤ 1 and non-decreasing sequence (qk , k ∈ N) satisfying the condition 1−p n Mn  p p−1 sup p QMj Mj = ∞, n∈N QMn j =1

does there exist a martingale f ∈ Hp such that sup Tn f p = ∞? n∈N

8.8 Final Comments and Open Questions

479

Open Problem Let f ∈ Hp , where 0 < p < 1/ (1 + α) , 0 < α ≤ 1 and (qk , k ∈ N) is a sequence of non-decreasing numbers satisfying the condition (8.132) for p = 1/(1 + α). Does there exist an absolute constant cα,p depending only on α and p such that p

∞ T f   k Hp k=1

k 2−(1+α)p

p

≤ cα,p f Hp ?

Open Problem Let f ∈ H1/(1+α), where 0 < α ≤ 1 and (qk , k ∈ N) be a sequence of non-decreasing numbers satisfying condition (8.132) for p = 1/(1 + α). Does there exist an absolute constant cα depending only on α such that 1/(1+α)

n 1  Tm f H1/(1+α) 1/(1+α) ≤ cα f H1/(1+α) ? log n m m=1

Chapter 9

Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces

9.1 Introduction In Chap. 7 we have proved the boundedness of σ ∗ from the Hardy space Hp to Lp for 1/2 < p ≤ ∞. In the present Chapter, we generalize these theorems to Hardy and Lebesgue spaces with variable exponents. For a measurable function p(·), the variable Lebesgue space Lp(·) consists of all measurable functions f 1 for which 0 |f (x)|p(x)dx < ∞. When p(·) is a constant, then we get back the usual Lp space. This topic needs essentially new ideas and is investigated very intensively in the literature nowadays (see e.g. Cruz-Uribe and Fiorenza [72], Diening et al. [83], Kempka and Vybíral [189], Nakai and Sawano [246, 286], Jiao et al. [173, 176, 178], Yan et al. [446] and Liu et al. [211, 212]). The interest in the variable Lebesgue spaces has increased since the 1990s because of their use in a variety of applications, such as in fluid dynamics, image processing, partial differential equations, variational calculus and harmonic analysis (see e.g. [1, 2, 11, 54, 68, 84, 158, 172, 210, 285, 374, 448]). The log-Hölder continuity condition is a very common condition in this theory (see e.g. Cruz-Uribe and Fiorenza [72] and Diening et al. [83]). Under this condition, the Hardy-Littlewood maximal operator is bounded on Lp(·)(R) whenever p− > 1. Nakai and Sawano [246] first introduced the Hardy space Hp(·)(R) with a variable exponent p(·) and established the atomic decompositions. Independently, CruzUribe and Wang [73] also investigated the variable Hardy space Hp(·)(R). Sawano [286] improved the results in [246]. Ho [165] studied weighted Hardy spaces with variable exponents. Similar results for the anisotropic Hardy spaces Hp(·)(R) can be found in Liu et al. [211, 212]. Summability in variable Lebesgue and Hardy spaces were studied in [211, 212, 333, 430–432, 434]. Martingale Hardy spaces with variable exponents were investigated in a great number of papers, such as [154, 155, 173, 175, 177–179, 435, 436, 441]. The atomic s decomposition of the Hardy space Hp(·) with variable exponent was proved in © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 L.-E. Persson et al., Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series, https://doi.org/10.1007/978-3-031-14459-2_9

481

482

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces

Jiao et al. [173], while in [178], the authors gave a systematic study of variable martingale Hardy spaces. The boundedness of σ ∗ from Hp to Lp was generalized: the boundedness from Hp(·) to Lp(·) for Walsh-Fourier series can be found in [178] while for Vilenkin-Fourier series in [438]. Martingale Musielak–Orlicz Hardy spaces were investigated in Xie et al. [180, 214, 442–444]. In this Chapter, we introduce three variable Hardy spaces containing Vilenkin martingales and generated by the (conditional) quadratic variation and by Doob’s s , H S and H M . We introduce two maximal function, they are denoted by Hp(·) p(·) p(·) new maximal operators and show that, under the log-Hölder continuity condition, they are bounded on Lp(·) if 1 < p− < ∞ (for more about boundedness see [334, 433, 435]). We use the notation p− = ess inf p(x) x∈Gm

and

p+ = ess sup p(x). x∈Gm

We give the atomic decomposition of these Hardy spaces and prove that they are all equivalent. Next we verify the convergence of the partial sums of the VilenkinFourier series in the Lp(·) -norm if 1 < p− < ∞. We prove also that σ ∗ is bounded from Hp(·) to Lp(·) under the conditions 1/2 < p− < ∞

and

1 1 − < 1. p− p+

It is also proved that these conditions are sharp. The condition p1− − p1+ < 1 is surprising because the corresponding results hold for Fourier series or Fourier transforms without this condition (see Liu et al. [211, 212] and Weisz [430, 434]). This gives a serious difference between the trigonometric Fourier analysis and Vilenkin-Fourier analysis. The proofs of these results for Vilenkin-Fourier series need essentially new ideas. Finally, the boundedness of σ ∗ implies some convergence results of the Fejér means as well.

9.2 Variable Lebesgue Spaces In this Section, we recall some basic notations on variable Lebesgue spaces and give some elementary and necessary facts about these spaces. The proofs are omitted because they can be found in the excellent books Cruz-Uribe and Fiorenza [72] and Diening et al. [83]. We are going to generalize the Lebesgue spaces Lp defined in Sect. 1.5. A measurable function p(·) : Gm → (0, ∞) is called a variable exponent. For any variable exponent p(·) and any measurable set A ⊂ Gm , we will use the notation p− (A) := ess inf p(x) x∈A

and

p+ (A) := ess sup p(x). x∈A

9.2 Variable Lebesgue Spaces

483

If A = Gm , then the numbers p− (A) and p+ (A) are denoted simply by p− and p+ . Denote by V the collection of all variable exponents p(·) satisfying 0 < p− ≤ p+ < ∞. In what follows, we use the symbol p = min{p− , 1}.

(9.1)

Definition 9.2.1 For p(·) ∈ V and a measurable function f , the modular functional ρp(·) is defined by 

1

ρp(·) (f ) :=

|f (x)|p(x) dμ(x)

0

and the Luxemburg quasi-norm is given by setting 

f p(·)

9

 f := inf η ∈ (0, ∞) : ρp(·) ≤1 . η

The variable Lebesgue space Lp(·) is defined to be the set of all measurable functions f such that ρp(·) (f ) < ∞ and equipped with the quasi-norm  · p(·) . It is easy to see that if p(·) is a constant, then we get back the Lp spaces. For p(·) ∈ V and f, g ∈ Lp(·) , we can easily prove that ηf p(·) = |η|f p(·) s |f | = f ssp(·) p(·)

(η ∈ C), (s ∈ (0, ∞))

and p

p

p

f + gp(·) ≤ f p(·) + gp(·) . The last inequality is a generalization of Minkowski’s inequality proved in Theorem 1.4.15. Details can be found in the monographs Cruz-Uribe and Fiorenza [72] and Diening et al. [83]. The function p (·) denotes the conjugate exponent function of p(·), i.e., 1 1 + =1 p(x) p (x)

(x ∈ Gm ).

The well known Hölder’s inequality given in Theorem 1.4.10 can be generalized for variable Lebesgue spaces (see Cruz-Uribe and Fiorenza [72, p.27] or Diening et al. [83, p.74]).

484

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces 

Theorem 9.2.2 Let p(·) ∈ V with p− ≥ 1. Then, for all f ∈ Lp(·) and g ∈ Lp (·) ,  Gm

|fg| dμ ≤ Cp(·) f p(·) gp (·) .

The next theorem generalizes (1.27) and can be found e.g. in Diening et al. [83, p.77]. Theorem 9.2.3 Let p(·) ∈ V with p− ≥ 1. Then 1 f p(·) ≤ sup 2 gp (·) ≤1

 Gm

|fg| dμ ≤ 2 f p(·) .

As we have seen in Sect. 1.3, Gm can be identified with the unique interval [0, 1) and the Haar measure μ with the Lebesgue measure λ. For simplicity, in what follows we will use these notations. Definition 9.2.4 We say that a function p(·) ∈ V satisfies the log-Hölder continuity condition, if there exists a positive constant Clog (p) such that, for any x, y ∈ [0, 1), |p(x) − p(y)| ≤

Clog (p) . log(e + 1/|x − y|)

(9.2)

The set of all these functions is denoted by C log . The following two lemmas were proved in Cruze-Uribe and Fiorenza [72] and Diening et al. [83]: Lemma 9.2.5 We have that p(·) ∈ C log if and only if there exists K > 1 such that for all intervals I ⊂ [0, 1), λ(I )p− (I )−p+(I ) ≤ K.

(9.3)

Lemma 9.2.6 If p(·) ∈ C log , then, for any interval I ⊂ [0, 1) and any x ∈ I , λ(I )1/p− (I ) ∼ λ(I )1/p(x) ∼ λ(I )1/p+ (I ) ∼ χI p(·) , where ∼ denotes the equivalence of the numbers. Remark 9.2.7 There exist a lot of functions p(·) satisfying (9.2). For concrete examples we mention the function a + cx with parameters a and c such that the function is positive (x ∈ [0, 1)). All positive Lipschitz functions with order 0 < α ≤ 1 also satisfy (9.2) and (9.3).

9.3 Doob’s Inequality in Variable Lebesgue Spaces

485

It is clear that (9.3) implies that there exists 0 < β < 1 such that λ(I )1/p+ (I )−1/p−(I ) ≤

1 . β

(9.4)

Actually, β = K −1/(p+(I )p− (I )) and (9.3) is equivalent to (9.4) because 0 < p− ≤ p+ < ∞. Note that K and β are depending on p(·).

9.3 Doob’s Inequality in Variable Lebesgue Spaces In this Section, we generalize Doob’s inequality to the variable Lebesgue space Lp(·) . A martingale f = (fn )n≥0 is called an Lp(·) -martingale if fn ∈ Lp(·) for all n ∈ N. In this case, we set f Lp(·) = sup fn p(·) . n≥0

If f Lp(·) < ∞, then f is said to be a Lp(·)-bounded martingale and it is denoted by f ∈ Lp(·) . For a martingale, we define Doob’s maximal maximal functions Mm , m ∈ N, and M of f by Mm (f ) = sup |fn |,

M(f ) = sup |fn |.

0≤n≤m

n≥0

The maximal function M(f ) is also denoted by f ∗ . For an integrable function f , we define the maximal operator in the same way as for the martingale (En f ). In the proof of Doob’s inequality (see Theorem 9.3.2), we need the following lemma. Recall that an interval of the form In (x) (n ∈ N, x ∈ [0, 1)) is called a Vilenkin interval. Lemma 9.3.1 Let p(·) ∈ C log with 1 ≤ p− ≤ p+ < ∞. Suppose that f ∈ Lp(·) with f p(·) ≤ 1 and f = f χ{|f |>1} . Then, for any Vilenkin interval I and for any p− (I ) ≤ r ≤ p+ (I ) (r < ∞) and 1 ≤ t ≤ p− ,

βt λ(I )

r

 |f (y)| dy

≤

I

1 λ(I )

t

 |f (y)|p(y)/t dy

,

I

where the constant 0 < β < 1 is defined in (9.4). Proof By Hölder’s inequality

βt λ(I )

r

 |f (y)| dy I

 ≤ β

t

1 λ(I )

t /p−(I ) r

 |f (y)| I

p− (I )/t

dy

.

(9.5)

486

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces

Inequality (9.4) implies that βλ(I )−1/p− (I ) ≤ λ(I )−1/p+ (I ) ≤ λ(I )−1/r . Since f = f χ{|f |>1} , 

 |f (y)|p(y)/t dy ≤ I

|f (y)|p(y) dy ≤ ρ(f ) ≤ 1. I

Therefore,  β

t

1 λ(I )

≤ λ(I )−t ≤ λ(I ) ≤ λ(I )

−t

−t

t /p−(I ) r

 |f (y)|

p− (I )/t

dy

I

 |f (y)|p− (I )/t dy





I

|f (y)|

p(y)/t

|f (y)|

p(y)/t

t r/p−(I )

t r/p−(I ) dy

I

t dy

,

I

The proof follows by combining this estimate with (9.5).



For a constant p, Doob’s inequality M(f )p ≤ Cp f p

(1 < p < ∞)

was proved in Theorem 2.3.6. We generalize this inequality to variable Lebesgue spaces as follows. The proof is based on the fact that every σ -algebra Fn is generated by finitely many Vilenkin intervals. Theorem 9.3.2 Let p(·) ∈ C log and f = (fn ) ∈ Lp(·) . If 1 < p− ≤ p+ < ∞, then M(f )p(·) ≤ Cp(·) f Lp(·) .

(9.6)

If 1 ≤ p− ≤ p+ < ∞, then sup λχ{M(f )>λ} p(·) ≤ Cp(·) f Lp(·) .

(9.7)

λ>0

Proof It is enough to prove the theorem for martingales of the type (f0 , f1 , . . . , fn , fn , . . .), where n ∈ N is arbitrary. For simplicity, we will write f instead of fn . By homogeneity, we may suppose that f p(·) = 1 and that f is non-negative. We

9.3 Doob’s Inequality in Variable Lebesgue Spaces

487

decompose f as f 1 + f 2 , where f 1 = f χ{f >1} ,

f 2 = f χ{f ≤1} .

Then f i p(·) ≤ 1 and ρ(f i ) ≤ 1, i = 1, 2. Moreover, by the convexity of the modular functional, 2ρ(αM(f )) ≤ ρ(2αM(f 1 )) + ρ(2αM(f 2 ))  1

p(x) 2αM(f 1 )(x) dx + 1, ≤

(9.8)

0

where α = β p− /4 < 1/4. We introduce the stopping times νk := inf{n ∈ N : fn1 > 2k }

(k ∈ Z).

It is clear that {νk < ∞} = {M(f 1 ) > 2k }. Since f 1 ∈ Lp(·), M(f 1 ) is almost everywhere finite and we may suppose that f 1 = 0, thus Mf 1 > 0 everywhere. Then {νk < ∞} = ∪j ∈N {νk = j } = ∪j ∈N ∪i Ik,j,i , where Ik,j,i ∈ Fj are disjoint Vilenkin intervals for a fixed k. For simplicity, for a fixed k, the reordering of the sets Ik,j,i are denoted by Bk,l (l = 1, 2, . . .). Hence, for a fixed k, Bk,j (j = 1, 2, . . .) are disjoint Vilenkin intervals, {νk < ∞} = ∪j Bk,j and 1 λ(Bk,j )

 f 1 dλ > 2k

(k ∈ Z).

Bk,j

For a fixed k, define the sets Ek,j as follows: Ek,1 := ({νk < ∞} \ {νk+1 < ∞}) ∩ Bk,1 , Ek,2 := (({νk < ∞} \ {νk+1 < ∞}) ∩ Bk,2 ) \ Ek,1 , Ek,3 := (({νk < ∞} \ {νk+1 < ∞}) ∩ Bk,3 ) \ Ek,2 ,

488

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces

etc. Then the sets Ek,j are disjoint for all k and j and ({νk < ∞} \ {νk+1 < ∞}) =



Ek,j .

j

We estimate the first term on the right hand side of (9.8) by 

1

1

p(x)

2αM(f )(x)



dx =

k∈Z {νk η/2} := F1 ∪ F2 .

dx

9.3 Doob’s Inequality in Variable Lebesgue Spaces

489

Since M(f 2 ) ≤ 1, F2 = ∅ if η > 2. If 0 < η ≤ 2, then ρ(η/2χF2 ) ≤ 1, and thus ηχF2 p(·) ≤ 2. On the other hand,  ρ(αηχF1 ) =

(αη)p(x) dx,

(9.9)

F1

where α = β/2 < 1/2. Let us introduce the stopping time ν := inf{n ∈ N : fn1 > η/2} and define the sets Bj and Ej similarly to the first half of the proof. Thus F1 = ∪j Bj ,

Ej ⊂ Bj ,

F1 = ∪j Ej

and 1 λ(Bj )

 f 1 dλ > η/2. Bj

Then  (αη)p(x) dx ≤ F1



 Ej

j

β λ(Bj )

p(x)

 f 1 (y) dP (y)

dx.

Bj

Applying Lemma 9.3.1 with r = p(x) and t = 1, we can see that  (αη)

p(x)

dx ≤

F1

 Ej

j

=

1 λ(Bj )

 λ(Ej )  λ(Bj )

j



1

≤

 |f 1 (y)|p(y) dy dx Bj

|f 1 (y)|p(y) dy

Bj

|f 1 (y)|p(y) dy ≤ 1.

0

This completes the proof of the theorem. Theorem 9.3.2 can be proved for Lp(·) -functions in the same way. Theorem 9.3.3 Let p(·) ∈ C log and f ∈ Lp(·). If 1 < p− ≤ p+ < ∞, then M(f )p(·) ≤ Cp(·) f p(·) .



490

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces

If 1 ≤ p− ≤ p+ < ∞, then sup λχ{M(f )>λ} p(·) ≤ Cp(·) f p(·) . λ>0

9.4 The Maximal Operator Us Besides the Doob’s maximal operator, we introduce two new maximal operators. For a martingale f = (fn ), the first one, Us , is defined by Us f (x) := sup

 mj −1 n−1

 Mj s 

x∈I j =0

Mn

l=0

   1  , f dλ n   λ(I j,l ) I j,l

where I is a Vilenkin interval with length Mn−1 , s is a positive constant and ˙ j−1 I j,l := I +lM +1 . ˙ denotes the addition on the corresponding Vilenkin group (see Recall that + Sect. 1.3). Of course, if f ∈ L1 , then we can write in the definition f instead of fn . Let us define Ik,n := [kMn−1 , (k + 1)Mn−1 )

(0 ≤ k < Mn , n ∈ N).

The definition can be rewritten to Us f := sup

M n −1

χIk,n (x)

n∈N k=0

 mj −1 n−1

 Mj s  j =0

Mn

l=0

      f dλ ,  n j,l  j,l  λ(I ) I 1

k,n

k,n

j,l

where Ik,n := (Ik,n )j,l . In the next theorem we will apply Theorem 5.5.1. Theorem 9.4.1 For all 0 < p ≤ ∞ and all 0 < s < ∞, we have Us f p ≤ Cp f Hp

(f ∈ Hp ).

Proof Observe that (9.10) holds for p = ∞. Indeed, Us f ∞ ≤ sup

 mj −1 n−1

 Mj s 

n∈N j =0

Mn

l=0

f ∞ ≤ C f ∞

(9.10)

9.4 The Maximal Operator Us

491

because of the fact that  n

 Mj s Mn

j =0

≤ C.

(9.11)

By Theorem 5.5.1 and interpolation, the proof will be complete if we show that the operator Us satisfies  |Us a(x)|p ≤ Cp

(9.12)

I

for each 0 < p ≤ 1 and p-atoms a with support I . Here I denotes the complement of I , i.e., I = [0, 1) \ I . Choose a p-atom a with support I , where I is a Vilenkin −1 −1 interval with length |I | = MK (K ∈ N). We can assume that I = [0, MK ). It is easy to see that  mj −1 n−1

 Mj s  Mn

j =0

l=0

   1  adλ = 0  j,l j,l λ(J ) J

if n ≤ K, where J is a Vilenkin interval with length Mn−1 . Therefore we can suppose −1 −1 ) and x ∈ J imply that J j,l ∩ [0, MK )=∅ that n > K. Observe that x ∈ [0, MK if j ≥ K. Thus J j,l a = 0 for j ≥ K. Hence, we may assume that j < K. The −1 −1 −1 −K ). ˙ same holds if j < K and x ∈ [Mj−1 +1 + MK , Mj ), because x +lMj +1 ∈ [0, 2 Hence |Us a(x)| ≤ sup χJ (x) n>K

1/p

≤ MK

K−1 

j =0

K−1 

j =0

Mj MK

Mj Mn

s

mj −1

χ[M −1

−1 j+1 ,Mj+1

+M −1 ) (x) K

 l=0

   1   adλ   j,l λ(J ) J j,l

s χ[M −1

−1 −1 j+1 ,Mj+1 +MK )

(x)

so that  |Us a(x)|p ≤ MK I

K−1 

j =0

Mj MK

sp

−1 MK ≤ Cp ,

and (9.12) holds, which completes the proof of the theorem.



492

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces

Since Hp is equivalent to Lp when 1 < p ≤ ∞, we obtain Corollary 9.4.2 For all 1 < p ≤ ∞ and all 0 < s < ∞, we have Us f p ≤ Cp f p

(f ∈ Lp ).

We generalize this result to Lebesgue spaces with variable exponents as follows. Theorem 9.4.3 Let p(·) ∈ C log with 1 < p− ≤ p+ < ∞ and 0 < s < ∞. If 1 1 − < s, p− p+

(9.13)

then Us f p(·) ≤ Cp(·) f p(·)

(f ∈ Lp(·) ).

Proof We assume that f p(·) ≤ 1/2. Then 

1

|Us f (x)|p(x) dx

0



1

≤ Cp(·) 

0 1

≤ Cp(·) 0

 |Us (f χ|f |≥1 )(x)|p(x) dx + Cp(·)

1 0

|Us (f χ|f | 1

j,l

λ(Ik,n )

(9.15)

Ik,n

for all j = 0, . . . , n − 1, k = 0, . . . , Mn − 1, n ∈ N. Let us denote by Ik,n,j,l,1 (respectively Ik,n,j,l,2 ) those points x ∈ Ik,n for which j,l

j,l

p(x) ≤ p+ (Ik,n )

(respectively p(x) > p+ (Ik,n )).

Then 

1

|Us f (x)|

p(x)

dx ≤ Cp(·)

0

2   m=1 0

1

 sup

M n −1 

(9.16)

χIk,n (x)

n∈N k=0

⎞p(x)

 mj −1  n−1

 Mj s  χIk,n,j,l,m (x) j =0

Mn

j,l

l=0

λ(Ik,n )

j,l

|f (t)| dt ⎠

dx

Ik,n

=: I + I I. Let q(x) =: p(x)/p0 > 1 for some 1 < p0 < p− . Since the sets Ik,n are disjoint for a fixed n and the function t #→ t q(x) is convex (x is fixed), we conclude 

1

I ≤ Cp(·)

 sup

0

M n −1 

n∈N k=0

χIk,n (x)

⎛ ⎞q(x) ⎞p0  mj −1  n−1

 Mj s  χIk,n,j,l,1 (x) ⎟ ⎝ |f (t)| dt ⎠ ⎠ dx j,l j,l Mn I λ(Ik,n ) k,n j =0 l=0

494

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces





1

≤ Cp(·)

sup 0

M n −1 

χIk,n (x)

n∈N k=0

q(x)⎞p0

 mj −1   n−1

 Mj s  χIk,n,j,l,1 (x) j,l

Mn

j =0

j,l

λ(Ik,n )

l=0

|f (t)| dt

⎠

dx

Ik,n

so that 

1

I ≤ Cp(·)



M n −1

sup 0

n−1

 j =0

χIk,n (x)

n∈N k=0

Mj Mn

 s m j −1

χIk,n,j,l,1 (x)

l=0

λ(Ik,n )

q+ (I j,l )



j,l

k,n

j,l

|f (t)| dt

Ik,n

⎞p0 ⎟ ⎠

dx.

Here we have used (9.15), the boundedness of (mj ) and the fact that q(x) ≤ j q+ (Ik,n ) on Ik,n,j,l,1 . Hence, by Lemma 9.3.1,  I ≤ Cp(·)

1

 sup

0

M n −1

(9.17)

χIk,n (x)

n∈N k=0

⎞ p0

 mj −1  n−1

 Mj s  χIk,n,j,l,1 (x) j =0

Mn

j,l

l=0

p0 ≤ Cp(·) Us (|f |q(·)) p0 ≤ Cp(·) |f |q(·)

λ(Ik,n )

j,l

|f (t)|q(t ) dt ⎠

dx

Ik,n

p0

p0

≤ Cp(·) . Next we choose 0 < r < s and observe that 

1

I I ≤ Cp(·)

sup 0

⎛ ⎝



M n −1

χIk,n (x)

n∈N k=0

  mj −1

 n−1

 Mj s−r  Mj r χIk,n,j,l,2 (x) j =0

Mn

l=0

Mn

j,l

λ(Ik,n )

j,l

Ik,n

⎞q(x)⎞p0 ⎟ |f (t)| dt ⎠ ⎠ dx

9.4 The Maximal Operator Us





1

≤ Cp(·)

sup 0

M n −1

495

χIk,n (x)

n∈N k=0

q(x) ⎞p0   mj −1 

 n−1

 Mj r χIk,n,j,l,2 (x) Mj s−r  ⎠ dx. |f (t)| dt j,l j,l Mn Mn Ik,n λ(Ik,n ) j =0 l=0 Thus, by Hölder’s inequality, ⎛



1

I I ≤ Cp(·)

⎝ sup

0



M n −1

j,l λ(Ik,n )

≤ Cp(·)

⎝ sup

0

|f (t)|

j,l Ik,n

⎛ 1

M n −1

j,l q− (Ik,n )

χIk,n (x)

n∈N k=0

j,l q(x)/q− (Ik,n )

Mn

Mn

j =0



χIk,n,j,l,2 (x)



χIk,n (x)

n∈N k=0

mj −1

  n−1

 Mj s−r  Mj rq(x)

Mn

l=0

q(x)/q−(I j,l ) k,n

dt

⎞p0 ⎟ ⎠

dx

mj −1

  n−1

 Mj s−r  Mj rq(x)

Mn

j =0

 χIk,n,j,l,2 (x)

j,l Ik,n

Mn

l=0

|f (t)|

j,l q− (Ik,n )

q(x)/q−(I j,l ) k,n

dt

⎞p0 ⎟ ⎠

dx.

j,l

Since |f | ≥ 1 or f = 0 and q(x) > q− (Ik,n ) on Ik,n,j,l,2 , we have j,l

q− (Ik,n ) ≤ q(t) < p(t)

j,l

t ∈ Ik,n

for all

and 



j,l

j,l Ik,n

|f (t)|q− (Ik,n ) dt ≤

j,l Ik,n

|f (t)|p(t ) dt ≤

1 . 2

We conclude that ⎛



1

I I ≤ Cp(·) 0

⎝sup

M n −1 

χIk,n (x)

n∈N k=0 j,l

q(x)/q−(Ik,n ) Mn χIk,n,j,l,2 (x)

mj −1

  n−1

 Mj s−r  Mj rq(x) j =0

Mn

 j,l

Ik,n

|f (t)|

l=0

 p0

j,l

q− (Ik,n )

Mn

dt

dx

(9.18)

496

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces

⎛



1

≤ Cp(·)

⎝sup

0

M n −1 

χIk,n (x)

n∈N k=0

mj −1  n−1

 Mj s−r  j =0

Mn

l=0

j,l

−rq(x)+q(x)/q−(Ik,n )−1 Mn χIk,n,j,l,2 (x)

rq(x)

Mj

j,l

p0



1 λ(Ik,n )

j,l

|f (t)|

q(t )

dt

dx.

Ik,n

For fixed k, n let Jj denote the Vilenkin interval with length Mj−1 and Ik,n ⊂ Jj . j,l ˙ −j −1 = Jj . Inequality (9.3) implies that Then Ik,n ⊂ Jj +2 −p(x)

Mj

j,l

−p− (Ik,n )

∼ Mj

x ∈ Ik,n .

for

It is easy to check that for x ∈ Ik,n,j,l,2 , rq(x)

Mj

rq(x)

= Mj

q(x)

Mj

rq(x)

≤ Cp(·)Mj

−q(x)

Mj

j,l

q− (Ik,n )

Mj

 rq(x)−

< Cp(·) Mj

−q(x)

Mj

j,l q(x)−q−(I ) k,n j,l q− (Ik,n )



= Cp(·) Mj

rq(x)− q(x) j,l +1 q− (Ik,n )





.

Furthermore, rq(x) −

 1 + 1 ≥ q(x) r − +1 j q− q− (Ik,n ) ) 1, if r −

≥ 1 q+ r − q− + 1, if r − q(x)

Let r0 := min 1, q+ r −

1 q−

1 q− 1 q−

≥ 0; < 0.

+ 1 . Then r0 > 0 if and only if 1 1 − < r. q− q+

(9.19)

9.4 The Maximal Operator Us

497

Hence, in view of (9.18),  I I ≤ Cp(·)

⎛ 1

0



Mj Mn 

≤ Cp(·)

⎝sup

M n −1

χIk,n (x)

n∈N k=0

 mj −1 n−1

 Mj s−r  Mn

j =0

rq(x)−q(x)/q−(I j,l )+1

j,l λ(Ik,n )

⎛ 1

0

⎝sup

M n −1

χIk,n (x)

n∈N k=0

mj −1

1

l=0

λ(Ik,n )

j,l

j,l Ik,n

|f (t)|q(t ) dt ⎠

dx

 n−1

 Mj s−r+r0 Mn

j =0

⎞ p0





(9.20)

⎞ p0



1

k,n

χIk,n,j,l,2 (x)

l=0

j,l

|f (t)|q(t ) dt ⎠

dx

Ik,n

p0 ≤ Cp(·) Us−r+r0 (|f |q(·) ) p0

p0 ≤ Cp(·) |f |q(·) p0

≤ Cp(·) . By combining (9.16), (9.17) and (9.20) we conclude that (9.14) holds. Since p0 can be arbitrarily near to 1 and r to s, inequality (9.19) proves the theorem with the range (9.13).  Remark 9.4.4 Inequality (9.13) and Theorem 9.4.3 hold if s ≥ 1, or more generally if p− > max(1/s, 1). The operator Us is not bounded on Lp(·) when (9.13) is not satisfied. More exactly, the following theorem holds. Theorem 9.4.5 Let p(·) ∈ C log with 1 < p− ≤ p+ < ∞ and 0 < s < ∞. If 1 1 >s − −1 p (I ˙ p− (I0,n +M1 ) + 0,n )

(9.21)

for all n ∈ N, then Us is not bounded on Lp(·) . Proof It is easy to see that 

1 0

 |Us f (x)|

p(x)



1

dx ≥ 0

χI0,n (x)

Mn−s

1 0,0 λ(I0,n )

p(x)

 0,0 I0,n

f (t) dt

dx.

498

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces

Defining 0,0 1/p− (I0,n )

f (t) := χI 0,0 (t)Mn

,

0,n

we have 0,0 1/p− (I0,n )

f p(·) = Mn

χI 0,0 p(·) ≤ C 0,n

by Lemma 9.2.6. Using (9.3), we can see that 

1

 |Us f (x)|

p(x)

−sp(x)

dx ≥

0

Mn

0,0 p(x)/p− (I0,n )

Mn

dx

I0,n



0,0 p+ (I0,n )(1/p− (I0,n )−s)

≥C

Mn

dx

I0,n 0,0 p+ (I0,n )(1/p− (I0,n )−s)

= CMn

Mn−1 ,

which tends to infinity as n → ∞ if (9.21) holds. The proof is complete.



9.5 The Maximal Operator Vα,s For a martingale f = (fn ), we define the next version of maximal operators, Vα,s , by Vα,s f (x) := sup



 mj −1 n−1  n−1

 Mj α Mj s 

x∈I j =0 i=j

Mn

Mi

l=0

   j,i,l λ(I ) 1

I j,i,l

  fn dλ ,

where I is a Vilenkin interval with length Mn−1 , α, s are positive constants and −1 −1 ˙ −1 ˙ I j,i,l := I +[lM j +1 , lMj +1 +Mi ).

Obviously, Vα,s f := sup

M n −1

χIk,n

n∈N k=0 j,i,l

where Ik,n := (Ik,n )j,i,l .



 mj −1 n−1  n−1

 Mj α Mj s  j =0 i=j

Mn

Mi

l=0

      f dλ ,  n j,i,l  j,i,l  λ(I ) I 1

k,n

k,n

9.5 The Maximal Operator Vα,s

499

Theorem 9.5.1 Suppose that 0 < p ≤ ∞ and 0 < α, s < ∞. Then Vα,s f p ≤ Cp f Hp

(f ∈ Hp ).

Proof The inequality holds for p = ∞ because Vα,s f

≤ sup

∞



 mj −1 n−1  n−1

 Mj α Mj s  Mn

n∈N j =0 i=j

Mi

f ∞ ≤ C f ∞ .

l=0

It is sufficient to prove that 

  Vα,s a(x)p ≤ Cp

(9.22)

I

for each 0 < p ≤ 1 and p-atoms a with support I . Let a be a p-atom with support −1 I = [0, MK ). If i ≤ K, then  I j,i,l

a dλ = 0.

Thus i > K and so n > K. Similarly to the proof of Theorem 9.4.1, j < K and −1 −1 −1 x ∈ [Mj−1 +1 , Mj +1 + MK ). Hence, in the case x ∈ [0, MK ), K−1 n−1    Vα,s a(x) ≤ sup χJ (x) n>K

j =0 i=K

mj −1

 l=0

≤



∞

K−1  j =0 i=K

≤

α

Mj Mi

s

    1  adλ χ[M −1 ,M −1 +M −1 ) (x)  j,i,l j+1 j+1 K λ(J ) J j,i,l

1/p MK χJ (x)

1/p MK

Mj Mn

K−1 

j =0

Mj MK

Mj MK

α

Mj Mi

s χ[M −1

−1 −1 j+1 ,Mj+1 +MK )

(x)

α χ[M −1

−1 −1 j+1 ,Mj+1 +MK )

(x),

where J is a Vilenkin interval with length Mn−1 . Consequently, 

K−1    Vα,s a(x)p ≤ MK I

j =0



Mj MK

and (9.22) is proved, which finishes the proof.

αp

−1 MK ≤ Cp ,



500

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces

Corollary 9.5.2 For all 1 < p ≤ ∞ and all 0 < α, s < ∞, we have Vα,s f p ≤ Cp f p

(f ∈ Lp ).

The generalization of this result to variable Lebesgue spaces reads as follows. Theorem 9.5.3 Let p(·) ∈ C log , 1 < p− ≤ p+ < ∞ and 0 < α, s < ∞. If 1 1 − < α + s, p− p+

(9.23)

then Vα,s f p(·) ≤ Cp(·) f p(·)

(f ∈ Lp(·) ).

Proof Similarly to the proof of Theorem 9.4.3, we may suppose again that f p(·) ≤ 1/2, |f | ≥ 1 or f = 0 and 

1 j,i,l

j,i,l

λ(Ik,n )

|f (t)| dt > 1.

Ik,n

We denote by Ik,n,j,i,l,1 (respectively Ik,n,j,i,l,2 ) those points x ∈ Ik,n for which j,i,l

j,i,l

p(x) ≤ p+ (Ik,n )

(respectively p(x) > p+ (Ik,n )).

Then 

1

|Vα,s f (x)|p(x) dx

0

≤ Cp(·)

2  

⎛ 1

⎝ sup

m=1 0



Mj Mi

M n −1

χIk,n (x)

n∈N k=0

s m j −1

=: I + I I.

(9.24)

χIk,n,j,i,l,m (x) j,i,l

l=0

λ(Ik,n )

 n−1  n−1

 Mj α j =0 i=j

Mn ⎞p(x)

 j,i,l

Ik,n

|f | dt ⎠

dx

9.5 The Maximal Operator Vα,s

501

Again, let q(x) =: p(x)/p0 > 1 for some 1 < p0 < p− . First we use convexity to conclude that ⎛ 

 mj −1  1 M n−1  n−1

n −1  Mj α Mj s  ⎝sup I ≤ Cp(·) χIk,n (x) Mn Mi 0 n∈N j =0 i=j

k=0



χIk,n,j,i,l,1 (x)

q(x)



l=0

⎞ p0

⎠ dx |f (t)| dt j,i,l j,i,l Ik,n λ(Ik,n ) ⎛ 

 mj −1  1 M n−1

n−1  n −1  Mj α Mj s  ⎝ χIk,n (x) ≤ Cp(·) sup Mn Mi 0 n∈N j =0 i=j

k=0



χIk,n,j,i,l,1 (x)

q+ (I j,i,l )



j,i,l λ(Ik,n )

l=0

k,n

|f (t)| dt

j,i,l Ik,n

⎞p0 ⎟ ⎠

dx

and then, according to Lemma 9.3.1,  I ≤ Cp(·)

⎛ 1

⎝t sup

0

M n −1

χIk,n (x)

n∈N k=0

χIk,n,j,i,l,1 (x)



 mj −1 n−1

n−1   Mj α Mj s  j =0 i=j

Mi

(9.25)

l=0

 p0



|f (t)| j,i,l j,i,l Ik,n λ(Ik,n ) p0 ≤ Cp(·) Vα,s (|f |q(·) ) p0 ≤ Cp(·) |f |q(·)

Mn

q(t )

dt

dx

p0

p0

≤ Cp(·) . We obtain for some 0 < α0 < α and 0 < r < s + α0 that  I I ≤ Cp(·) 0

⎛ 1

⎝sup

M n −1

n∈N k=0

χIk,n,j,i,l,2 (x) j,i,l

λ(Ik,n )

⎛ 

 mj −1 n−1

n−1   Mj α Mj s  χIk,n (x) ⎝ Mn Mi j =0 i=j

l=0

q(x) ⎞p0

 j,i,l

Ik,n

|f (t)| dt

⎠

dx

502

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces

⎛



1

≤ Cp(·)

⎝sup

0

 l=0

1

≤ Cp(·)

χIk,n (x)

n∈N k=0

mj −1 



M n −1

Mj Mi

 sup

0

r



 n−1  n−1

 Mj α−α0 Mj α0 +s−r j =0 i=j

χIk,n,j,i,l,2 (x) j,i,l

j,i,l

λ(Ik,n ) M n −1



Mn

Mi

q(x)⎞p0 ⎠ dx |f (t)| dt

Ik,n

χIk,n (x)

n∈N k=0

mj −1



  n−1

n−1   Mj α−α0 Mj α0 +s−r  Mj rq(x) j =0 i=j



Mn

Mi

χIk,n,j,i,l,2 (x)

l=0



j,i,l λ(Ik,n )

Mi

q(x)/q−(I j,i,l ) k,n

j,i,l

|f (t)|q− (Ik,n ) dt

j,i,l Ik,n

⎞ p0 ⎟ ⎠

dx

and, hence, 

1

I I ≤ Cp(·)

 sup

0

M n −1

χIk,n (x)

n∈N k=0

mj −1



  n−1

n−1   Mj α−α0 Mj α0 +s−r  Mj rq(x) j =0 i=j

Mn

j,i,l q(x)/q− (Ik,n )

Mi 

Mi

j,i,l Ik,n

|f (t)|

l=0

j,i,l q− (Ik,n )

χIk,n,j,i,l,2 (x)

Mi

q(x)/q−(I j,i,l ) k,n

dt

⎞p0 ⎟ ⎠

Since  j,i,l Ik,n

|f (t)|

j,i,l

q− (Ik,n )

 dt ≤

j Ik,n

|f (t)|p(t ) dt ≤

1 , 2

dx.

9.5 The Maximal Operator Vα,s

503

we can conclude that 

1

I I ≤ Cp(·)

 sup

0

M n −1

χIk,n (x)

n∈N k=0



  mj −1

n−1  n−1

 Mj α−α0 Mj α0 +s−r  Mj rq(x) Mn

j =0 i=j

Mi

Mi

l=0

j,i,l

q(x)/q−(Ik,n )−1 Mi χIk,n,j,i,l,2 (x)



1 j,i,l

λ(Ik,n )

j,i,l

|f (t)|

 p0

j,i,l

q− (Ik,n )

dt

dx.

Ik,n

Therefore, similarly to the proof of Theorem 9.4.3, we get that ⎛



1

I I ≤ Cp(·)

⎝ sup

0 mj −1

 l=0

1

≤ Cp(·) 0 mj −1



χIk,n (x)

n∈N k=0 

Mj Mi





 n−1  n−1

 Mj α−α0 Mj α0 +s−r Mn

j =0 i=j 

rq(x)−

q(x) j,i +1 q− (Ik,n )

⎝ sup

M n −1

χIk,n (x)

n∈N k=0

j,i,l l=0 λ(Ik,n )

⎞ p0

1

j,i,l

Ik,n

⎟ |f (t)|q(t ) dt ⎠

dx



 n−1

n−1   Mj α−α0 Mj α0 +s−r+r0 j =0 i=j

Mn

Mi

⎞p0



1

Mi



j,i,l λ(Ik,n )

⎛



M n −1

j,i,l Ik,n

|f (t)|q(t ) dt ⎠

dx,

whenever (9.19) holds. Note that r0 was defined just before (9.19). Thus p0 p0 I I ≤ Cp(·) Vα−α0 ,α0 +s−r+r0 (|f |q(·)) ≤ Cp(·) |f |q(·) ≤ Cp(·) . p0

p0

(9.26)

By combining (9.22), (9.24), (9.25) and (9.26) we can conclude that 

1

|Vα,s f (x)|p(x) dx ≤ 2Cp ≤ cp .

0

Since r can be arbitrarily near to s + α0 and α0 to α, this completes the proof.



Remark 9.5.4 Inequality (9.23) and Theorem 9.5.3 hold if p− > max(1/(α+s), 1). The operator Vα,s is not bounded on Lp(·) if (9.23) does not hold.

504

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces

Theorem 9.5.5 Let p(·) ∈ C log with 1 < p− ≤ p+ < ∞ and 0 < α, s < ∞. If 1 1 >α+s − −1 p (I ˙ p− (I0,n +M1 ) + 0,n ) for all n ∈ N, then Vα,s is not bounded on Lp(·) . Proof Choosing j = 0 and i = n − 1, the theorem can be proved in the same way as Theorem 9.4.5 was proved. We omit the details. 

9.6 Variable Martingale Hardy Spaces Let us first recall some definitions. For a martingale f = (fn )n≥0 , dn f = fn − fn−1

(n ≥ 0)

denote the martingale differences, where f−1 = 0. We may suppose also that f0 = 0. For a martingale, we define the square functions Sm and S and the conditional square functions sm and s of f , respectively, as follows: Sm (f ) :=

 m 

1/2 |dn f |

2

,

S(f ) :=

n=0

∞ 

1/2 |dn f |

2

n=0

and  sm (f ) :=

m 

1/2 En−1 |dn f |

2

,

s(f ) :=

n=0

∞ 

1/2 En−1 |dn f |

2

,

n=0

where E−1 := E0 . For an integrable function f , we define the operators in the same way as for the martingale (En f ). s M , HS Definition 9.6.1 The variable martingale Hardy spaces Hp(·) p(·) and Hp(·) associated with the variable Lebesgue spaces Lp(·) are defined by M Hp(·) := {f = (fn )n≥0 : f H M = M(f )Lp(·) < ∞}, p(·)

S Hp(·)

:= {f = (fn )n≥0 : f H S = S(f )Lp(·) < ∞}

s Hp(·)

s := {f = (fn )n≥0 : f Hp(·) = s(f )Lp(·) < ∞}.

p(·)

9.6 Variable Martingale Hardy Spaces

505

If p(·) = p is a constant, then the above definitions of variable Hardy spaces coincide with the classical definitions in [104] and [400]. Theorem 9.3.2 implies immediately the next corollary. M is equivalent Corollary 9.6.2 If p(·) ∈ C log and 1 < p− ≤ p+ < ∞, then Hp(·) to Lp(·) with the inequalities

f Lp(·) ≤ f H M ≤ Cp(·) f Lp(·) . p(·)

Next, we will see that the Lebesgue space Lp(·) is also equivalent to these spaces. M is equivalent Corollary 9.6.3 Let p(·) ∈ C log . If 1 < p− ≤ p+ < ∞, then Hp(·) to Lp(·) with the inequality

f p(·) ≤ f Lp(·) ≤ f H M ≤ Cp(·) f p(·) . p(·)

M ⊂ Lp(·) with the inequality If 1 ≤ p− ≤ p+ < ∞, then Hp(·)

f p(·) ≤ f H M . p(·)

Proof If f ∈ Lp(·) , then f is integrable and so M(f ) = M(F ), where F = (En f ). We have seen in Theorem 9.3.3 that f Hp(·) = F Hp(·) = M(f )p(·) ≤ Cp(·) f p(·) . M , then f ∈ L p(·) ⊂ Lp− , the On the other hand, if f = (fn ) ∈ Hp(·) p(·) . Since L martingale convergence theorem implies that there exists f∞ such that

lim fn = f∞

n→∞

a.e. and in the Lp− -norm

and fn = En (f∞ ). By Fatou’s lemma, f∞ p(·) ≤ (fn )Lp(·) ≤ (fn )H M , p(·)

which proves the first inequality of the corollary. The second inequality of the corollary can be proved as the first part was proved, so the proof is complete.  Note that for an integrable function f , we use the same symbol for the function f and for the martingale (En f ).

506

9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces

9.7 Atomic Decomposition of Variable Hardy Spaces In this Section, we generalize the atomic decomposition for variable Hardy spaces. First of all we introduce more general atoms than in Theorem 5.3.2. The original definition of the p-atom (see (5.3)) is equivalent to the following one: there exists a Vilenkin interval I such that  a dμ = 0, M(a)∞ ≤ μ (I )−1/p , supp (a) ⊂ I. (9.27) I

Now let I ∈ FN be a Vilenkin interval and τ = NχI . Then (9.27) is equivalent to En a = 0

for all n ≤ τ

and −1 M(a)∞ ≤ χI −1 p = χ{τ