Advances in Harmonic Analysis and Partial Differential Equations [1st ed.] 9783030582142, 9783030582159

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Trends in Mathematics

Vladimir Georgiev Tohru Ozawa Michael Ruzhansky Jens Wirth Editors

Advances in Harmonic Analysis and Partial Differential Equations

Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference.

More information about this series at http://www.springer.com/series/4961

Vladimir Georgiev • Tohru Ozawa • Michael Ruzhansky • Jens Wirth Editors

Advances in Harmonic Analysis and Partial Differential Equations

Editors Vladimir Georgiev Department of Mathematics University of Pisa Pisa, Italy

Tohru Ozawa Department of Applied Physics Waseda University Tokyo, Japan

Michael Ruzhansky Department of Mathematics Ghent University Gent, Belgium

Jens Wirth Department of Mathematics University of Stuttgart Stuttgart, Germany

ISSN 2297-0215 ISSN 2297-024X (electronic) Trends in Mathematics ISBN 978-3-030-58214-2 ISBN 978-3-030-58215-9 (eBook) https://doi.org/10.1007/978-3-030-58215-9 Mathematics Subject Classification: 35-XX, 42Bxx, 42Cxx, 43Axx, 47Gxx, 35Q55, 35L70, 42B37 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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. 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

Harmonic analysis and the analysis of partial differential equations are two active fields of mathematical research with many close interactions and emerging new ideas. The aim of this volume is to bring together a diverse range of current research topics, providing insights in novel approaches and highlighting some of the inspiring connections between seemingly different areas of pure and applied mathematics. We also hope to set new impulses for future interactions. Most of the contributions in this volume are related to talks given and results presented at the 12th ISAAC Congress in Aveiro in July/August 2019. The special session Harmonic Analysis and PDEs attracted 43 participants from 15 countries. Pisa, Italy Tokyo, Japan Gent, Belgium Stuttgart, Germany June 2020

Vladimir Georgiev Tohru Ozawa Michael Ruzhansky Jens Wirth

v

Contents

Local Smoothing of Fourier Integral Operators and Hermite Functions . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Ramesh Manna and P. K. Ratnakumar On (λ, μ)-Classes on the Engel Group .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Marianna Chatzakou

1 37

Gelfand Triples for the Kohn–Nirenberg Quantization on Homogeneous Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Jonas Brinker and Jens Wirth

51

A Multiplicity Result for a Non-Homogeneous Subelliptic Problem with Sobolev Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Annunziata Loiudice

99

The Dixmier Trace and the Noncommutative Residue for Multipliers on Compact Manifolds . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 121 Duván Cardona and César Del Corral On the Focusing Energy-Critical 3D Quintic Inhomogeneous NLS . . . . . . . . 165 Yonggeun Cho and Kiyeon Lee Lifespan of Solutions to Nonlinear Schrödinger Equations with General Homogeneous Nonlinearity of the Critical Order .. . . . . . . . . . . . . . . . . 197 Hayato Miyazaki and Motohiro Sobajima Spectral Theory for Magnetic Schrödinger Operators in Exterior Domains with Exploding and Oscillating Long-Range Potentials . . . . . . . . . . 209 Kiyoshi Mochizuki Simple Proof of the Estimate of Solutions to Schrödinger Equations with Linear and Sub-linear Potentials in Modulation Spaces .. . . . . . . . . . . . . . 245 Keiichi Kato

vii

viii

Contents

Remark on Asymptotic Order for the Energy Critical Nonlinear Damped Wave Equation to the Linear Heat Equation via the Strichartz Estimates .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 253 Takahisa Inui On Uniqueness for the Generalized Choquard Equation.. . . . . . . . . . . . . . . . . . . 263 Vladimir Georgiev and George Venkov Characterization of the Ground State to the Intercritical NLS with a Linear Potential by the Virial Functional . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 279 Masaru Hamano and Masahiro Ikeda Well-Posedness for a Generalized Klein-Gordon-Schrödinger Equations . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 309 Jishan Fan and Tohru Ozawa

Contributors

Jonas Brinker Instut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Stuttgart, Germany Duván Cardona Ghent University, Department of Analysis, Logic and Discrete Mathematics, Ghent, Belgium Marianna Chatzakou Imperial College London, London, UK César Del Corral Universidad de Los Andes, Department of Mathematics, Bogotá, Colombia Jishan Fan Department of Applied Mathematics, Nanjing Forestry University, Nanjing, P.R. China Vladimir Georgiev Dipartimento di Matematica, Università di Pisa, Pisa, Italy Faculty of Science and Engineering, Waseda University, Tokyo, Japan Institute of Mathematics and Informatics at BAS, Sofia, Bulgaria Masaru Hamano Department of Mathematics, Graduate School of Science and Engineering, Saitama University, Saitama, Japan Takahisa Inui Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, Japan Masahiro Ikeda Center for Advanced Intelligence Project, RIKEN, Saitama, Japan Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan Keiichi Kato Department of Mathematics, Tokyo University of Science, Tokyo, Japan Annunziata Loiudice Department of Mathematics, University of Bari, Bari, Italy Ramesh Manna Harish-Chandra Research Institute, Jhusi, Allahabad, India

ix

x

Contributors

Hayato Miyazaki Teacher Training Courses, Faculty of Education, Kagawa University, Takamatsu, Kagawa, Japan Kiyoshi Mochizuki Emeritus, Department of Mathematics, Chuo University, Kasuga, Tokyo, Japan Tohru Ozawa Department of Applied Physics, Waseda University, Tokyo, Japan P. K. Ratnakumar Harish-Chandra Research Institute, HBNI, Jhunsi, Allahabad, India Motohiro Sobajima Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda-shi, Chiba, Japan George Venkov Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Sofia, Bulgaria Jens Wirth Instut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Stuttgart, Germany

Local Smoothing of Fourier Integral Operators and Hermite Functions Ramesh Manna and P. K. Ratnakumar

(Dedicated to Prof. S. Thangavelu)

Abstract We prove local smoothing estimates for the Fourier integral operators of the form  Ff (x, t) = ei(x·ξ +t |ξ |) a(x, t, ξ ) fˆ(ξ ) dξ R2

for a wide class of symbols a ∈ S m (R2 × R × R2 ), m ≤ 0. Our result generalises the local smoothing estimate of Mockenhaupt, Seeger and Sogge, and is a global result with respect to the space variable. The novelty in our approach is the use of harmonic analysis of Hermite functions in the study of Fourier integral operators. Keywords Fourier integral operator · Wave front set · Local smoothing · Littlewood-Paley square function · Hermite expansion

1 Introduction A Fourier integral operator is an integral operator of the form 

 Ff (x) =

ξ ∈RM

eiφ(x,y,ξ ) a(x, y, ξ ) f (y) dy dξ, y∈Rn

f ∈ S(Rn )

(1)

R. Manna Harish-Chandra Research Institute, Jhunsi, Allahabad, India e-mail: [email protected] P. K. Ratnakumar () Harish-Chandra Research Institute, HBNI, Jhunsi, Allahabad, India e-mail: [email protected] © Springer Nature Switzerland AG 2020 V. Georgiev et al. (eds.), Advances in Harmonic Analysis and Partial Differential Equations, Trends in Mathematics, https://doi.org/10.1007/978-3-030-58215-9_1

1

2

R. Manna and P. K. Ratnakumar

x ∈ RN , where the phase function φ is real valued and positively homogeneous of degree 1 with respect to the ξ variable and smooth for ξ = 0. The amplitude function a ∈ S m (RN ×Rn ×RM ), the symbol class of order m ∈ R: i.e. a is a smooth function β on RN × Rn × RM satisfying the estimate |∂x,y ∂ξα a(x, y, ξ )| ≤ Cα,β (1 + |ξ |)m−|α| for some constant Cα,β , for all multi-indices α, β. The basic theory of Fourier integral operators was developed by Hörmander in [13] in 1971, soon after the work of Eskin [9], who has actually studied such operators as degenerate elliptic pseudo differential operators. Hörmander established the local L2 regularity estimate for such operators in terms of certain geometric conditions on the phase function, when the amplitude function a ∈ S 0 . The global L2 regularity estimate has been obtained by K. Asada and D. Fujiwara under certain assumptions on amplitude and phase functions, see [1]. This result has been extended to a larger class of Fourier integral operators by Ruzhansky and Sugimoto in [20]. The local Lp regularity estimates has been proved by Seeger, Sogge and Stein [21] for Fourier integral operators with amplitude function a ∈ S m for the range m ≤ −(n − 1)|1/p − 1/2|, 1 < p < ∞. When p = 2 this shows that a can be in S 0 . Also letting p tend to 1 or ∞ in the above inequality, we see that the local Lp boundedness for 1 < p < ∞, requires that a ∈ S m with − n−1 2 < m ≤ 0, as S μ ⊂ S ν if μ < ν. The above range of p is also optimal in view of the result of Peral [16] and Miyachi [17] on wave equation, which corresponds to the phase function φ(x, t, ξ ) = x · ξ + t|ξ |, x, ξ ∈ Rn , t > 0 and a ≡ 1. The global Lp estimate has been obtained by Coriasco and Ruzhansky, under additional assumptions, see [5]. In the context of wave equation, C. D. Sogge observed that averaging over t ∈ [1, 2], results in a gain of regularity in Lp for 2 < p < ∞, see [23]. In fact, Sogge showed that there is an (p) > 0 such that the following estimate 

2



t =1 R2

σ 2

|(I − ) Ff (x, t)| dxdt p

 p1

≤ cσ,p f Lp (R2 )

(2)

  holds for all σ ∈ C with Re(σ ) < p1 − 12 + (p), 2 < p < ∞. Compared to the result of [16, 17, 21] this amounts to a gain in regularity for Ff by (p). The estimate (2) is called the local smoothing estimate of order (p). Local smoothing estimates were previously obtained for Schrödinger equation in [22], see also [29],[6]. Mockenhaupt, Seeger and Sogge later gave an estimate for (p) for the Fourier integral operators with amplitude function of the form a(x, t, ξ ) = ρ1 (x, t) a1 (ξ ) 1 with ρ1 ∈ Cc∞ (R2 × R). They showed that (p) ≤ 2p for p ≥ 4 and (p) ≤ 12 ( 12 − < p ≤ 4, see [18]. It is conjectured that (p) = p1 for p ≥ 4, see [23]. They have also extended their results to manifolds in [19]. Recently some improvement on regularity has been obtained by Lee and Vargas [15], as a consequence of their study on cone multipliers associated to wave equation in R3 . For more on cone 1 p ) for 2

Local Smoothing of Fourier Integral Operators and Hermite Functions

3

multipliers and its connection with the local smoothing estimates, we refer to Wolff [30] and the references therein. The main result in this paper concerns the local smoothing estimate for Fourier integral operators of the form  Gf (x, t) =

R2

ei(x·ξ +t |ξ |) a(x, t, ξ ) fˆ(ξ ) dξ, f ∈ S(R2 )

(3)

with a general amplitude function a ∈ S m (R2 × R × R2 ), m ≤ 0, depending on (x, t) and ξ variables with some mild decay assumption with respect to the (x, t) variables. Here fˆ denotes the Fourier transform of f . Since we are interested in the local (in t-variable) smoothing estimate, it is enough to work with the Fourier integral operators of the form  Ff (x, t) = ρ1 (t)

R2

ei(x·ξ +t |ξ |) a(x, t, ξ ) fˆ(ξ ) dξ, f ∈ S(R2 )

(4)

for ρ1 ∈ Cc∞ (R). The local smoothing estimate for Gf then follows from the estimate for Ff, by a suitable choice of 0 ≤ ρ1 ∈ Cc∞ (R). The local smoothing estimate is given in terms of the fractional Lp Sobolev p spaces. Recall that the Lp Sobolev space of order α ≥ 0 is defined by Lα (Rn ) = −α/2 p n p n (− +I ) L (R ), the Sobolev space of L functions on R with ‘α derivatives’ p in Lp , see [25]. Lα becomes a Banach space with norm f Lpα = (− +I )α/2 f p . p Note that Lα is also defined for complex α, and are spaces of tempered distributions when Re(α) < 0. k,p We also need to consider the Hermite Sobolev spaces WH (R3 ), k ≥ 0, 1 ≤ 2 2 p ≤ ∞. Let H = − x,t + |x| + t denote the Hermite operator on R3 , where k,p

x,t is the Laplacian in the (x, t) variable. Then WH (R3 ) consists of tempered distributions u on R3 , for which H k uLp (R3 ) < ∞. In fact the only assumption we make on the amplitude function a(x, t, ξ ) ∈ S m , m ≤ 0 is that it satisfies the following Hermite-Sobolev estimate: ∂ξα a(·, ·, ξ )W 4,∞ (R2 ×R) := H 4 ∂ξα aL∞ (R2 ×R) ≤ Cα (1 + |ξ |)m−|α|

(5)

H

for all multi indices α, with a constant Cα independent of ξ . Our main result is the following local smoothing estimate, which extends the result obtained in [18] to much more general class of amplitude functions a(x, t, ξ ), in particular leading to global result with respect to the space variable. Theorem 1.1 Let Gf be the Fourier integral operator given by (3) with amplitude function a ∈ S m , m ≤ 0 satisfying (5). Then for any compact t-interval I , Gf satisfies the estimate: Gf Lp (R2 ×I ) ≤ Cσ f Lp

2 m−σ (R )

(6)

4

R. Manna and P. K. Ratnakumar p

for all f ∈ Lm−σ (R2 ) with a constant Cσ depending on the length of I , where  Re(σ ) < 12 ( p1 − 12 ), Re(σ )
0 (1 + |n|)3+1/4 2j/2 f L∞ (R2 ) (1 + |n|)3+1/4

(12) (13)

6

R. Manna and P. K. Ratnakumar

for all j ∈ N, n ∈ N30 , with constants C, C independent of j and n. The estimate (13) follows by standard arguments expressing Fnj as a convolution operator with an L1 kernel, see Proposition 3.1 in Sect. 3. The estimate (12) is subtle and requires more sophisticated arguments. First, by a wavefront set analysis, we can identify the region where Fnj f has rapid decay. In fact, by Remark 1.1, we see that the wave front set of the distributions Fnj f, j ∈ N given by (10) is actually contained in the conic set C = {(x, t, ξ, |ξ |) : |x| = t, ξ ∈ R2  {0}}.

(14)

Note that each Fnj , j ≥ 1, n ∈ N30 are integral operators with a distribution kernel having singularities “along the direction” τ = |ξ | in the frequency domain. So it is natural to split the kernel localising around the wave front set and away from it and analyze separately. This leads to the two operators Qδ and Rδ defined as follows. Choose an even function ψ ∈ Cc∞ (−2, 2) such that ψ = 1 in [−1, 1]. This gives a cutoff function ψ δ supported near the cone |ξ | = τ in R3 by  ψ δ (ξ, τ ) = ψ

|ξ | − τ δ

 , (ξ, τ ) ∈ R2 × R, δ ≥ 1.

(15)

Let Qδ and Rδ denote the multiplier operators on R 3 with multipliers ψ δ and 1 −ψ δ respectively: n δ n  Q δ (Fj f )(ξ, τ ) = ψ (ξ, τ ) Fj f (ξ, τ ), n δ n  R δ (Fj f )(ξ, τ ) = [1 − ψ (ξ, τ )] Fj f (ξ, τ ).

(16)

Since Fnj f = Qδ (Fnj f ) + Rδ (Fnj f ), the Lp estimate for Fnj f follows from the corresponding estimates for Qδ (Fnj f ) and Rδ (Fnj f ). We require these estimates with appropriate decay in n ∈ N30 . The estimate for Rδ (Fnj f ) is easy and follows via standard kernel estimate, see Proposition 4.1 in Sect. 4. The estimate for Qδ (Fnj f ) is more delicate and uses Littlewood–Paley type argument as in [18]. Here we give a more direct proof based on a duality argument and a refined estimate for square function based purely on angular decomposition.

3 Some Auxiliary Estimates In this section we prove the estimates for various oscillatory integrals that we require. We start with showing that the Fourier integral operators Ff with compactly supported amplitude functions are infinitely smoothing. For ρ1 ∈ Cc∞ (I ) and

Local Smoothing of Fourier Integral Operators and Hermite Functions

7

b ∈ Cc∞ (R2 ) set  F0 f (x, t) = ρ1 (t)

R2

ei(x·ξ +t |ξ |) b(ξ ) fˆ(ξ ) dξ, f ∈ S(R2 ).

(17)

Theorem 3.1 Let F0 f (x, t) be as in (17), with supp b ⊂ {ξ ∈ R2 : |ξ | ≤ 2}. Then the operator f → F0 f (·, t) is a smoothing operator. In fact, for each σ ∈ C the estimate (I − x )σ F0 f Lp (R2 ×R) ≤ Cσ f Lp (R2 ) , 1 < p < ∞

(18)

holds for all f ∈ Lp (R2 ) with Cσ = sup|α|≤2 ∂ α b(ξ )∞ . Proof For each t ∈ R and σ ∈ C, the operator Ttσ : f (x) → (I − x )σ F0 f (·, t) is a multiplier operator on L2 (R2 ) with multiplier function Mtσ (ξ ) = ρ1 (t) eit |ξ | (1 + |ξ |2 )σ b(ξ ), ξ ∈ R2

(19)

as b ∈ Cc∞ (R2 ). A simple computation shows that |ξ ||α| |∂ξα Mtσ (ξ )| ≤ C ρ1 (t) sup ∂ α b(ξ )∞ |α|≤2

(20)

for |α| ≤ 2, with a constant C independent of t. Hence by the Mihlin multiplier theorem (see [2]), applied to Ttσ for each t, we get the estimate (I − x )σ F0 f (·, t)Lp (R2 ) ≤ C ρ1 (t) sup ∂ α b(ξ )∞ f Lp (R2 ) , |α|≤2

(21)

for 1 < p < ∞. A further integration over t ∈ R after taking pth power will give the required estimate.

Next we discuss a further decomposition of the operators Fnj in terms of the angular variable as in [10, 21], which is useful to deal with Qδ Fnj . For fixed j ≥ 1, let N = N(j ) be the largest integer less than or equal to 2j/2 , so that 2j/2 − 1 < N ≤ 2j/2 . We now choose N equally spaced points ξ0 , ξ1 , . . . , ξN−1 on the unit circle S 1 = {ξ ∈ R2 : |ξ | = 1} with ξ0 = e1 := (1, 0). In fact, we take ξν = Oξ0 for 1 ≤ ν ≤ N − 1, where O is the rotation in the counter clockwise direction by an angle 2πν/N. 2 With N = N(j ) as above, let {χν }N−1 ν=0 be a partition of unity on R  {0} with respect to the angular variable with the following properties: χν (ξ ) = χ0 (O −1 ξ ), 1 ≤ ν ≤ N − 1

(22)

8

R. Manna and P. K. Ratnakumar

where O is the rotation by an angle 2πν/N, and jk

|∂ξk1 χ0 (ξ )| ≤ Ck , |∂ξk2 χ0 (ξ )| ≤ Ck N k ≈ Ck 2 2 for |ξ | = 1

(23)

for all k ∈ N, with a constant Ck independent of ν (hence independent of j ). Here and elsewhere, by A(s) ≈ B(s) we mean there exist positive constants c1 , c2 such that c1 A(s) ≤ B(s) ≤ c2 A(s). Note that the functions χν are homogeneous functions of degree zero on R2 {0}. An explicit construction of such a partition of unity is carried out in Appendix. Using the homogeneous partitions of unity {χν }ν , we define the operators  Fnj,ν f (x, t) = ρ1 (t) ei(x·ξ +t |ξ |) ρ0 (2−j |ξ |) an (ξ ) χν (ξ ) fˆ(ξ ) dξ (24) R2

for 0 ≤ ν ≤ N − 1, j ∈ N and n ∈ N30 . Note that Fnj f = ˜ n given by need to consider the Fourier integral operators F

N−1 ν=0

Fnj,ν f . We also

j,ν

˜ n f (x, t) = ρ1 (t) F j,ν

 R2

ei(x·ξ +t |ξ |) ρ(2−j |ξ |) an (ξ ) χν (ξ ) fˆ(ξ ) dξ, j ∈ N (25)

where ρ is a smooth non negative function such that ρ0 = ρ 2 . We have  n n ˜ Fj,ν f (x, t) = (x − y, t) f (y) dy K˜ j,ν y∈R2

(26)

where an n (x, t) := K˜ j,ν (x, t) = ρ1 (t) K˜ j,ν



ei(x·ξ +t |ξ |) ρ(2−j |ξ |) an (ξ ) χν (ξ ) dξ.

(27)

ξ

The following kernel estimate is a refinement of the estimate obtained in [21], and is crucial in our argument for dealing with general amplitude functions depending on (x, t). Let {an (ξ )}n∈N3 be a family of symbols satisfying 0

|∂ α an (ξ )| ≤ Cα

(1 + |ξ |)−|α| , (1 + |n|)3+1/4

(28)

for all multi-indices α ∈ N20 , n ∈ N30 with constant Cα independent of n. Lemma 3.1 Let {an (ξ )}n∈N3 be a family of symbols in S 0 (R2 ) satisfying (28). Then 0 the Kernels K˜ n given by (27) satisfy the estimates j,ν

n |K˜ j,ν (x, t)| ≤

C 23j/2 ρ1 (t) j (T x + te1 ), (1 + |n|)3+1/4

(29)

Local Smoothing of Fourier Integral Operators and Hermite Functions

9

−k

−k 1 + 2j |x2 |2 where j (x) := j,k (x) = 1 + 22j |x1 |2 , k ≥ 1, j ∈ N and the constant C = Ck is independent of j, ν and n. Here T ∈ SO(2) is such that T ξν = e1 , 0 ≤ ν ≤ N − 1. n (x, t) by oscillatory Proof We first consider the case ξν = ξ0 = e1 and estimate K˜ j,0 integral techniques. By (27) n (x, t) = ρ1 (t) K˜ j,0

 R2

ei(x·ξ +t |ξ |) ρ(2−j |ξ |) an (ξ ) χ0 (ξ ) dξ.

(30)

   Consider the differential operator L = I − 22j ∂ξ21 I − 2j ∂ξ22 as in [21], so that k  k  1 + 2j |x2 |2 ei(x+t e1).ξ , k ∈ N. Lk ei(x+t e1)·ξ = 1 + 22j |x1 + t|2

(31)

Re writing ei(x·ξ +t |ξ |) as eit (|ξ |−ξ1 ) ei(x+t e1)·ξ and using the above formula, we get  −k  −k 1 + 2j |x2 |2 eit (|ξ |−ξ1 ) Lk ei(x+t e1)·ξ . ei(x·ξ +t |ξ |) = 1 + 22j |x1 + t|2 (32) Using this formula in (30) and an integration by parts shows that n (x, t) = j (x + te1 ) Aaj n (x, t) K˜ j,0

(33)

−k

−k 1 + 2j |x2 |2 and where j (x) = 1 + 22j |x1 |2  Aaj (x, t)

=

R2

  ei[x·ξ +t ξ1] Lk eit (|ξ |−ξ1 ) ρ(2−j |ξ |) a(ξ ) χ0 (ξ ) dξ

(34)

for a ∈ S 0 (R2 ). Note that the integrand in (34) is supported on the set E = supp χ0 ∩ {ξ : 2j −1 ≤ |ξ | ≤ 2j +1 }.

(35)

To complete the proof for ν = 0, we need to show that |Aaj n (x, t)| ≤

Ck 23j/2 . (1 + |n|)3+1/4

(36)

This will follow from (34) with a = an , once we verify the following; • The measure of E is bounded by a constant times 23j/2

k it (|ξ |−ξ ) −j 1 • L e ρ(2 |ξ |) an (ξ ) χ0 (ξ ) ≤ Ck /(1 + |n|)3+1/4 for some constant Ck independent of j and n.

10

R. Manna and P. K. Ratnakumar

The first statement is clear. In fact for all ξ ∈ E, we have |ξ2 | ≤ ξ1 sin(2π/N)  2j/2 as ξ1 ≤ 2j +1 . To verify the second statement, we observe that Lk is a linear combination of various derivatives (22j ∂ξ21 )k1 (2j ∂ξ22 )k2 with k1 + k2 ≤ 2k. Note that each of the above derivative of the functions ρ(2−j |ξ |) are uniformly bounded in j . In view of (23) and the fact that χ0 (ξ ) is homogeneous of degree zero, the above derivatives of χ0 (ξ ) are uniformly bounded in j ∈ N. Also in view of (28), the above derivatives falling on an gives the bound Ck /(1 + |n|)3+1/4 , with Ck independent of j and n. All the above derivatives applied to eit (|ξ |−ξ1 ) also give functions bounded uniformly in j on E, in view of Lemma 3.2 below. n for general ν, first note that χ (ξ ) = χ (O −1 ξ ) by (22) where To estimate K˜ j,ν ν 0 O ∈ SO(2) is such that ξν = Oe1 . Thus using the change of variable ξ → Oξ anν an n in (27), we see that K˜ j,ν (x, t) := K˜ j,ν = K˜ j,0 (O −1 x, t) where anν (ξ ) = an (Oξ ). Thus in view of (33) and (34) we have ν

a n K˜ j,ν (x, t) = j (O −1 x + te1 ) Aj n (O −1 x, t).

(37)

aν

n Notice that the estimate for |Aj,0 (y, t)| depends on the derivatives of anν = an ◦ O, and these derivatives have the bound twice as that of an given by (28). Hence the proof follows with T = O −1 , once we prove the following lemma.



Lemma 3.2 Let h(ξ ) = |ξ | − ξ1 for ξ = (ξ1 , ξ2 ) ∈ R2 , then we have kj

|∂ξk1 h(ξ )| ≤ Ak 2−kj , |∂ξk2 h(ξ )| ≤ Bk 2− 2 , for k ∈ N,

(38)

on the set E = supp χ0 ∩ {ξ = (ξ1 , ξ2 ) : 2j −1 ≤ |ξ | ≤ 2j +1 }, j ≥ 1. Proof This lemma has already been proved in [26], for more general function h(ξ ) = (x, ξ ) − ξ (x, ξ¯ ) · ξ corresponding to a general phase function . In our special case, we give a proof based on geometric reasoning. Writing ξ = |ξ |(cos θ, sin θ ), we see that ∂ξ1 h(ξ ) = |ξξ1| − 1 = −2 sin2 (θ/2). Since | sin θ | ≤ |θ | and |θ | ≤ 2π/N on the support of χ0 , we have |∂ξ1 h(ξ )| ≤

2π 2 2π 2 ≤ , ξ ∈E N2 2j

(39)

as N ≥ 2j/2 , which proves the case k = 1. To deal with the case k > 1, we write ∂ξk1 h(ξ ) = ∂ξk−1 g(ξ ), where g = ∂ξ1 h which is a function homogeneous of degree 1 zero on R2 , hence ∂ξk−1 g is homogeneous of degree 1 − k. It follows that 1 ∂ξk1 h(ξ ) = |ξ |1−k (∂ξk−1 g)(ξ/|ξ |). 1

(40)

Now note that g(ξ ) = −2 sin2 (θ/2) := g(θ ˜ ) as computed above and also ∂ξ1 = − sin θ ∂θ on functions homogeneous of degree zero. An easy induction argument shows that (− sin θ ∂θ )k−1 g(θ ˜ ) = Pk (cos θ ) sin2 θ where Pk is a polynomial of

Local Smoothing of Fourier Integral Operators and Hermite Functions

11

degree k − 1. Now for ξ = (r cos θ, r sin θ ) ∈ E, we have |θ | ≤ 2π/N and hence |Pk (cos θ ) sin2 θ | ≤ ck | sin2 θ | ≤ ck 4π 2 N −2 ≈ Ck 2−j for some constant Ck independent of j . It follows from (40) that, for k > 1     k ∂ξ1 h(ξ ) ≤ Ck 2−kj

(41)

as |ξ | ≈ 2j on E. Since ∂ξ2 h(ξ ) = |ξξ2| = sin θ , the required inequality follows as above on E, for k = 1. For k ≥ 2, note that ∂ξk2 h(ξ ) = ∂ξk2 |ξ |. Since the function g1 (ξ ) = |ξ | is homogeneous of degree 1, these derivatives are homogeneous functions of degree 1 − k. It follows that |∂ξk2 h(ξ )| ≤ Ck |ξ |1−k ≤ Ck |ξ |−k/2 on E, for k ≥ 2 and hence the required inequality holds on E.

n can also be expressed in Remark 3.1 We note that the estimate for the kernel K˜ j,ν the form. n (x, t)| ≤ |K˜ j,ν

 −k  −k C23j/2 2j 2 j ⊥ 2 1 + 2 |x, ξ  + t| 1 + 2 |x, ξ | ν ν (1 + |n|)3+1/4 (42)

where ξν is as in Lemma 3.1 and ξν⊥ is a unit vector perpendicular to ξν . In fact, n the function j (x) defining K˜ j,ν in Lemma 3.1 is a function of 2j |x1| and 2j/2 |x2 |. Now for T ∈ SO(2), (T x)1 := T x, e1  = x, ξν  if T ξν = e1 . Similarly (T x)2 = x, ξν⊥ , where T ξν⊥ = e2 . It follows that −k  −k  1 + 2j |x, ξν⊥ |2 . j (T x + te1 ) = 1 + 22j |x, ξν  + t|2

(43)

Hence (42) follows from Lemma 3.1 and the fact that |ρ1 (t)| ≤ 1. Now we prove the L∞ estimate (13) for Fnj mentioned in Sect. 2. Proposition 3.1 Let Fnj be the operator given by (10) with amplitude function an as in (28). Then Fnj satisfies the inequality Fnj f L∞ (R3 ) ≤

C 2j/2 f L∞ (R2 ) (1 + |n|)3+1/4

(44)

with a constant C independent of j ∈ N and n ∈ N30 . n n Proof We have Fnj = N−1 ν=0 Fj,ν , where Fj,ν is the operator given by (24) which is a convolution operator (in the x-variable, for each t) with kernel  n (x, t) = ρ1 (t) Kj,ν

R2

ei(x·ξ +t |ξ |) ρ0 (2−j |ξ |) an (ξ ) χν (ξ ) dξ.

(45)

12

R. Manna and P. K. Ratnakumar

n differs from K ˜ n defined by (27), only in the power of ρ: ρ0 = ρ 2 . Note that Kj,ν j,ν n Hence Kj,ν (·, t) also satisfies the pointwise estimate as in Lemma 3.1. Hence we have n (·, t)L1 (R2 ) ≤ Kj,ν

C1 |ρ1 (t)| (1 + |n|)3+1/4

(46)

with C1 independent of j, ν, n and t. It follows that Fj,ν f (·, t)L∞ (R2 ) ≤

C |ρ1 (t)| f L∞ , (1 + |n|)3+1/4

(47)

for 0 ≤ ν ≤ N − 1. Summing over ν, this gives the required estimate after a t

integration, observing that there are N = N(j ) ≈ 2j/2 terms in the sum.

4 Lp -Estimates for Rδ (Fnj f ) Recall that Rδ was defined in (16) as a multiplier operator on R3 with multiplier 1 − ψ δ (ξ, τ ). The estimate for Rδ (Fnj f ) relies on the rapid decay of its Fourier transform. The following lemma is the key ingredient for the same. Lemma 4.1 For j ∈ N and 1 ≤ δ < 2j/2 consider the set Bjδ = {(ξ, τ ) ∈ R2 × R : 2j −1 < |ξ | ≤ 2j +1 , |τ − |ξ || > δ}.

(48)

Then for each 0 < ≤ 12 , the estimate |τ − |ξ || > Cj, ,δ (|τ | + |ξ |) holds for all (ξ, τ ) ∈ Bjδ with Cj, ,δ =

1 2

  2 3

(49)

δ . 2j

Proof The required inequality clearly holds when τ ≤ 0, as τ − |ξ | = −(|τ | + |ξ |) and δ < 2j/2 . So we assume τ > 0. We write Bjδ = B1 ∪ B2 where B1 = {(ξ, τ ) ∈ Bjδ : τ > 2|ξ |}, B2 = {(ξ, τ ) ∈ Bjδ : τ ≤ 2|ξ |}

(50)

−|ξ || We show that inf(ξ,τ )∈Bi (τ|τ+|ξ |) ≥ Cj, ,δ for i = 1, 2. Since τ > 2|ξ | on B1 , we have

τ − |ξ | 1−θ |τ − |ξ || = = τ 1− (τ + |ξ |) (τ + |ξ |) (1 + θ )

(51)

Local Smoothing of Fourier Integral Operators and Hermite Functions

where θ = (ξ, τ ) ∈ B1 ,

|ξ | τ




|τ − |ξ || 1 > (τ + |ξ |) 2

1 2

and 1 + θ
. 3 2 3 2j

(52)

On the other hand on B2 , we have δ δ δ |τ − |ξ || > ≥ ≥ j 6− (τ + |ξ |) (τ + |ξ |) (3|ξ |) 2 which dominates

δ 1 2j 2

  2 3

(53)

since ≤ 1/2.



Proposition 4.1 For j ∈ N, let Rδ be as in (16) with δ ≥ 1, and Fnj f be as in (10) with amplitude function an satisfying (28). Then for each > 0, there exist a constant C independent of j and n, such that the inequality Rδ (Fnj f )Lp (R2 ×R) ≤

C [Cj, ,δ ]−3/ f Lp , (1 + |n|)3+1/4

(54)

holds for all f ∈ Lp (R2 ), 1 ≤ p < ∞ with Cj, ,δ as in Lemma 4.1. Proof Since f → Rδ (Fnj f ) is a linear map, it is enough to estimate (54) for f ∈ S(R2 ) by density of S(R2 ) in Lp (R2 ), 1 ≤ p < ∞. In view of (16) and (10), we have n δ −j R δ (Fj f )(ξ, τ ) = [1 − ψ (ξ, τ )] fˆ(ξ ) an (ξ ) ρ0 (2 |ξ |) ρˆ1 (τ − |ξ |).

(55)

Thus by the Fourier inversion formula and expanding fˆ we see that  Rδ (Fnj f )(x, t) =  =

R2 ×R

R2

n ei(x·ξ +t τ ) R δ (Fj f )(ξ, τ ) dξ dτ

f (y) Kδj (x

(56) − y, t) dy

where,  Kδj (x, t) =

R2 ×R

ei(x·ξ +t τ ) [1 − ψ δ (ξ, τ )] an (ξ ) ρ0 (2−j |ξ |) ρˆ1 (τ − |ξ |) dξ dτ. (57)

14

R. Manna and P. K. Ratnakumar

Thus in view of Young’s inequality (see [11]), it is enough to prove the estimate Kδj (·, t)L1 (R2 ) 

C (Cj, ,δ )−3/ (1 + |n|)3+1/4 (1 + |t|2 )

(58)

for each t ∈ R, with C independent of j, n and t. Now observe that for all N ∈ N, (1 + |x|2 )N (1 + |t|2 )N ei(x·ξ +t τ ) = (I − ξ )N (I − ∂τ2 )N ei(x·ξ +t τ ).

(59)

Hence an integration by parts in (57) shows that (1 + |x|2 )N (1 + |t|2 )N Kδj (x, t)  = ei(x·ξ +t τ ) (I − ξ )N (I − ∂τ2 )N bj (ξ, τ ) dξ dτ,

(60)

ξ,τ

where bj,n (ξ, τ ) = [1 − ψ δ (ξ, τ )] an (ξ ) ρ0 (2−j |ξ |) ρˆ1 (τ − |ξ |). Note that (I −

ξ )N (I − ∂τ2 )N bj,n (ξ, τ ) is a sum of terms that involves various partial derivatives of order up to 4N, of the functions ψ δ (ξ, τ ), an (ξ ), ρ0 (2−j |ξ |) and ρˆ1 (τ − |ξ |). Each derivative on ψ δ brings in a negative power of δ ≥ 1 and hence all these derivatives are bounded uniformly in δ. In the same way, all the above derivatives acting on ρ0 (2−j |ξ |) are bounded uniformly in j ∈ N. By assumption an and all its 1 derivatives are bounded by a constant times (1+|n|) 3+1/4 . Also, since ρˆ1 is a Schwartz class function we have |∂ α ρˆ1 (y)| ≤ CM,N (1 + |y|)−M for |α| ≤ N, for some constant CM,N . It follows that for each N, M ∈ N, there is a constant CM,N independent of , j, δ and n such that |(I − ξ )N (I − ∂τ2 )N bj (ξ, τ )| ≤ ≤

CM,N (1 + |τ − |ξ ||)−M (1 + |n|)3+1/4 CM,N (1 + Cj, ,δ (τ + |ξ |) )−M (1 + |n|)3+1/4 (61)

for |τ − |ξ || > δ, by Lemma 4.1. Note that the integral in (57) and hence in (60) is actually over the set |τ − |ξ || > δ, as 1 − ψ δ (ξ, τ ) = 0 on |τ − |ξ || ≤ δ, hence Lemma 4.1 is applicable. Using (61) in (60), the right hand side of (60) is bounded by CM,N (1 + |n|)3+1/4

 R2 ×R

[1 + Cj, ,δ (|τ | + |ξ |) ]−M dξ dτ

CM,N = (Cj, ,δ )−3/ (1 + |n|)3+1/4

 R2 ×R

[1 + (|τ | + |ξ |) ]−M dξ dτ (62)

Local Smoothing of Fourier Integral Operators and Hermite Functions

15

by a change of scale in the variables ξ and τ . Choosing M > M := [3/ + 1], the last integral is finite, and (60) translates to the inequality Kδj (x, t) ≤

(Cj, ,δ )−3/ CM( ),N . (1 + |n|)3+1/4 (1 + |x|2 )N (1 + |t|2 )N

(63)

Choosing N = 2, we see that Kj (·, t) ∈ L1 (R2 ) and the estimate (58) holds, which completes the proof.



5 A Square Function Estimate In this section, we prove the Lp estimate for a square function based on angular decomposition. This estimate for 4/3 ≤ p < 2 is new to the best of our knowledge, and the estimate for p = 4/3 is crucial in our proof of L4 estimate for Qδ (Fnj f ). Given j ∈ N and δ ≥ 1, consider the family of Fourier multiplier operators δ }N−1 on R3 given by {Tν,j ν=0 −j δ T ν,j g (ξ, τ ) = ρ(2 |ξ |) χ˜ ν (ξ ) ψ



|ξ | − τ δ

 g(ξ, ˆ τ ), g ∈ S(R3 )

(64)

where ρ is as in (25), χ˜ ν is a homogeneous function (smooth and compactly supported as a function on S 1 ), such that χ˜ ν χν = χν for χν given by (22) and ψ is as in (15). Consider the square function N−1 1 2 2

 δ  S(g)(x, t) = g . Tν,j

(65)

ν=0

For the square function S(g), we establish the following Lp boundedness result. Theorem 5.1 Let Sg be as in (65). Then the inequality SgLp (R3 ) ≤ C 2j/8 j b δ 1/4 gLp (R3 ) , 4/3 ≤ p ≤ 4

(66)

holds for all g ∈ S(R3 ) with constants C and b independent of j ∈ N and δ ≥ 1. Note that the result was previously known for the range 2 ≤ p ≤ 4, see [18]. However, our approach for estimating Qδ (Fnj f ) requires the square function estimate for p = 4/3. In fact, we first establish the case p = 4, which will be used to establish the case 4/3 ≤ p < 4. The proof of the following square function estimate is essentially based on the approach of Cordoba [3]. In fact we use a reduction to a Kakeya type maximal operator as in [18]. Such control by Kakeya type maximal operator was originally used by C. Fefferman in [10], as far as we know.

16

R. Manna and P. K. Ratnakumar

δ be as in (64). Then the following square function estimate Proposition 5.1 Let Tν,j holds   N−1  12  

   δ |Tν,j g|2  ≤ C δ 1/4 j b gL4 (R3 ) (67)     ν=0  4 3 L (R )

for all g ∈ S(R3 ) with constants C and b independent of j . Proof In view of (64) we have  δ g(x, t) = Tν,j

R3

δ (x − y, t − s) g(y, s) dyds, k˜j,ν

(68)

where δ (x, t) = k˜j,ν



   |ξ | − τ dξ dτ ei(x·ξ +t τ ) χ˜ ν (ξ ) ρ(2−j |ξ |) ψ δ R3

(69)

∨

= δψ (δt) Kj,ν (x, t), 

with Kj,ν (x, t) = R2 ei(x·ξ +t |ξ |) χ˜ ν (ξ ) ρ(2−j |ξ |) dξ. Note that Kj,ν is same as K˜ n in (27) with an ≡ 1. Hence by the same argument as in Lemma 3.1, we see j,ν

δ satisfies the pointwise estimate as in (29). It follows that that Kj,ν and hence k˜j,ν

 R2 ×R

δ |k˜j,ν (x, t)| dxdt ≤ C

(70)

with a constant C independent of j and δ. Thus an application of Cauchy–Schwarz inequality in (68) yields  δ g(x, t)|2 |Tν,j

≤C

R3

δ |g(y, s)|2 |k˜j,ν (x − y, t − s)|dy ds,

(71)

δ g = T δ g , where g is given by the with C independent of j and δ. Note that Tν,j ν ν,j ν Fourier transform:

˜ ν (ξ )  g (ξ, s), gν (ξ, s) = 

(72)

˜ ν denoting the characteristic function of the support of χ˜ ν . Thus summing with  over ν in (71) with g replaced by gν , squaring, integrating and taking square root

Local Smoothing of Fourier Integral Operators and Hermite Functions

17

leads to the inequality   N−1  12 2  

  δ |Tν,j g|2      4  ν=0

L (R3 )

⎛ ≤C⎝



 R3

N−1

R3 ν=0

2 δ |gν (y, s)|2 |k˜j,ν (x − y, t − s)| dy ds

⎞1/2 dxdt ⎠

  N−1   

  δ = C sup  |gν (y, s)|2 |k˜j,ν (x − y, t − s)| dy ds h(x, t) dxdt  . 3 3   h 2 =1 R R ν=0

L

(73) δ (x, t) = K (x) δψ ∨ (δt), using the change of variable t → t + s, we see Since k˜j,ν j,ν that  δ |k˜j,ν (x − y, t − s)h(x, t)| dx dt R3

 ≤ sup ν

(74) R3

|Kj,ν (x − y, t)| |hs (x, t)| dx dt

where hs (x, t) = δψ ∨ (δt)h(x, t +s). Thus an interchange of integrals in (73) shows that the right hand side of (73) is at most a constant times       

  sup  |gν (y, s)|2 sup |Kj,ν (x − y, t)| |hs (x, t)| dxdt dy ds  3 3   ν R h 2 =1 R ν

L

⎡ ≤C⎣



 R3

 sup

hL2 =1



2 |gν (y, s)|2

⎤1/2 dy ds ⎦

×

ν

  sup  3 ν

R

R3

1 2 2  |Kj,ν (x − y, t)| |hs (x, t)| dxdt  dy ds (75)

by Cauchy–Schwarz inequality in the variables (y, s). We first show that the second term on the right hand side of (75) is bounded by C δ 1/2 j 3/2 . Note that the kernel Kj,ν satisfies the pointwise estimate (29) as mentioned above and hence

18

R. Manna and P. K. Ratnakumar

by Remark 3.1, Kj,ν also satisfies the estimate (42). Hence we have the maximal inequality 

  sup  2 ν 

R

R3

2  |Kj,ν (x − y, t)| |hs (x, t)| dxdt  dy ≤ Cj 3 hs 2L2 (R3 )

(76)

with C independent of j . This is a restatement of the maximal inequality (1.11) in [18], in view of Lemma 1.4 with δ = 2−j/2 as per the notation in the same paper. Since hs (x, t) = δψ ∨ (δt)h(x, t + s), integrating with respect to the s-variable over R on both sides gives 

  sup  3 ν

2  |Kj,ν (x − y, t)| |hs (x, t)| dxdt  dy ds R3    ∨  δψ (δt) h(x, t + s)2 dxdt ds ≤ C j3

R

s∈R



= C j3 δ

t ∈R

R3

|ψ ∨ (t)|2



 R3

(77)

|h(x, s)|2 dxds dt

≤ C δ j 3 h2L2 (R3 ) for each h ∈ L2 (R3 ). Taking square root and supremum over hL2 (R3 ) = 1 leads to the required estimate for the second term. Also the first term on the right hand side of (75) is bounded by C j 2b1 g2L4 (R3 ) with some constants C and b1 independent ˜ ν in (72) is independent of s, we can of j . In fact, since the multiplier function  appeal to the two dimensional square function estimate of Cordoba [4] for each s to get 

 2  

   |gν (y, s)|2  dy ≤ Cj 4b1 g(·, s)4L4 (R2 )  R2   ν

(78)

with C and b1 independent of s. From this the assertion follows by integration over s ∈ R. It follows that the right hand side of (75) is bounded by δ 1/2 j 3/2+2b1 g2L4 (R3 ) . Using this in (73) and taking square root, the required estimate follows with b = 3/4 + b1 .

Now we prove the square function estimate given by Theorem 5.1. Proof of Theorem 5.1 We use the Rademacher function argument as in Stein [25] (page 106), to reduce the square function estimate to a multiplier problem. Recall that the Rademacher functions {rk }k≥0 are functions on R defined as follows. Let r0 be the periodic function on R with period 1 defined by r0 (s) = χ[0,1/2] (s) − χ(1/2,1) (s), for 0 ≤ s < 1.

(79)

Local Smoothing of Fourier Integral Operators and Hermite Functions

19

k s). The Rademacher functions have the following For k ∈ N, define rk (s) = r0 (2 interesting property: if F (s) = ν aν rν (s) ∈ L2 ([0, 1]) then F ∈ Lp ([0, 1]) for all p ∈ (1, ∞). In fact, we have

c1 F p ≤ F 2 ≤ c2 F p

(80)

for positive constants c1 , c2 depending only on p (and not on the particular function F ), see [25, page 277]. For each s ∈ [0, 1), set P (s, x, t) =

N−1

δ rν (s) Tν,j g(x, t).

(81)

ν=0

By the orthonormality of the Rademacher functions rν , we have 

1

|P (s, x, t)|2 ds = |Sg(x, t)|2 .

(82)

0

Hence in view of (80), we have 

1

|Sg(x, t)| ≤ Cp

1/p |P (s, x, t)|p ds

(83)

0

for 1 < p < ∞, with a constant Cp independent of (x, t) ∈ R3 . It follows that 

p

R3



|Sg(x, t)|p dx dt ≤ Cp

 R3

[0,1)

|P (s, x, t)|p ds dxdt.

(84)

In view of (81) and (64), we have P (s, x, t) = Ts g(x, t), where Ts is the multiplier operator on R3 , defined by T ˜ δ,s g (ξ, τ ) s g(ξ, τ ) = m j (ξ, τ ) with m ˜ δ,s j (ξ, τ ) = 

N−1 ν=0

rν (s) χ˜ ν (ξ ) ρ(2−j |ξ |) ψ p

R3

|Sg(x, t)|p dx dt ≤ c2

 [0,1)



|ξ |−τ δ

(85)  . Thus (84) reads as

 R3

|Ts g(x, t)|p dx dt ds.

(86)

Hence the Lp boundedness of S for 4/3 ≤ p ≤ 4, will follow once we show that the operators Ts are uniformly bounded in Lp for the above range of p. This will follow from the next proposition.



20

R. Manna and P. K. Ratnakumar

Proposition 5.2 Let Ts be the multiplier operator given by (85). Then Ts satisfy the estimate Ts gLp ≤ C 2j/8 δ 1/4 j b gp

(87)

for all g ∈ Lp (R3 ), 4/3 ≤ p ≤ 4, with constants C and b independent of δ, j and s. δ δ Proof We have Ts g = N−1 ν=0 rν (s)Tν,j g, where Tν,j is as in (64). We first prove the estimate for a related operator, acting on vector valued functions. For s ∈ [0, 1], consider the operator T˜s : L2 (R3 : RN ) → L2 (R3 ) given by T˜s (h) =

N−1

δ rν (s) Tν,j (hν ), h = (h0 , h1 , . . . , hN−1 )

(88)

ν=0 δ is as in (64), h ∈ S(R3 ) and r , ν = 0, 1, · · · , N − 1 are Rademacher where Tν,j ν ν functions as above. Then the following estimates hold

 1/2    √  δ  2  T˜s h2 ≤ 5  |T h | ν,j ν    ν  2     1/2 

  δ  . T˜s h∞ ≤ C2j/4  |Tν,j hν |2    ν 

(89)

(90)

∞

The estimate (90) follows from (88) by Cauchy–Schwarz inequality with respect to the sum over ν as |rν | = 1 and the fact that N ≤ 2j/2 . The estimate (89) uses Plancheral theorem. Note that  ˜s h(ξ, τ )|2 = |T

N−1

ν,ν  =0

≤

N−1

 δ δ rν (s)rν  (s) T ν,j hν (ξ, τ ) Tν  ,j hν  (ξ, τ ) (91)

 

  δ h (ξ, τ ) T δ h  (ξ, τ ) . T  ν,j ν  ν  ,j ν

ν=0 |ν−ν  |≤2

This follows from the fact that for any given ν, the support of χν intersects the  δ gT δ g ≡ 0 supports of χ  only for |ν − ν  | ≤ 2 by the choice of χ , and hence T ν

ν

ν,j

ν  ,j

only for |ν − ν  | ≤ 2. It follows that the last expression is a sum of five terms of the form  N−1

   δ h  (ξ, τ ) , with ν − 2 ≤ ν  ≤ ν + 2. T δ hν (ξ, τ ) T  ν,j  ν  ,j ν

ν=0

(92)

Local Smoothing of Fourier Integral Operators and Hermite Functions

21

Setting ν  = ν + l where |l| ≤ 2, by Cauchy–Schwarz inequality we see that N−1



   T δ hν (ξ, τ ) T δ  h (ξ, τ ) ν+l,j ν+l  ν,j 

ν=0

   N−1 2 1/2 N−1 2 1/2



     δ δ ≤ Tν,j hν (ξ, τ ) Tν+l,j hν+l (ξ, τ ) ν=0

(93)

ν=0

δ  Observe from (91) that, when ν = N − 1 the terms T ν  ,j hν that matters are only  those with ν = N − 2, N − 1, 0, 1, 2. Similar interpretation holds for the cases when ν + l ∈ {0, 1, . . . , N − 1}. Hence we see that the second term on the right  2 1/2 N−1    δ hand side of the above inequality is same as . It follows ν=0 Tν,j hν (ξ, τ )

that  N−1

 N−1  2    δ h (ξ, τ ) . T δ hν (ξ, τ ) T δ  T h (ξ, τ ) ≤   ν+l,j ν+l ν,j ν  ν,j 

ν=0

(94)

ν=0

Now using (94) in (91) and integrating both sides yields the estimate (89). Interpolating (89) and (90) (see [12, Theorem 1.19]), we get the estimate T˜s hp ≤ 51/p 2

j 1 1 2(2−p)

 1/2     δ  2   , 2 ≤ p ≤ ∞. |T h | ν,j ν    ν 

(95)

p

Note that for g ∈ S(R3 ), we have Ts (g) = T˜s (h) with h = (g, g, . . . , g), in view of (85) and (88). Hence the inequality (95) with p = 4 gives  1/2   

  j/8  δ 2  Ts g4 ≤ C2  |Tν,j g|   ν 

4

(96)

≤ C2j/8 δ 1/4 j b g4 by Proposition 5.1. Since Ts given by (85) is a multiplier operator, the above estimate holds for the dual index p = 4/3 as well. Hence the required estimate follows by Riesz–Thorin interpolation theorem between these two estimates.



22

R. Manna and P. K. Ratnakumar

6 Lp -Estimates for Fnj f The main difficulty in proving Lp estimate for Fnj f lies in the L4 estimate for Qδ (Fnj f ). We estimate Qδ (Fnj f )4 by duality, which requires the Lp estimate for the square function discussed in the previous section for p = 4/3. We start with the following n

˜ f be as in (16) and (25) respectively. Then Proposition 6.1 Let Qδ (Fnj f ) and F j,ν the inequality

Qδ (Fnj f )L4 (R3 ) ≤ C δ 1/4 j b 2j/8

  N−1  12   n   ˜ f |2  |F   j,ν    ν=0 

(97)

,

L4 (R3 )

holds for all f ∈ S(R2 ) with constants C and b independent of j and n. Proof We use duality to estimate the L4 norm. For H ∈ L4/3 (R3 ), writing N−1 n Qδ (Fj f ) = ν=0 Qδ (Fnj,ν f ), we have  Qδ (Fnj f ), H  =

R3

Qδ (Fnj,ν f )(x, t) H (x, t) dxdt.

(98)

ν

By Parseval’s theorem for the Fourier transform, and in view of (16) we see that 

 R3

Qδ (Fnj,ν f )(x, t) H (x, t) dxdt

=

R3

 =

R3

n   F j,ν f (ξ, τ ) Qδ (H )(ξ, τ ) dξ dτ

(99) ˜ n f )(x, t) T δ H (x, t) dxdt (F j,ν ν,j

n

δ ˜ where F j,ν is as in (25) and Tν,j is the multiplier operator given by (64). Now summing over ν and using Cauchy–Schwarz inequality with respect to sum over ν on the right hand side of (99), followed by an application of Hölder’s inequality yields

 1  

2   ˜ n f |2  | F |Qδ (Fnj f ), H | ≤  j,ν    ν 

4

 1    2 2     δ    Tν,j H     ν 

.

(100)

4/3

By Theorem 5.1, the second term on the right hand side of (100) is bounded by C δ 1/4 j b 2j/8 H 4/3 . Taking supremum over H 4/3 ≤ 1, the required estimate follows.

Next we proceed to estimate the square function appearing in Proposition 6.1, following similar arguments as in the proof of Proposition 5.1.

Local Smoothing of Fourier Integral Operators and Hermite Functions

23

˜ n f be as in (25) with amplitude function an satisfying (28). Proposition 6.2 Let F j,ν Then the square function estimate   N−1  12   n   ˜ f |2  |F   j,ν    ν=0 

≤C

j b+3/4 f L4 (R2 ) (1 + |n|)3+1/4

(101)

L4 (R3 )

holds for all f ∈ S(R2 ) with constants C, b independent of j and n. Proof Let ν denote the characteristic function of the support of χν defined in (22) ˜n f = F ˜ n fν where fν (ξ ) = ν (ξ ) fˆ(ξ ). It follows so that χν = ν χν . Then F j,ν j,ν that  n ˜ n f (x, t) = F (x − y, t) fν (y) dy, (102) K˜ j,ν j,ν R2

n as in (27). Using Cauchy–Schwarz inequality in (102) and summing over with K˜ j,ν ν, we get N−1

˜ fν (x, t)|2 |F j,ν n

ν=0

≤

 ν

≤

R2

n |fν (y)|2 |K˜ j,ν (x − y, t)| dy

C (1 + |n|)3+1/4



R2

  R2

n |K˜ j,ν (x − y, t)| dy



n |fν (y)|2 |K˜ j,ν (x − y, t)| dy

ν

(103)     n (·, t) since K˜ j,ν

L1 (R2 )

≤

C (1+|n|)3+1/4

for some constant C independent of t, j and

n, in view of Lemma 3.1. Squaring, integrating and taking square root in (103) leads to the inequality ⎛ 2 ⎞1   N−1 2 

 n ⎝  |F˜ j,ν fν |2 ⎠      ν=0  4

L (R3 )

     

C   2 n ≤ sup  |fν (y)| |K˜ j,ν (x − y, t)| dy g(x, t) dxdt  3+1/4  (1 + |n|) R2 ν gL2 =1  R3     

C   2 n ≤ sup  |fν (y)| |K˜ j,ν (x − y, t)| |g(x, t)| dxdt dy   (1 + |n|)3+1/4 g 2 =1  R2 ν R3 L

(104)

24

R. Manna and P. K. Ratnakumar

where we used duality in the above equality for the L2 (dxdt) norm as in (73), and Fubini’s theorem in the last step. By Cauchy–Schwarz inequality in y variable, the term inside the modulus sign is at most ⎡  ⎢ ⎣

⎛

N−1

R2

⎝

⎞2

⎤1/2

⎥ |fν (y)|2 ⎠ dy ⎦

ν=0



  sup  2 ν

R

R3

2  12  n ˜ |Kj,ν (x − y, t)| |g(x, t)| dxdt  dy .

(105) Note that the first term above is  1/2   N−1 2 

 2   |fν |    ν=0  4 L

≤ C[log N]2b f 2L4 (R2 )

(106)

(R2 )

for constants b > 0 and C independent of N, as shown by A. Cordoba in [4]. Since N ≈ 2j/2 , we see that the first term above is at most Cj 2b f 2L4 , with C and b n given by (42), we have independent of j . In view of the pointwise estimate for K˜ j,ν the maximal inequality 

  sup 

R2 ν

R3

2  1 2  n |K˜ j,ν (x − y, t)| |g(x, t)| dxdt  dy ≤

C j 3/2 gL2 (R3 ) (1 + |n|)3+1/4 (107)

as in (76), with C independent of j and n. Using these estimates in (104) and taking the square root, the proof follows.

Proposition 6.3 Let Qδ Fnj f be as in (16) with δ ≥ 1. Then the estimate Qδ Fnj f 4 ≤ C

δ 1/4 2j/8 j 2b+3/4 f L4 (R2 ) (1 + |n|)3+1/4

(108)

holds for all f ∈ L4 (R2 ), with constants C, b > 0 independent of j, n and δ. Proof The proof follows from Propositions 6.1 and 6.2, by the density of S(R2 ) in L4 (R2 ).

Proposition 6.4 Let Fnj f be as in (10) with amplitude function satisfying (28). Then for each 0 < ≤ 1/2, there exists a constant C independent of j and n such that

Local Smoothing of Fourier Integral Operators and Hermite Functions

25

the estimate Fnj f 4 ≤ C

2j (3 +1/8) f L4 (R2 ) (1 + |n|)3+1/4

(109)

holds for all f ∈ L4 (R2 ). Proof In view of (16), we have Fnj f = Rδ (Fnj f ) + Qδ (Fnj f ). For each 0 < ≤ 1/2, by Proposition 6.3 with δ = 2 j , we have Qδ Fnj f 4 ≤ C ≤

2j ( /4+1/8) j 2b+3/4 f L4 (R2 ) (1 + |n|)3+1/4

 Cb,

2j (3 +1/8) f L4 (R2 ) (1 + |n|)3+1/4

(110)

by writing j 2b+3/4 ≤ Cb, 211 j/4. For the same choice of and δ, Proposition 4.1 gives Rδ (Fnj f )L4 (R2 ×R) ≤

C  f L4 . (1 + |n|)3+1/4



The required estimate follows from these two estimates. The above results lead to the following

Lp

estimate for

(111)

Fnj f .

Theorem 6.1 Let Fnj f be as in (10) with amplitude function satisfying (28). Then for each 0 < ≤ 1/2, the estimate Fnj f p ≤ C

2(1/2−3/2p)j 212 j/p f Lp (R2 ) (1 + |n|)3+1/4

(112)

holds for all f ∈ Lp (R2 ), 4 ≤ p ≤ ∞ with a constant C independent of j and n. Proof By Riesz–Thorin interpolation, Propositions 6.4 and 3.1 yields Fnj f Lp (R2 ×R) ≤

C 2j (3 +1/8)(1−t ) 2tj/2 f Lp (R2 ) , (1 + |n|)3+1/4

for 4 ≤ p ≤ ∞, where p1 = gives the required estimate.

1−t 4 .

Substituting t = 1 −

4 p

(113)

on the right hand side



26

R. Manna and P. K. Ratnakumar

7 Local Smoothing Estimate In this section we give the proof of Theorem 1.1. We will show that the local smoothing estimate for the Fourier integral operator F will follow from the Lp Sobolev estimate for Fnj f via a summability argument. This Lp Sobolev estimate in turn follows from the Lp estimates for Fnj f given by Theorem 6.1. A key step in this reduction is the following Lemma. Lemma 7.1 For σ ∈ C, j ∈ N and ρ ∈ Cc∞ ([ 12 , 2]), define fσ,j by −j 2 σ/2 , f ∈ S(R2 ). f σ,j (ξ ) = fˆ(ξ ) ρ(2 |ξ |) (1 + |ξ | )

(114)

Then the estimate fσ,j Lp (R2 ) ≤ Cσ 2j Re(σ ) f Lp (R2 ) , 1 ≤ p ≤ ∞

(115)

holds for all f ∈ S(R2 ) for Re(σ ) ≤ 0, with a constant Cσ independent of j . Proof We have, fσ,j (x) = f ∗ kσ (x),

(116)

where  kσ (x) =

R2

eix·ξ ρ(2−j |ξ |) (1 + |ξ |2 )σ/2 dξ 

= 22j

(117) R2

ei2

j x·ξ

ρ(|ξ |) (1 + 22j |ξ |2 )σ/2 dξ.

Note that the above integral is actually over the annulus 1/2 ≤ |ξ | ≤ 2 by the support property of ρ. Hence an integration by parts and the fact that Re(σ ) ≤ 0 shows that |kσ (x)| ≤ Cσ 2j Re(σ )

22j . (1 + |2j x|2 )2

Hence kσ 1 ≤ Cσ 2j Re(σ ) and the proof follows by Young’s inequality. We also consider the Fourier ˜ n f (x, t) = ρ1 (t) F j

˜n integral operator F j

 R2

(118)



given by

ei(x·ξ +t |ξ |) ρ(2−j |ξ |) an (ξ ) fˆ(ξ ) dξ

(119)

Local Smoothing of Fourier Integral Operators and Hermite Functions

27

˜ n and Fn given by (10) differs only in the power of ρ, where ρ 2 = ρ0 . Note that F j j N−1 ˜ n n ˜ ˜ n f is as in (25). It follows that F ˜ nf and we have Fj f = ν=0 Fj,ν f where F j,ν j also satisfies the same norm estimates as in Theorem 6.1: n

˜ f p ≤ C F j

2(1/2−3/2p)j 212 j/p f Lp (R2 ) , 4 ≤ p ≤ ∞ (1 + |n|)3+1/4

(120)

with constant C independent of j and n. In fact, the L4 and L∞ estimates for Fnj f involve the bound for ρ0 and its derivatives, which in turn depend only on the bound for ρ and its derivatives since ρ0 = ρ 2 , as seen in Lemma 3.1, Propositions 4.1, 6.1, and 6.2. Hence the assertion. Theorem 7.1 Let Fnj f be as in (10) with the amplitude function an ∈ S m , m ≤ 0 (1+|ξ |) satisfying |∂ξα an (ξ )| ≤ Cα (1+|n|) 3+1/4 for all multi-indices α, with Cα independent m−|α|

of n ∈ N30 . Then for each > 0, there exist a constant C > 0 independent of j and n such that the estimate (I − x )(σ −m)/2Fnj f Lp (R2 ×R) ≤

C 2j θ f Lp (R2 ) , j ∈ N (1 + |n|)3+1/4

(121)

holds for all f ∈ Lp (R2 ), 4 ≤ p ≤ ∞, whenever Re(σ ) ≤ 0, where θ = 12 /p + (1/2 − 3/2p) + Re(σ ). Proof First we consider the case m = 0. Set L = (I − x )1/2 . Then we have ˜ n (fσ,j ) Lσ (Fnj f ) = F j

(122)

˜ n and fσ,j are as in (119) and Lemma 7.1 respectively. This follows by where F j taking Fourier transform in the x-variable and using the fact that ρ0 = ρ 2 . Thus in view of (122), and (120) with fσ,j together with Lemma 7.1, we see that Lσ (Fnj f )Lp (R2 ×R) ≤ =

C 2(1/2−3/2p)j 212 j/p 2j Re(σ ) f Lp (R2 ) (1 + |n|)3+1/4 C 2j θ f Lp (R2 ) (1 + |n|)3+1/4 (123)

where θ = 12 /p + (1/2 − 3/2p) + Re(σ ). This settles the case m = 0. Now if an ∈ S m , m < 0, then L−m Fnj is a Fourier integral operator with amplitude function (1 + |ξ |2 )−m/2 an (ξ ) ∈ S 0 . Since Lσ −m Fnj = Lσ (L−m Fnj ), the required estimate follows from the case m = 0, hence the proof.



28

R. Manna and P. K. Ratnakumar

Remark 7.1 Note that (I − x )(σ −m)/2Fnj f = Fnj ((I − x )(σ −m)/2 f ) as in (122). Hence the inequality in Proposition 7.1 can be re written as Fnj f Lp (R2 ×R) ≤

C 2j θ f Lp (R2 ) . m−σ (1 + |n|)3+1/4

(124)

Now we proceed to prove the local smoothing estimate for general Fourier integral operators of the form  Gf (x, t) =

ei(x·ξ +t |ξ |)a(x, t, ξ ) fˆ(ξ ) dξ

(125)

where a ∈ S m (R2 × R × R2 ), m ≤ 0. First note that the Lp estimate for Gf on R2 × I will follow from the Lp estimate for the Fourier integral operator given by (4):  Ff (x, t) = ρ1 (t) ei(x·ξ +t |ξ |)a(x, t, ξ ) fˆ(ξ ) dξ (126) by choosing ρ1 ∈ Cc∞ (R) which is non negative and identically equal to 1 on the compact time interval I . The key idea in the proof is to expand the general amplitude function a(x, t, ξ ) in terms of Hermite functions in (x, t) variables for each fixed ξ , to write it as a sum of Fourier integral operators as in (8) with amplitude function independent of x and t. Recall that the one dimensional Hermite functions hn are given by hn (x) = $

(−1)n √ 2n n! π



 2 x d n −x 2 e 2 , x ∈ R. e dx n

(127)

The k-dimensional Hermite functions are defined by the tensor product of the one dimensional Hermite functions: i.e., for x ∈ Rk and n = (n1 , n2 , . . . , nk ) we have hn (x) = ki=1 hni (xi ), x = (x1 , x2 , . . . , xk ). Then the functions hn ∈ L2 (Rk ) and we have hn 2 = 1 for each multi-index n ∈ Nk0 . The functions hn are eigenfunctions of the Hermite operator H = − x + |x|2 with eigenvalue 2|n| + k, for the multi index n = (n1 , n2 , . . . , nk ). The family {hn }n∈Nk form an orthonormal basis for L2 (Rk ), hence every L2 function has an 0

L2 convergent Hermite expansion. For more details on the Hermite functions and Hermite polynomials, we refer to [14, 28]. Proof of Theorem 1.1 We expand a(x, t, ξ ) in terms of Hermite functions in (x, t) variables, for each fixed ξ ∈ R2 : a(x, t, ξ ) =

n∈N30

an (ξ ) hn (x, t), (x, t) ∈ R2 × R

(128)

Local Smoothing of Fourier Integral Operators and Hermite Functions

29

where an denotes the Fourier–Hermite coefficient given by  an (ξ ) =

R3

(129)

a(x, t, ξ ) hn (x, t) dxdt

and hn stands for the normalised Hermite function on R3 discussed above. It is easy to see that the series given by (128) converges absolutely in (x, t, ξ ) if a(·, ·, ξ ) belongs to the Hermite–Sobolev space WH4,∞ (R3 ) for each ξ , with the Hermite– Sobolev norm aW 4,∞ := H 4 aL∞ (dx,dt ) bounded uniformly in ξ . In fact, using H

H 4 hn = (2|n| + 3)4 hn

(130)

in (129) and an integration by parts together with the fact that a ∈ S m (R2 × R × R2 ), m ≤ 0 shows that |∂ξα an (ξ )| ≤ H 4 ∂ξα aL∞ (R2 , dxdt )

hn L1 (R3 , dxdt ) (2|n| + 3)4

.

(131)

In view of Lemma 1.5.2 in [28], we have hn L1 (R3 ) ≈ (n1 n2 n3 )1/4 ≤ |n|3/4 for n = (n1 , n2 , n3 ) ∈ N30 . Thus we get |∂ξα an (ξ )| ≤

C H 4 ∂ξα a(·, ·, ξ )L∞ (R2 , dxdt ) (1 + |n|)3+1/4

(132)

for all α ∈ N20 . In particular, when α = 0 we see that |an (ξ )| ≤

C , n ∈ N30 (1 + |n|)3+1/4

(133)

with a constant C independent of n and ξ , provided H 4 a(·, ·, ξ )L∞ (R2 , dxdt ) is uniformly bounded in ξ . Since hn ∞ is bounded uniformly in n ∈ N30 , (see Lemma 1.5.2 in [28]), it follows that the Hermite expansion (128) converges absolutely and uniformly in (x, t) and ξ variables. In view of (128) and the decomposition (11), we have for f ∈ S(R2 ) Ff (x, t) =

∞



hn (x, t) Fnj f (x, t)

(134)

n∈N30 j =0

where Fnj f is as in (10). The above step involves an interchange of integral and sum, which is justified by dominated convergence theorem whenever f ∈ S(R2 ), and the fact that n,j |an (ξ )| ρ(2−j |ξ |) is bounded uniformly in ξ , in view of (133). Since

30

R. Manna and P. K. Ratnakumar

hn ∞ is bounded uniformly in n ∈ N30 as observed above, taking Lp norm on both sides of (134) yields ∞



Ff Lp (R2 ×R) ≤

n∈N30

j =0

Fnj f Lp (R2 ×R) .

(135)

β

Note that H 4 ∂ξα a is a linear combination of terms of the form (x, t)γ ∂x,t ∂ξα a for |β| + |γ | ≤ 8. This does not satisfy the decay condition of the symbol. However by assumption (5) we have H 4 ∂ξα aL∞ dxdt ≤ Cα (1 + |ξ |)m−|α| for all α. Thus from (132) we see that an satisfies the estimate |∂ξα an (ξ )| ≤ Cα

(1 + |ξ |)m−|α| (1 + |n|)3+1/4

(136)

for each α, with Cα independent of n. Hence by Theorem 7.1 and Remark 7.1, we get for each n ∈ N30 and j ∈ N Fnj f Lp (R3 ) ≤

C 2j θ f Lp (R2 ) m−σ (1 + |n|)3+1/4

(137)

for 4 ≤ p ≤ ∞, where θ = 12 /p + (1/2 − 3/2p) + Re(σ ). The same estimates holds for j = 0 as well, in view of Theorem 3.1 with b(ξ ) = ϕ0 (ξ ) an (ξ ), and (136), together with the commutativity of (I − x ) with Fn0 , where ϕ0 = j ≤0 ρ0 (2−j |ξ |). 3 1 Since θ < 0 whenever Re(σ ) < 2p − 12 − 12 /p and n∈N3 (1+|n|) 3+1/4 < ∞, 0 n p 3 we see that n,j Fj f is absolutely summable in L (R ) and Ff Lp (R2 ×R) ≤ for Re(σ ) < σ =

3 2p

∞

n∈N30 j =0

−

1 2

Fnj f Lp (R2 ×R) ≤ C ,σ f Lp

2 m−σ (R )

− 12 /p, where

C ,σ = C

n∈N30

∞

1 2j θ < ∞. (1 + |n|)3+1/4

Note that > 0 is arbitrary, and σ → Re(σ )
0, and so the natural gradation of its Lie algebra l4 appears as l4 = V1 ⊕ V2 ⊕ V3 ,

(8)

where V1 = span{X1 , X2 } , V2 = span{X3 } and V3 = span{X4 }, and is such that [Vi , Vj ] ⊂ Vi+j , i = j . Finally, let us note that, from the general theory of homogeneous Lie groups, the Lebesgue measure on R4 is invariant with respect to the left and right invariant translation on B4 , that is the Lebesgue measure on R4 is the Haar measure for B4 and we can formulate as   · · · dx1 dx2 dx3 dx4 = · · · dx1 dx2 dx3 dx4 . (9) B4

R4

3 Group Representation and Quantization of the Fourier Transform The representations of the Engel group B4 are the infinite dimensional unitary (equivalence classes of) representations of B4 . Parametrised by λ = 0 and μ ∈ R, following [2, p. 333], they act on L2 (Rn ). We denote them by πλ,μ , and realise

1 For 2 For

smooth vectors X, Y in Rn , we define the Lie-bracket [X, Y ] := Y X − XY . V , W spaces of vector fields, we denote by [V , W ] the set {[v, w] : v ∈ V , w ∈ W }.

40

M. Chatzakou

them as    μ λ h(u + x1 ) , πλ,μ (x1 , x2 , x3 , x4 )h(u) = exp i − x2 + λx4 − λx3 u + x2 u2 2λ 2

(10) for h ∈ L2 (R), u ∈ R. The group Fourier transform of a function f ∈ L1 (B4 ) is by definition the linear endomorphism on L2 (R) FB4 (f )(πλ,μ ) ≡ fˆ(πλ,μ ) ≡ πλ,μ (f ) :=

 B4

f (x)πλ,μ (x)∗ dx .

(11)

Rigorous computations show that fˆ(πλ,μ )h(u) can be written as   R4

   μ λ x2 − λx4 + λx3 (u − x1 ) − x2 (u − x1 )2 f (x1 , x2 , x3 , x4 ) exp i 2λ 2  × h(u − x1 ) dx1 dx2 dx3 dx4

=(2π)−2



 R4

R4

 FR4 (f )(ξ, η, τ, ω) · eix1 ξ · eix2 η · eix3 τ · eix4 ω

   μ λ x2 − λx4 + λx3 (u − x1 ) − x2 (u − x1 )2 × exp i 2λ 2  × h(u − x1 ) dx1 dx2 dx3 dx4 dξ dη dτ dω    = − (2π)

R R

eix1 ξ

 λ μ , λ(x1 − u), λ)h(u − x1 ) dx1 dξ × F(f )R4 (ξ, (u − x1 )2 − 2 2λ     λ μ , −λv, λ)h(v) dv dξ , =(2π) ei(u−v)ξ F(f )R4 (ξ, v 2 − 2 2λ R R (12) for h ∈ L2 (R) and u ∈ R, that is FB4 (f )(πλ,μ ) = Op[af,λ,μ (·, ·)] ,

(13)

λ μ , −λv, λ) . af,λ,μ (v, ξ ) = (2π)2 FR4 (f )(ξ, v 2 − 2 2λ

(14)

where

On (λ, μ)-Classes on the Engel Group

41

Here the Fourier transform FR4 is defined via: FR4 f (ξ ) = (2π)−2

 R4

f (x)e−ixξ dx

(ξ ∈ R4 , f ∈ L1 (R4 )) ,

(15)

and Op denotes the Kohn–Nirenberg quantization, that is for a smooth symbol a on R × R the operator Op(a)f (u) = (2π)

−1

  R R

ei(u−v)ξ a(v, ξ )f (v) dv dξ ,

(16)

for f ∈ S(R) and u ∈ R. We note that for the case of the Heisenberg group Hn the group Fourier transform has been computed in [3] as being the operator $ √ n FHn (f )(πλ ) = (2π) 2 OpW [FR2n+1 (f )( |λ|·, λ·, λ)] ,

(17)

where OpW denotes the Weyl-quantization, i.e., OpW (a)f (u) = (2π)−n



 Rn

  u+v ei(u−v)ξ a ξ, f (v) dv dξ , 2 Rn

(18)

for f ∈ S(Rn ) and u ∈ Rn , where πλ denotes the Schrödinger representations of the Heisenberg group Hn , Going back to our case, of one keeps the same notation πλ,μ for the infinitesimal representation, we compute that: πλ,μ (X1 ) = ∂u = Op(iξ ) , πλ,μ (X2 ) =

  iμ i  2 μ iλu2 − , λu − = Op 2 λ 2 2λ

(19)

πλ,μ (X3 ) = −iλu = Op(−iλu) , πλ,μ (X4 ) = iλ = Op(iλ) , thus d2 1  2 μ 2 πλ,μ (L) = πλ,μ (X1 )2 + πλ,μ (X2 )2 = λu − − du2 4 λ   2  1 μ λu2 − = −Op ξ 2 + . 4 λ

(20)

42

M. Chatzakou

With our choice of notation, the Plancherel measure of the Engel group B4 is (2−3 π −4 )dλ dμ, in the sense that following expression for the Plancherel formula  B4

|f (x1 , x2 , x3 , x4 )|2 dx1 dx2 dx3 dx4 =2

−3 −4

π





λ=0 μ∈R

(21) πλ,μ (f )2HS dμ dλ ,

holds for any f ∈ S(R), where  · HS denotes the Hilbert-Schmidt norm of an operator on L2 , that is AHS := T r(A∗ A). The last allows for an extension of the group Fourier transform to L2 (B4 ), and in particular formula (21) holds true for any f ∈ L2 (B4 ). Indeed, by using (13) the operator πλ,μ (f ) has integral kernel  Kf,λ,μ (u, v) = 2π

λ μ , −λv, λ) dξ , ei(u−v)ξ FR4 (f )(ξ, v 2 − 2 2λ R

(22)

or equivalently 3 λ μ , −λv, λ) , Kf,λ,μ (u, v) = (2π) 2 FR3 (f )(v − u, v 2 − 2 2λ

(23)

where the Fourier transform is taken with respect to the second, the third and the fourth variable of f . Integrating the L2 (R × R)-norm of Kf,λ,μ (or the HilbertSchmidt norm of πλ,μ (f )) against dλ, dμ we obtain 

  

R\{0} R R R



= (2π)

  

3

 = (2π)

|Kf,λ,μ (u, v)|2 du dv dμ dλ

R\{0} R R

  

λ μ , −2λ, λ)|2 du dv dλ dμ |FR3 (f )(u − v, v 2 − 2 2λ R

3 R\{0} R R R

|FR3 (f )(x1 , w2 , w3 , w4 )|2

1 dw2 dw3 dw4 dx1 , 2 (24)

where the constant 12 comes from the calculation of the determinant of the Jacobian matrix of the linear transformation F (u, v, λ, μ) = (w1 = u − v, w2 = λ2 v 2 − μ 3 2λ , w3 = −λv, w4 = λ). Finally, the Plancherel formula on R in the variable

On (λ, μ)-Classes on the Engel Group

43

(w2 , w3 , w4 ) with dual variable (x2 , x3 , x4 ) gives 

 R\{0} R3

|Kf,λ,μ (u, v)|2 dv du dμ dλ  =2 π

(25)

2 3

|f (x1 , x2 , x3 , x4 )| dx1 dx2 dx3 dx4 , 2

R4

and the last implies (21).

4 Difference Operators Difference operators on the setting of a compact Lie group introduced in [5] as acting on Fourier coefficients, while on graded Lie groups in [4]. In the setting of the Engel group B4 this yields the definition of the difference operators xi as:

xi κ(π ˆ λ,μ ) := πλ,μ (xi κ) ,

i = 1, · · · , 4 ,

(26)

for suitable distributions κ on B4 . To find the explicit expressions of the difference operators xi we make use of the following property: For X and X˜ being a left and a right invariant vector field, respectively, in the Lie algebra l4 , and for a distribution κ on B4 we have πλ,μ (Xκ) = πλ,μ (X)πλ,μ (κ) ,

˜ = πλ,μ (κ)πλ,μ (X) . πλ,μ (Xκ)

(27)

Notice that the right invariant vector fields that generate l4 can be calculated as: X˜ 1 = ∂x1 − x2 ∂x3 − x3 ∂x4 , X˜ 2 = ∂x2 , X˜ 3 = ∂x3 , X˜ 4 = ∂x4 .

(28)

Proposition 4.1 For suitable distribution κ on B4 we have:

x1 κ(π ˆ λ,μ ) =

i (πλ,μ (X3 )πλ,μ (κ) − πλ,μ (κ)πλ,μ (X3 )) , λ

(29)

where πλ,μ (X3 ) = −iλu, and

x2 κ(π ˆ λ,μ ) =

2λ ∂μ πλ,μ (κ) . i

(30)

44

M. Chatzakou

Proof Since πλ,μ (X4 ) = iλ, and X˜ 3 − X3 = X4 x1 , we have 1 1 πλ,μ (X4 x1 κ) = ((X˜ 3 − X3 )κ) iλ iλ i = (πλ,μ (X3 )πλ,μ (κ) − πλ,μ (κ)πλ,μ (X3 )) . λ

πλ,μ (x1 κ) =

(31)

Now, for the difference operator corresponding to x2 , we differentiate the group Fourier transform of κ as in (12) at h with respect to μ and get ∂μ {πλ,μ (κ)h(u)} ' = ∂μ

  μ x2 − λx4 κ(x) exp i 2λ R4    ( λ 2 × exp i λx3 (u − x1 ) − x2 (u − x1 ) h(u − x1 )dx 2   μ  = x2 − λx4 κ(x) exp i 2λ R4      λ i × exp i λx3 (u − x1 ) − x2 (u − x1 )2 x2 dx , h(u − x1 ) 2 2λ (32)

or in terms of difference operators,  ∂μ πλ,μ (κ) = πλ,μ

i x2 κ 2λ

 =

i

x πλ,μ (κ) . 2λ 2

(33)



Proposition 4.2 For a suitable distribution κ we have: ˆ λ,μ )

x3 κ(π =

i ( x2 πλ,μ (κ)πλ,μ (X3 ) + πλ,μ (κ)πλ,μ (X1 ) − πλ,μ (X1 )πλ,μ (κ)) , λ (34)

where x2 |πλ,μ is given in Proposition 4.1 and πλ,μ (X1 ) = ∂u , πλ,μ (X3 ) = −iλu.

On (λ, μ)-Classes on the Engel Group

45

Proof Since X1 − X˜ 1 − x2 X3 = ∂x4 x3 we have πλ,μ (x3 κ) =

1 1 (X4 x3 κ) = ((X1 − X˜ 1 − x2 X˜ 3 )κ) iλ iλ

1 (πλ,μ (X1 )πλ,μ (κ) − πλ,μ (κ)πλ,μ (X1 ) − x2 πλ,μ (κ)πλ,μ (X3 ) iλ i = ( x2 πλ,μ (κ)πλ,μ (X3 ) + πλ,μ (κ)πλ,μ (X1 ) − πλ,μ (X1 )πλ,μ (κ)) , λ (35)

=



completing the proof. Proposition 4.3 For a suitable distribution κ on B4 we have:  ( μ u2 '

+ π (κ)h(u) ( x4 πλ,μ (κ))h(u) = i∂λ {πλ,μ (κ)h(u)} − x λ,μ 2 2 2λ2 ( ' ( ' + u x3 πλ,μ (κ)h(u) − x3 x1 πλ,μ (κ)h(u) 

( 1' ( '

x2 2x1 πλ,μ (κ)h(u) , + u x2 x1 πλ,μ (κ)h(u) − 2 (36) where the difference operators xi |πλ,μ , i = 1, 2, 3, are given in Propositions 4.1 and 4.2, respectively. Proof Differentiating the group Fourier transform of κ as in (12) at h with respect to λ yields '

 μ  x2 − λx4 κ(x) exp i 2λ R4    ( λ 2 × exp i λx3 (u − x1 ) − x2 (u − x1 ) h(u − x1 ) dx 2     μ λ x2 − λx4 + λx3 (u − x1 ) − x2 (u − x1 )2 κ(x) exp i = 2λ 2 R4 ( '  μ x2 × h(u − x1 ) i − 2 x2 − x4 + x3 (u − x1 ) − (u − x1 )2 dx . 2λ 2 (37)

∂λ {πλ,μ (κ)h(u)} = ∂λ

46

M. Chatzakou

Rewriting the above formula in terms of difference operators we obtain ∂λ {πλ,μ (κ)h(u)}  ( ' (   μ u2 '

x2 πλ,μ (κ)h(u) − x4 πλ,μ (κ)h(u) + =i − 2 2λ 2 ( ' ( ' + u x3 πλ,μ (κ)h(u) − x3 x1 πλ,μ (κ)h(u)

(38)

' ( 1' (

x2 2x1 πλ,μ (κ)h(u) , + u x2 x1 πλ,μ (κ)h(u) − 2



completing the proof. For example we have

x1 πλ,μ (X1 ) = −I,

x1 πλ,μ (X2 ) = x1 πλ,μ (X3 ) = x1 πλ,μ (X4 ) = 0

(39a)

x2 πλ,μ (X1 ) = x2 πλ,μ (X3 ) = x2 πλ,μ (X4 ) = 0, (39b)

x2 πλ,μ (X2 ) = −λI

x3 πλ,μ (X1 ) = x3 πλ,μ (X4 ) = 0,

x3 πλ,μ (X2 ) = −λu + u,

(39c)

x3 πλ,μ (X3 ) = −I,

x4 πλ,μ (X1 ) = x4 πλ,μ (X4 ) = 0,

x4 πλ,μ (X2 ) =

u2 μ (1 − λ) + , 2 2λ

(39d)

x4 πλ,μ (X4 ) = −I, where the difference operators xi πλ,μ (Xj ) can be understood as the group Fourier transform of the distribution xi Xj δ0 .

5 Quantization and Symbol Classes In this note, we may slightly change the notation of the symbol introduced in [4]. We keep the notation x = (x1 , x2 , x3 , x4 ) ∈ B4 ,

(40)

On (λ, μ)-Classes on the Engel Group

47

to denote the coordinates of an element in the Engel group B4 , and we may denote by σ (x, λ, μ) := σ (x, πλ,μ ) ,

(x, λ, μ) ∈ B4 × R \ {0} × R ,

(41)

the symbol σ parametrised by (x, λ, μ). In addition, if the multi-index α ∈ N40 is written as α = (α1 , α2 , α3 , α4 ) ,

αi ∈ N0 ,

(42)

then the homogeneous degree of α is given by: [α] = α1 + α2 + 2α3 + 3α4 .

(43)

x α = x1α1 x2α2 x3α3 x4α4 ,

(44)

For each α we may write:

so that the corresponding difference operator can be defined as:

α

= αx11 αx22 αx33 αx44 .

(45)

Finally for the vector field X we write Xα to denote the following composition of vector fields: X1α1 X2α2 X3α3 X4α4 .

(46)

m Following [4] we define the symbol classes Sρ,δ (B4 ), where 0 ≤ δ < ρ ≤ 1 and m ∈ R, as the set of symbols σ for which the following quantities are finite: m ,a,b,c := σ Sρ,δ

sup λ∈R\{0},μ∈R,x∈B4

m ,a,b,c , σ (x, λ, μ)Sρ,δ

a, b, c ∈ N0 ,

(47)

m ,a,b,c is defined as where σ (x, λ, μ)Sρ,δ

sup

[a]≤a [β]≤b,|γ |≤c

πλ,μ (I − L)

ρ[α]−m−δ[β]+γ 2

α

γ

Xβ σ (x, λ, μ)πλ,μ (I − L)− 2 op . (48)

There is a natural quantization on any type-I Lie group introduced by Taylor [6] that can be served as the analogue of the Kohn–Nirenberg quantization on Rn . In particular, the quantization, i.e., the mapping σ → Op(σ ) produces operators

48

M. Chatzakou

m (B )) on S(B ) associated with a symbol σ (for example in the class of symbols Sρ,δ 4 4 given by:

Op(σ )φ(x) = 2−3 π −4



 λ=0 μ∈R

) * T r πλ,μ (x)σ (x, λ, μ)πλ,μ (φ) dμ dλ . (49)

Here we have used our notation for the description of the dual, as well as for the symbol and the Plancherel measure, see (21). Let us note that by (13), we see that for the symbol σ quantized as: σ (x, λ, μ) = Op(aκx ,λ,μ ) ,

(50)

then its symbol that is given by λ μ , −λv, λ) , aκx ,λ,μ (v, ξ ) = (2π)2 FR4 (κx )(ξ, v 2 − 2 2λ

(51)

shall be called the (λ, μ)-symbol, where {κx (y)} is the kernel of the symbol {σ (x, λ, μ)}, i.e., σ (x, λ, μ) = πλ,μ (κx ) .

(52)

The above, together with the property of the Fourier transform ˆ λ,μ )πλ,μ (x) = FB4 (φ(x·))(πλ,μ ) , φ(π

(53)

and the properties of the trace yield the following alternative formula for the quantization given in (49): Op(σ )(φ)(x) = 2

−3 −4

π





λ=0 μ∈R

) * T r Op(aκx ,λ,μ )Op(aφ(x·),λ,μ) dμ dλ . (54)

The last formula shows that the quantization formula (49) can be expressed in terms of composition of quantization of symbols in the Euclidean space. Similarly, for the case of the Heisenberg group Hn , (17) implies that the operator Op(σ ) on S(Hn ) involves “Euclidean objects”, and in particular: Op(σ )φ(x)    $ √ = cn T r OpW (ax,λ )OpW [FR2n+1 (φ(x·))( |λ|·, λ·, λ)] |λ|n dλ , λ=0

(55)

On (λ, μ)-Classes on the Engel Group

49

where the symbol ax,λ (also called the λ-symbol) given by: $ √  ax,λ (ξ, u) = FR2n+1 (κx )( |λ|ξ, λu, λ) ,

(56)



where {κx (y)} is the kernel of the symbol σ , is such that σ (x, λ) := σ (x, πλ ) = OpW (ax,λ ) .

(57)

For our notation, especially for the Plancherel measure cn |λ|n on Hn , see [4, Chapter 6]. In contrast with the case of the Engel group B4 , in the setting of the Heisenberg group Hn , one can renormalise ax,λ as $ √ ax,λ (ξ, u) := a˜ x,λ ( |λ|ξ, λu) ,

(58)

m (H ) by the property and therefore, one can characterise the symbol classes Sρ,δ n that these λ-symbols belong to some Shubin spaces, called λ-type version of the usual Shubin classes, leading to sufficient criteria for ellipticity and hypoellipticity of operators on Hn in terms of the invertibility properties of their λ-symbols, see [2, Chapter 6].

Acknowledgments We wish to thank Professor Michael Ruzhansky for the useful discussions and suggestions that helped to improve the present work.

References 1. Bahouri, H., Fermanian-Kammerer, C., Gallagher, I.: Phase-space analysis and pseudodifferential calculus on the Heisenberg group. Astérisque 342 (2012). See also revised version of March 2013 of arXiv:0904.4746 2. Dixmier, J.: Sur les représentations unitaires des groupes de Lie nilpotents. III. Can. J. Math 10, 321–348 (1957) 3. Fischer, V., Ruzhansky, M.: A pseudo-differential calculus on the Heisenberg group. C. R. Math. Acad. Sci. Paris 352(3), 197–204 (2014) 4. Fischer, V., Ruzhansky, M.: Quantization on Nilpotent Lie Groups, vol. 314. Progress in Mathematics. Birkhäuser/Springer (2016). Open access book 5. Ruzhansky, M., Turunen, V.: Pseudo-Differential Operators and Symmetries: Background Analysis and Advanced Topics. Pseudo-Differential Operators: Theory and Applications, vol. 2. Birkäuser, Basel (2010) 6. Taylor, M.E.: Noncommutative microlocal analysis. I. Memoirs of the American Mathematical Society, vol. 52. American Mathematical Society, Providence (1984)

Gelfand Triples for the Kohn–Nirenberg Quantization on Homogeneous Lie Groups Jonas Brinker and Jens Wirth

Abstract We study the action of the group Fourier transform and of the Kohn– Nirenberg quantization (Fischer and Ruzhansky, Quantization on Nilpotent Lie Groups. Birkäuser, Boston, 2016) on certain Gelfand triples for homogeneous Lie groups G. Even for the Heisenberg group G = H there seems to be no simple intrinsic characterization for the Fourier image of the Schwartz space of rapidly decreasing smooth functions S(G), see (Geller, J Funct Anal 36(2), 205– 254, 1980; Astengo et al., Stud Math 214, 201–222, 2013). But we may derive a simple characterization of the Fourier image for a certain subspace S∗ (G) of S(G). We restrict our considerations to the case, where G admits irreducible unitary representations, that are square integrable modulo the center Z(G) of G, and where dim Z(G) = 1. This enables us to use an especially applicable characterization of these irreducible unitary representations that are square integrable modulo Z(G) (Moore and Wolf, Trans Am Math Soc 185, 445–445, 1973; M˘antoiu and Ruzhansky, J Geom Anal 29(2), 2823–2861, 2018; Gröchenig and Rottensteiner, J Funct Anal 275(12), 3338–3379, 2018). Also, Pedersen’s machinery (Pedersen, Invent. Math. 118, 1–36, 1994) combines very well with this setting (M˘antoiu and Ruzhansky, J Geom Anal 29(2), 2823–2861, 2018). Starting with S∗ (G), we are able to construct distributions and Gelfand triples around L2 (G, μ) and its   Fourier image L2 (G, μ), such that the Fourier transform becomes a Gelfand triple isomorphism. In this context we show for the Fourier side, that multiplication of distributions with a large class of vector valued smooth functions is possible and well behaved. Furthermore, we rewrite the Kohn–Nirenberg quantization as an isomorphism for our new Gelfand triples and prove a formula for the Kohn– Nirenberg symbol, which is known from the compact group case (Ruzhansky and Turunen, Pseudo-Differential Operators and Symmetries. Birkäuser, Boston, 2010). Keywords Homogeneus Lie groups; Group Fourier transform; Kohn-Nirenberg quantization; Distribution spaces

J. Brinker () · J. Wirth Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Stuttgart, Germany e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 V. Georgiev et al. (eds.), Advances in Harmonic Analysis and Partial Differential Equations, Trends in Mathematics, https://doi.org/10.1007/978-3-030-58215-9_3

51

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1 Introduction There is an imbalance between the Schwartz space of rapidly decreasing smooth  for simply connected nilpotent Lie functions S(G) and its Fourier image S(G)  is a space of operator valued functions on the irregroups G. For the space S(G) ducible unitary representations of G, that is not easily characterized without relying on the Fourier transform itself, see [1, 2]. Of course we can see S(G) as a projective limit of suitable Hilbert spaces, which results in a corresponding representation of  as a projective limit. But this approach is rather cumbersome, if we want S(G)  Though, if G is also a homogeneous to identify multiplication operators on S(G).  can be parametrized (up to a group with one-dimensional center, then the dual G null set of the Plancherel measure) by R \ {0} = R× . The setting of homogeneous Lie groups with square integrable (modulo the kernel) representations seems to be convenient in general, see e.g. [3, 4]. Now the main idea is to define a subspace S∗ (G) of S(G), such that its Fourier image can be identified with a tensor product of rapidly decreasing functions on R× and a well behaved space of operators isomorphic to L(S (Rn ), S(Rn )). Here we choose S∗ (G), such that S∗ (G) is still densely embedded into L2 (G, μ). Hence, we may construct a Gelfand triple G∗ (G) from S∗ (G) and L2 (G, μ) and show that the Fourier transform acts as a Gelfand triple isomorphism. By using the theory of vector valued distributions of L. Schwartz and related results in [5], we may define multiplication operators on the Fourier side. We will employ the concept of polynomial manifolds, which were used in [6], for the corresponding spaces S(R× ) and OM (R× ) of smooth functions on R× with growth conditions. Using the Fourier transform on S∗ (G), we will also examine the Kohn– Nirenberg quantization, defined in [7]. We incorporate our new function spaces into Gelfand triples of symbols and operators, on which the Kohn–Nirenberg quantization acts as a Gelfand triple isomorphism. Finally, we will prove a formula for the Kohn–Nirenberg symbol of operators A ∈ L(OM (G)), that is motivated by the corresponding formula a(x, ξ ) = ξ ∗ · A(ξ ) for symbols of operators A ∈ L(D(H )) for compact Lie groups H from [8]. The paper is structured as follows: Sects. 1.1 and 1.2 are dedicated to recalling common facts about harmonic analysis and functional analysis. At the end of Sect. 1.2 we cite theorems about the continuation of bilinear maps to completions of topological tensor products, which will be of importance for the multiplication of vector valued functions and distributions. In Sect. 1.3 we define Gelfand triples with additional real structure and discuss direct sums, tensor products and kernel maps in this context. Section 2 is dedicated to the definition of new Gelfand triples for the Fourier transform, by using polynomial manifolds and Pedersen’s quantization procedure. We start by recalling basic concepts from the theory of vector valued functions and by defining polynomial manifolds. Then, in Sect. 2.1, we pay special attention to

Gelfand Triples for Homogeneous Lie Groups

53

the polynomial manifold R× , the space of Schwartz functions S(R× ) and the dual ˆ E. In the succeeding subsection we recall the characterization space of S(R× ) ⊗ of irreducible representations by the orbit method and Pedersen’s machinery and apply both to homogenous Lie groups G with irreducible representations, that are square integrable modulo the center Z(G). In Sect. 2.3, using the methods developed in the previous two subsections, we define an adjusted group Fourier transform Fπ . Finally, in Sect. 2.4, we define the test function space S∗ (G), define the corresponding Gelfand triple G∗ (G) and characterize the Gelfand triple G(R× ; π), that is isomorphic to G∗ (G) via the adjusted Fourier transform. We finish this section by discussing multiplication operators on S(R× ; π), the Fourier side of S∗ (G). In the last section we show, in what way the Kohn–Nirenberg quantization extends to a Gelfand triple isomorphism from the space of operator valued test ˆ S(R× ; π). Finally in Sect. 3.2 we prove the formula for the Kohn– functions S(G) ⊗ Nirenberg symbol of an operator A ∈ L(OM (G)). Now we start by reminding ourselves of some standard notations and concepts.

1.1 Generalities By G we will always denote a simply connected nilpotent lie group. Since G is diffeomorphic to its Lie algebra g via the exponential map, we will model G to be g as set. I.e. G = g is a Lie algebra and a group with multiplication (x, y) → xy given by the Baker–Campbell–Hausdorff formula. We will use the symbol G or g depending on which property we want to emphasise. The center of G = g will be denoted by Z(G) = z. We will denote the space of left invariant differential operators on G by u(gL ). We choose a fixed Haar measure μ on the group g = G. This measure μ is both a Haar measure with respect to the multiplication and addition. Note, that for each P ∈ u(gL ), there is a unique left invariant differential operator P t , such that 



ϕ P t ψ dμ,

(P ϕ) ψ dμ = G

for all ϕ, ψ ∈ D(G),

(1)

G

where D(M) denotes the space of smooth compactly supported functions on a manifold M. Let E, F be locally convex spaces (always assumed to be Hausdorff) over C. In general, we will denote the strong dual space of E, by E  . The dual pairing between E and E  , will be denoted by e , e := e (e) for e ∈ E  , e ∈ E. Similarly we will equip the space of continuous operators from E to F with the topology of uniform convergence on bounded sets of E and denote it by L(E, F ), resp. L(E) for E = F . For any subspace A ⊂ E we denote its annihilator by A◦ . If E and F are Hilbert spaces, the Hilbert-Schmidt operators from E to F will be denoted by HS(E, F ) (again HS(E) := HS(E, E)). As usual A∗ is the adjoint of A ∈ L(E). The trace of

54

J. Brinker and J. Wirth

some nuclear operator A ∈ L(E), will be denoted by Tr[A]. Furthermore HS(E, F ) is equipped with the inner product (A, B) → Tr[AB ∗ ]. Often we need to integrate vector valued functions. For this purpose we will use the concept of weak integrals. Definition 1.1 Suppose (X, ν) is a measure space and E is a locally convex vector space. We will call a function f : X → E is integrable, iff there is some e ∈ E, such that for each e ∈ E  we have e ◦ f ∈ L1 (X, ν) and  e ◦ f dν. (2) e (e) = X



The element X f dμ := e is called integral over f . Usually we will just say, that X f dμ converges in E. From this definition automatically follows, that 

 f dν =

A X

A ◦ f dν.

(3)

X

for any continuous linear or antilinear operator A : E → F into another locally convex space F . Here, the integrability of f implies the integrability of A ◦ f . We denote by Irr(G) the set of strongly continuous, unitary and irreducible representations of the group G. The dual of G is the quotient of Irr(G) under the  The Plancherel equivalence relation of unitary equivalence and is denoted by G. measure to μ, will be  μ. The representation space of some π ∈ Irr(G) is denoted by Hπ and the corresponding space of smooth vectors will be denoted by Hπ∞ . It is equipped with a Fréchet topology defined by the seminorms v → π(P )vH π ,

for P ∈ u(gL ),

 where π(P )v := Px π(x)v x=0 .

(4)

Finally we will write Hπ−∞ for the strong dual space of Hπ∞ , i.e. the dual space of Hπ∞ equipped with the topology of uniform convergence on bounded sets of Hπ∞ . The group Fourier transform is defined by 

ϕ(x)π(x)∗ dμ(x),

FG ϕ(π) :=

for ϕ ∈ S(G), π ∈ Irr(G),

(5)

G

and its inverse reads  ϕ(x) := Tr[π(x) FG ϕ(π)] d μ([π]),  G

for ϕ ∈ S(G), x ∈ G.

(6)

Notice, that for each π ∈ Irr(G) and each ϕ ∈ S(G) the operator FG ϕ(π) is nuclear  = FG S(G) with the final topology induced on Hπ [9, Theorem 4.2.1]. Let S(G) by FG . Since FG is injective, this is the unique topology that makes FG : S(G) →

Gelfand Triples for Homogeneous Lie Groups

55

 an isomorphism. The space L2 (G,    S(G) μ) is defined to be the completion of S(G) with respect to the inner product  L2 (G, ( ϕ , ψ)  μ) :=

  G

 ∗ ] d Tr[ ϕ (π) ψ(π) μ([π]),

  ∈ S(G). for  ϕ, ψ

(7)

The Fourier transform extends to a unitary operator between L2 (G, μ) and   L2 (G, μ). See e.g. [7] for more details.

1.2 Tensor Products By E ⊗ F we denote the algebraic tensor product between E and F . Their complete ˆ ε F and their complete projective injective tensor product will be denoted by E ⊗ ˆ π F . If Ej and Fj , j = 1, 2, are locally convex spaces and tensor product by E ⊗ Aj ∈ L(Ej , Fj ), then ˆ ε E2 → : F1 ⊗ ˆ ε F2 A 1 ⊗ A 2 : E1 ⊗

(8)

denotes the tensor product map of A1 and A2 . The linear map A1 ⊗A2 is continuous and is even an isomorphism, if A1 and A2 are isomorphisms [10, Proposition 43.7]. Notice, A1 ⊗ A2 can also be defined, if A1 and A2 are continuous anti-linear operators. Later, we will need the following Lemma. Lemma 1.1 Let E, F and G be locally convex spaces, then ˆ ε F, G ⊗ ˆ ε F ) : A → A ⊗ 1 L(E, G) → L(E ⊗

(9)

is continuous. ˆ ε F, G ⊗ ˆ ε F ) is induced by seminorms of the form Proof The topology on L(E ⊗ A → sup p(Az)

(10)

z∈B

ˆ ε F. ˆ ε F and p is a continuous seminorm on G ⊗ where B is a bounded set in E ⊗ For p it is sufficient to take any semi norm of the form p(z) := sup q((1 ⊗ φ)(z))

(11)

φ∈V

where V is an equicontinuous subset of F  and q a continuous seminorm on G [10, ˆ ε F is bounded, Definition 43.1 and Proposition 36.1]. Notice that the set B ⊂ E ⊗

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J. Brinker and J. Wirth

iff for all equicontinuous sets W ⊂ F  and all continuous seminorms r on E sup sup r(1 ⊗ φ(z)) < ∞.

(12)

φ∈W z∈B

In general, a subset of E is bounded, iff all continuous seminorms r are bounded on the set. Hence the set BV := ∪φ∈V (1 ⊗ φ)(B) is a bounded subset of E. We arrive at sup p((A ⊗ 1)z) = sup sup q((A ⊗ φ)(z)) z∈B

z∈B φ∈V

= sup

sup

q(Ae) = sup q(Ae),

φ∈V e∈(1⊗φ)B

(13)

e∈BV

where the right hand side defines a continuous seminorm on L(E, G).



If either E or F are nuclear, both tensor product topologies result in the same ˆ F for either of the complete tensor locally convex space and we may just write E ⊗ products of E and F [10, Theorem 50.1]. If both E and F are nuclear Fréchet spaces, ˆ F [10, Proposition 50.1] and [11, Chapter III corollary to 6.3]. One then so is E ⊗ reason for the usage of nuclear spaces is the following abstract kernel theorem [10, Propositions 50.5–50.7]. Theorem 1.1 If F is a complete locally convex space and E is a nuclear Fréchet ˆ E ∼ space or dual to a nuclear Fréchet space, then F ⊗ = L(E, F ) via the extension of the canonical map





fj ⊗ ej → e → fj ej (e) .

j

ˆ E Suppose additionally both E and F are Fréchet spaces, then F  ⊗ via the extension of the map

fk ⊗ ek →

k

(14)

j



j

fj ⊗ ej →



fk (fj ) ek (ej ) .

ˆ E) , (F ⊗

(15)

j,k

The multiplication between spaces of smooth functions and spaces of distributions is rarely a continuous bilinear map. But often it is hypocontinuous. Here, a bilinear map u : E × F → G is defined to be hypocontinuous between the locally convex spaces E, F and G, if for all bounded sets BE ⊂ E and BF ⊂ F , the two sets of linear maps {u(e, ·) | e ∈ BE } are equicontinuous.

and {u(·, f ) | f ∈ BF }

(16)

Gelfand Triples for Homogeneous Lie Groups

57

Linear maps on tensor factors can easily be combined to construct a linear map on the complete tensor product. The situation for bilinear maps is not as simple. However, in the context of nuclear spaces, we may use the following theorem, which is an amalgamation of a proposition of C. Bargetz and N. Ortner and a corollary of L. Schwartz. Theorem 1.2 Let H, K and L be complete nuclear locally convex spaces with nuclear strong duals and let E, F and G be complete locally convex spaces. Suppose that u : H × K → L and b : E × F → G

(17)

are two hypocontinuous bilinear maps. Suppose furthermore, that either one of the three properties • H and E are Fréchet spaces • H and E are strong duals of Fréchet spaces • the bilinear map b is continuous are fulfilled. Then there is a unique hypocontinuous bilinear map b u:

ˆ E) × (K ⊗ ˆ F) → L⊗ ˆ G, (H ⊗

(18)

that fulfils the consistency property b u (S

⊗ e, T ⊗ f ) = u(S, T ) ⊗ b(e, f ).

(19)

Proof For the cases, where H and E are both Fréchet or both duals to Fréchet spaces, this statement can be found in [5, Proposition 1]. For the case, where b is continuous, we find the statement in [12, Corollair and Remarques on page 38]. However, in the sources the notation H(E) := HεE for the ε-product of Schwartz is used. The nuclearity and completeness of H and the completeness of E make sure, ˆ E by [13, Satz 10.17 and Satz 11.18], which fits our notation. that HεE = H ⊗

Examples of spaces fulfilling the conditions for H, K and L, are S(Rn ) S (Rn ), OM (Rn ) [14, Chapitre II Théorème 16], L(S(Rn )) L(S (Rn )) and n m ˆ OM (R ) ⊗ L(S(R )) [15]. ˆ 2 F . By a The Hilbert space tensor product of E with F will be denoted by E ⊗ slight abuse of notation, we will also denote 

ˆ 2 F2 → F1 ⊗ ˆ 2 E2 A 1 ⊗ A 2 : E1 ⊗

(20)

to be the tensor product map of continuous linear maps Aj between the Hilbert spaces Ej and Fj . If A1 and A2 are unitary, then so is A1 ⊗ A2 .

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1.3 Gelfand Triples Gelfand triples are a convenient setting for both distributions and the Fourier transform. We start by defining the class of Gelfand triples we are going to use. Definition 1.2 A Gelfand triple (with a real structure) is a tuple of spaces G = (E, H, E  ), and a structure map C : H → H fulfilling the following properties: (a) E is a nuclear Fréchet space, with strong dual E  (b) H is a Hilbert space, with dense and continuous embedding E → H (c) C is antiunitary, C2 = 1 and C|E : E → E is a homeomorphism. The map C will be called the real structure of G. Notice, that by the definition of Gelfand triples we automatically have a continuous dense embedding H  → E  dual to the embedding E → H . Classically Gelfand triples are defined without the structure map C [16]. Here E and H are antilinearly → H  . But this embedded into E  via the Fréchet-Riesz isomorphism R : H − approach would be unwieldy, because we will use Gelfand triples in concert with tensor products. Since there is no canonical unitary map between H and H  , we are going to use C to fix one. The corresponding induced embedding I : H → E  is defined via R◦C

I : H −−→ H  → E  .

(21)

The structure map C has a natural continuation to a homeomorphism C : E  → E  by (Ce )(e) := e (Ce) for e ∈ E, e ∈ E  . Hence C induces compatible real structures on E, H and E  . In the sequel we will always use the term Gelfand triple for the concept of a Gelfand triple with real structure equipped with embeddings as described above. Definition 1.3 Let Gj = (Ej , Hj , Ej ), j = 1, 2, be Gelfand triples. For a map T : E  → E  , we write T : G1 → G2 ,

(22)

if T (E1 ) ⊂ E2 and T (H1 ) ⊂ H2 with respect to the above described embeddings. We will call T a Gelfand triple isomorphism, if T |E1 : E1 → E2 , T : E1 → E2 are homeomorphisms and T |H1 : H1 → H2 is unitary. The above definition implies, that writing T : G1 → G2 is equivalent to saying, that the diagram

is commutative.

E1

H1

T

T

E2

H2

E1 T

E2

Gelfand Triples for Homogeneous Lie Groups

59

Now we will describe, how we may construct new Gelfand triples. Definition 1.4 Let Gj = (Ej , Hj , Ej ) be Gelfand triples with structure maps Cj ˆ E2 for j = 1, 2, 3, 4. Using the identifications E1 ⊕ E2 (E1 ⊕ E2 ) and E1 ⊗     (E1 ⊕ E2 ) resp. L(E2 , E1 ) L(E2 , E1 ) via Theorem 1.1 we may define the following Gelfand triples. The sum resp. tensor product of G1 and G2 is defined by ⎛

⎞ E1 ⊕ E2 G1 ⊕ G2 := ⎝H1 ⊕ H2 ⎠ E1 ⊕ E2

⎛

⎞ ˆ E2 E1 ⊗ ˆ 2 H2 ⎠ resp. G1 ⊗ G2 := ⎝H1 ⊗  ˆ E2 E1 ⊗

(23)

with structure maps C1 ⊕ C2 resp. C1 ⊗ C2 . Here C1 ⊗ C2 is the continuation of H1 ⊗ H2 → H1 ⊗ H2 :



h1,j ⊗ h2,k →

j,k

(C1 h1,j ) ⊗ (C2 h2,k ).

(24)

j,k

The operator Gelfand triple from G2 to G1 is defined as ⎞ L(E2 , E2 ) L(G2 , G1 ) := ⎝HS(H2 , H1 )⎠ L(E2 , E1 ) ⎛

(25)

with structure map T → C1 T C2 . Let us now discuss, why G1 ⊗ G2 and L(G2 , G1 ) are indeed Gelfand triples. → H2 determined by C2 , we Notice, that by using the unitary isomorphism I2 : H2 − get an isomorphism resp. a unitary isomorphism ˆ π H2 H1 ⊗

N(H2 , H1 )

resp. HS(H2 , H1 )

ˆ 2 H2 , H1 ⊗

(26)

by extending the linear map K−1 : H1 ⊗ H2 → N(H2 , H1 ), where K−1 (h1 ⊗ h2 )(h˜ 2 ) := h1 · (I2 h2 , )(h˜ 2 )

(27)

and N(H2 , H1 ) is the space of nuclear operator from H2 to H1 . These isomorphisms, together with Theorem 1.1, can be used to construct the following chain of continuous maps with dense ranges ˆ E2 E1 ⊗ K

L(E2 , E1 )

ˆ π H2 H1 ⊗ K−1

N(H2 , H1 )

ˆ 2 H2 H1 ⊗ K

HS(H2 , H1 )

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We get a commutative diagram by completing the rows of this diagram. This both shows, that L(G2 , G1 ) and G1 ⊗ G2 are Gelfand triples, and proves the following Lemma. Lemma 1.2 Suppose G1 and G2 are the Gelfand triples from above, then the canonical linear map K−1 : E1 ⊗ E2 → L(E2 , E1 ),

where

K−1 (e1 ⊗ e2 )(e2 ) := e1 · e2 (e2 ),

(28)

induces a Gelfand triple isomorphism K : L(G2 , G1 ) → G1 ⊗ G2 . Since we choose G simply connected and nilpotent, the space Hπ∞ is a nuclear Fréchet space [9, Corollary 4.1.2] for any π ∈ Irr(G). With respect to π we may define the Gelfand triples ⎛

⎞ Hπ∞ G(π) := ⎝ Hπ ⎠ Hπ−∞

⎛ ⎞ L(Hπ−∞ , Hπ∞ ) and Gop (π) := L(G(π), G(π)) = ⎝ HS(Hπ ) ⎠ , L(Hπ∞ , Hπ−∞ ) (29)

if we associate a real structure Cπ to G(π). Of course, this real structure is not unique. Instead, we will define IrrR (G) to be pairs consisting of π ∈ Irr(G) and an associated real structure Cπ on G(π). Usually we will just write π ∈ IrrR (G) and mean, that we took a choice of Cπ for π.

2 Polynomial Manifolds and Gelfand Triples for the Fourier Transform We will need polynomial manifolds for the Pedersen quantization, a generalization of the Weyl quantization, and for the generalizations of the spaces S(Rn ; E) and OM (Rn ; E) for a complete locally convex space E, see [14, 17]. For it will be convenient to have one notion of Schwartz functions and slowly increasing functions, that can be applied to simply connected nilpotent Lie groups, Lie algebras, coadjoint orbits, and that is also compatible with their relation to each other. Furthermore this will lead to a notion of Schwartz functions and slowly increasing function on R× = R \ {0}, that we will rely on heavily. Before we define polynomial manifolds, let us fix some basic notation and recall a few definitions from the theory of vector valued smooth functions. As usual we will just write S(Rn ) := S(Rn ; E) and OM (Rn ) := OM (Rn ; E) for scalar valued case E = C.

Gelfand Triples for Homogeneous Lie Groups

61

A function f : Rn → E is differentiable, if for each j = 1, . . . , n and each x ∈ Rn the limit 1 ∂j f (x) := lim (f (x + tej ) − f (x)) t →0 t

(30)

exists in E, where (ej )j is the standard basis in Rn , and each partial derivative, ∂j f , is a continuous function. f is called smooth, if partial derivatives of arbitrary order are again differentiable. I.e. for all α ∈ Nn0 the functions ∂ α f = ∂1α1 · · · ∂nαn f exist and are continuous. Denote by P(Rn ) the vector space of polynomial functions from Rn to C and by DiffP (Rn ) the set of differential operators with polynomial coefficients on Rn . The space of E-valued Schwartz functions, S(Rn ; E), is the space of smooth functions ϕ : Rn → E, such that sup p(P ϕ(x)) < ∞

(31)

x∈Rn

for each continuous seminorm p on E and each P ∈ DiffP (Rn ). The above expression also defines a set of seminorms which define the topology on S(Rn ; E). The scalar valued slowly increasing functions, OM (Rn ), is the space of smooth functions f : Rn → C, such that for each P ∈ DiffP (Rn ) there is a real valued q ∈ P(Rn ), such that |Pf | ≤ q. The space of E-valued slowly increasing functions, OM (Rn ; E), is the space of smooth functions f : Rn → E, such that [ϕ → f · ϕ] ∈ L(S(Rn ), S(Rn ; E)),

(32)

equipped with the subspace topology in L(S(Rn ), S(Rn ; E)). For E = C the two given definitions are equivalent [10, Theorem 25.5]. Furthermore the above defined spaces of E-valued functions have a very useful ˆ E characterization by tensor products. To be precise, we have S(Rn ; E) = S(Rn ) ⊗ ˆ E as topological vector spaces. and OM (Rn ; E) = OM (Rn ) ⊗ The definition given below is a slight generalization of the polynomial manifolds used by Pedersen in [6]. Definition 2.1 Suppose now M is an n-dimensional smooth manifold with finitely many connected components. An atlas A of M will be called a polynomial atlas, iff each two charts (φ, U ), (ψ, V ) ∈ A fulfil (i) U , V are connected components of M and φ(U ) = ψ(V ) = Rn , (ii) and if U = V , then φ ◦ ψ −1 is a polynomial function on Rn . Two polynomial atlases A, A are said to be equivalent, iff A ∪ A is a polynomial atlas. A polynomial structure is an equivalence class of polynomial atlases. Together with a polynomial structure M will be called a polynomial manifold. A chart of a polynomial structure of M will be called a polynomial chart on M.

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The following list provides some basic examples of polynomial manifolds: • • • •

a finite dimensional vector space, with respect to the linear charts a finite dimensional affine spaces, with respect to the affine linear charts a simply connected nilpotent Lie group, with respect to the exponential map a coadjoint orbit to a simply connected nilpotent Lie group [6]

By using polynomial charts, we may generalize polynomials and definitions that depend on the set of polynomials. Definition 2.2 Suppose M, N are polynomial manifolds and E a complete locally convex space. For X(•) ∈ {P(•), S(•; E), OM (•; E)} we define X(M) := {f : M → E | f ◦ φ −1 ∈ X(Rn ) for all polynomial charts φ}

(33)

We equip S(M; E) resp. OM (M; E) with the projective topology from the maps f → f ◦ φ −1 into S(Rn ; E) resp. OM (Rn ; E). As usual we set S(M) := S(M; C) and OM (M) := OM (M; C). The set of polynomial differential operators on M is defined to be

+ , ϕ → P (ϕ ◦ φ) ◦ φ −1 ∈ DiffP (Rn ) DiffP (M) := P ∈ L(D(M)) : . for all polynomial charts φ

(34)

A function f : M → N will be called polynomial resp. slowly increasing, iff ψ ◦f ◦ φ −1 is a polynomial resp. slowly increasing (φ, U ) on M and (ψ, V ) on N with U ⊂ f −1 (V ). The function f will be called polynomial resp. tempered diffeomorphism, iff f is bijective and both f and f −1 are polynomial resp. slowly increasing. As for smooth manifolds, we may construct new polynomial manifolds by ˙ of polynomial manifolds N, M with the same dimension disjoint unions M ∪N and products M × N of arbitrary polynomial manifolds N, M. The corresponding ˙ is induced by the polynomial charts on M and N. On polynomial structure on M ∪N M × N we choose the canonical polynomial structure defined by combining charts φ on M and ψ on N to polynomial charts (φ, ψ) on M × N. Directly from our definition follows, that similar to the Euclidean case ˙ S(M ∪N) = S(M) ⊕ S(N)

ˆ S(N) ⊗ ˆ E, and S(M × N; E) = S(M) ⊗

(35)

where E is a complete locally convex space. The identities also hold, if we exchange S with OM . Similar identities are true for P(M) and DiffP (M). We will call a Radon measure ν on Rn tempered, iff it is equivalent to the dν Lebesgue measure dx and the Radon–Nikodym derivatives dx dν and dx are slowly increasing almost everywhere. A Radon measure on a polynomial manifold Rn will be called tempered, if each pushforward by a polynomial chart is tempered.

Gelfand Triples for Homogeneous Lie Groups

63

Definition 2.3 Suppose M is a polynomial manifold and ν a tempered measure on M. Then G(M, ν) is defined to be the Gelfand triple S(M) → L2 (M, ν) → S (M),

(36)

equipped with the real structure defined by the usual complex conjugation ϕ → ϕ. If f : M1 → M is a tempered diffeomorphism, then for each φ ∈ S (M) the pull back ℘f φ(ϕ) := φ(ϕ ◦ f −1 ) is well defined and induces a Gelfand triple isomorphism G(M, ν) → G(M1 , ν ◦ f ).

(37)

Indeed, we defined tempered measures and polynomial manifolds in such a way, that we have a very simple Gelfand-Triple isomorphism G(M, ν)

k -

G(Rn , dx),

(38)

j =1

given by pullbacks and multiplications with slowly increasing functions, provided that M is a n-dimensional polynomial manifold with k connected components.

2.1 The Polynomial Manifold R× For us the two most important examples of polynomial manifolds are the half lines R+ and R− . Here the polynomial structure is induced by the chart σ : λ → |λ| − $ 1/|λ|. On R+ the inverse reads σ −1 (y) = (y + y 2 + 4)/2. Lemma 2.1 If we extend each function in S(R± ) by zero to the whole real line, then S(R± ) = {ϕ ∈ S(R) | ϕ ≡ 0 on R∓ }

(39)

and S(R± ) carries the subspace topology with respect to S(R). Proof We will prove the statement for the R+ case, for R− the proof is analogous. Since σ is a polynomial diffeomorphism from R+ to R, the map ϕ → ϕ ◦ σ

(40)

is a linear homeomorphism between D(R) and D(R+ ) resp. between S(R) and S(R+ ). Hence D(R+ ) is dense in S(R+ ). Let us define S+ (R) := {ϕ R+ | ϕ ∈ S(R) with ϕ ≡ 0 on R− },

(41)

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J. Brinker and J. Wirth

equipped with the subspace topology with respect to S(R). Let f : R → [0, 1] be smooth, such that supp f ⊂ R+ and f ≡ 1 on [1, ∞). For each ϕ ∈ S+ (R) and α ∈ N0 we have ∂ α ϕ(x) = o(x N ) for x → 0 of arbitrary high order N ∈ N. Hence each α, β ∈ N0 there is some C1 , C2 > 0 and N > α with sup |x β ∂xα (f (n x) ϕ(x) − ϕ(x))|

x∈R+

≤ C1

0 n almost the same inequality as above can be used. We assume now n ≥ k. For 0 < x < 1 and each m ∈ N |ϕ(x)/x m | = |

1 xm



x 0

1 (x − t)m−1 (m) ϕ (t) dt| ≤ sup |ϕ (m) (y)|. (m − 1)! m! y

(46)

Gelfand Triples for Homogeneous Lie Groups

65

Hence

n! 1 x k−j −1 |ϕ (n−j ) (x)| |x k ∂xn ( ϕ(x))| ≤ x (n − j )! n

j =0

≤

n

j =0

≤

n

j =0

n! x −n−1 |ϕ (n−j ) (x)| (n − j )!

(47)

1 sup |ϕ (2n+1−j )(y)|, (n − j )!(n + 1) y

1 for all 0 < x < 1, n ≤ k and ϕ ∈ D(R+ ). In conclusion m ∈ L(S+ (R)) and thus also B ∈ L(S+ (R)). Due to the continuity of A and B we arrive at

S+ (R) → S(R+ ),

(48)

i.e. the S+ (R)-topology is finer than the S(R+ )-topology. For the reverse embedding we will transport our situation to the whole real line by ϕ → ϕ ◦ σ −1 ,

(49)

which is an isomorphism D(R+ ) → D(R) and S(R+ ) → S(R). We denote the image of S+ (R) by this map by S⊕ (R) and equip it with the transported S+ (R)topology. Then S⊕ (R) is a space of smooth functions on R with D(R) → S⊕ (R) → S(R),

(50)

where both embeddings are dense. The topology in S⊕ (R) is induced by seminorms of the form S⊕ (R) → R : ϕ → sup |C k E j ϕ(y)|,

k, j ∈ N0 ,

(51)

y∈R

where Cϕ := (∂(ϕ ◦ σ )) ◦ σ −1 and Eϕ := (m(ϕ ◦ σ )) ◦ σ −1 = σ −1 · ϕ. The operator C can be rewritten as   2 $ Cϕ(y) = 1 + ϕ  (y) =: ψ(y) · ϕ  (y), ϕ ∈ S⊕ (R), y ∈ R. 2 2 (y + y + 4) (52)

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Because σ −1 , ψ ∈ OM (R), both C and E have extensions in L(S(R)). Thus S⊕ (R) = S(R) and finally S+ (R) = S(R+ ).

The most important property of S(R± ) (next to being a closed subspace of S(R)) is stated in the following corollary. Corollary 2.1 The map x → |x|v is in OM (R± ) for each v ∈ R. 1 Proof The continuity m was already shown in the proof to the last lemma with inequalities (47). Of course mϕ(x) := xϕ(x) defines a continuous operator on S(R± ), as well. The derivatives of x → |x|v can be bounded by terms of the form x → x k for k ∈ Z, which concludes the proof.



We now find a characterisation for the functions in OM (R± × M; E). This space will be of importance later on, when we examine the Fourier image of S(G) in further detail and when we want to discuss the integral formula for the Kohn– Nirenberg quantization. Corollary 2.2 A smooth function f : R± × M → E is in OM (R± × M; E), iff for each k ∈ N0 , each P ∈ DiffP (M) and each continuous seminorm p on E, there exists an l ∈ N and an q ∈ P(M), such that p(∂λk Px f (λ, x)) ≤ (1 + |λ|l + |λ|−l )q(x). Proof We know, that OM (R+ × M; E) is the space of all smooth functions f on R+ , such that [ϕ → f · ϕ] ∈ L(S(R± × M), S(R± × M; E)).

(53)

We prove the statement for R+ , then the other statement follows at once, since R− is isomorphic to R+ by x → −x. Also, it is enough to consider M = Rn , as the more general case follows by just using polynomial coordinate charts. Suppose f ∈ OM (R+ × Rn ; E). Because f induces a continuous multiplication operator and because S(R+ ) is a subspace of S(R), for each k ∈ N0 , α ∈ Nn0 and each continuous seminorm p on E, there is some m ∈ N and C > 0 with sup

λ∈R+ ,x∈M

p(∂λk ∂xα (f (λ, x)ϕ(λ, x))) ≤ C max

sup

|β|,l≤m λ∈R+ ,x∈M

(1 + |λ|m )(1 + |x|2 )m |∂λl ∂xβ ϕ(λ, x)|, (54)

for all ϕ ∈ S(R+ × Rn ). We choose some ϕ ∈ S(R+ × Rn ), such that ϕ ≡ 1 on some neighbourhood around (λ, x) = (1, 0), and define ϕa,y (x) := ϕ(xa −1, x − y)

Gelfand Triples for Homogeneous Lie Groups

67

for a > 0, y ∈ Rn . Then p(∂ (k,α)f (a, y))

 = p(∂λk ∂xα (f (λ, x)ϕa,y (λ, x)))(λ,x)=(a,y)

≤ C max

sup

(1 + |λ|m )(1 + |x|2 )m |∂λl ∂xβ ϕa,y (λ, x)|

= C max

sup

a −l (1 + |aλ|m )(1 + |x + y|2 )m |∂λl ∂xβ ϕ(λ, x)|

|β|,l≤m λ∈R+ ,x∈Rn |β|,l≤m λ∈R+ ,x∈Rn

≤ C  (1 + a m + a −m )(1 + |y|2 )m , (55) where k, α, m and C are as above. Of course this implies, that for each k ∈ N0 , P ∈ DiffP (Rn ) and each continuous seminorm p on E, there exists an l ∈ N and a q ∈ P(Rn ), such that p(∂λk Px f (λ, x)) ≤ (1 + |λ|l + |λ|−l )q(x).

(56)

Now for the converse implication. Let f : R+ ×Rn → C be any smooth function, such that for p, k and P we find m and q for the inequality (56). Then for arbitrary ϕ ∈ S(R+ × M), sup

λ∈R+ ,x∈Rn

(1 + |λ|k )(1 + |x|2)k p(∂ α (f ϕ)(λ, x)) ≤C

sup

λ∈R+ ,x∈Rn

(1 + λk )(1 + |x|2)k



|∂ α−β f (λ, x) ∂ β ϕ(λ, x)|

β≤α

≤ C  sup (1 + |x|2 )k+m (1 + λk+m + λk−m ) x∈R+



|∂ β ϕ (j ) (x)|.

β≤α

(57) 1 Since m is a continuous operator on S(R+ ), the last line defines a continuous seminorm on S(R+ × Rn ). Thus the operator ϕ → f · ϕ is continuous.



From the polynomial structures on R+ and R− , we construct the polynomial manifold R× = R+ ∪˙ R− . Its Schwartz space S(R× ) = S(R+ ) ⊕ S(R− ) can be seen as the closed subspace of S(R) of functions f , which vanish of arbitrary order in 0, i.e. ∂ k f (0) = 0 for all k ∈ N0 . The dual space and the Fourier image of S(R× ) will play an important role in the coming discussion. The first statement requires no further proof.

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Lemma 2.2 The image of S(R× ) under the Fourier transform on R, FR , is S∗ (R), which is defined to be the subspace of Schwartz functions f with vanishing moments of arbitrary order, i.e.  R

f (x) p(x) dx = 0,

for all p ∈ P(R).

(58)

The next Lemma is less obvious. It is an extension of the well know fact, that S∗ (R), as a vector space, can be identified with the quotient S (R)/P(R) e.g. [18, Proposition 1.1.3]. Lemma 2.3 Let E be a nuclear Fréchet space and E0 (R) the space of distributions on R with support in {0}. Then ˆ E) (S(R× ) ⊗

ˆ E  )/(E0 (R) ⊗ E  ), (S (R) ⊗

(59)

ˆ E. especially E0 (R) ⊗ E  is a closed subspace of S (R) ⊗ Proof First we will prove, that Z := E0 (R) ⊗ E  is a closed subspace of X ˆ E  , where X := S(R) ⊗ ˆ E. The family (∂ k δ0 )k∈N0 is a basis for E0 (R) S (R) ⊗ where δ0 is the delta distribution. We use Lemma 1.1 on the sequence PN of projections onto the subspaces spanned by {δ0 , . . . , ∂ N δ0 } and conclude, that Z is sequentially dense in its closure Z. Furthermore we realize, that for any φ ∈ Z there is a sequence (ek ) ⊂ E  , such that φ = lim φN := lim N→∞

N→∞

N

(∂ k δ0 ) ⊗ ek .

(60)

k=0

Because X is a Fréchet space and Z ⊂ X , we can apply the Banach-Steinhaus Theorem. Hence there exists a continuous seminorm q on E and M ∈ N, such that |φN (f )| ≤ max supxM q(∂xk f (x)) k≤M x∈R

(61)

ˆ E and all N ∈ N. for all functions f ∈ X = S(R) ⊗ Now suppose there is one l > M, such that el = 0. Let us define the sequence of Schwartz functions fm (x) := eimx ψ(x)e/ml−1 , where ψ is a Schwartz function equal to one near zero and e ∈ E with el (e) = 1. We arrive at  l   (im)k   m→∞  |φl (fm )| =  e (e) −−−−→ ∞. m(l−1) k  k=0

(62)

Gelfand Triples for Homogeneous Lie Groups

69

But also sup max supxM q(∂xk fm (x)) < ∞,

m∈N k≤M x∈R

(63)

which is a contradiction. Hence φ ∈ Z, i.e. φ is in the finite span of the ∂ k δ and ek . Now let Y := Z ◦ be the polar of Z. Because X is reflexive, we may identify Y ⊂ X. Since Z is a closed subspace, we also have Y ◦ = Z ◦◦ = Z. Since ∂ k δ0 ⊗ e ∈ Z for all k ∈ N0 , e ∈ E  and (∂ k δ0 ⊗ e )(ϕ) = e (∂ k ϕ(0)),

for ϕ ∈ X = S(R; E),

(64)

ˆ E. it is quite obvious, that Y = S(R× ) ⊗ Since E is a nuclear Fréchet space, X is a nuclear Fréchet space. That also means, that X is an (FS) space. I.e. it is the projective limit X1 ← X2 ← · · · ← X

(65)

of a sequence of Banach spaces (Xk )k with compact maps Xk ← Xk+1 [11, Chapter 3, Corollary 3 to Theorem 7.3]. Notice that the maps Xk ← Xk+1 are weakly compact, too. Now we may conclude the proof, by using Theorem 13 of [19]. The theorem states, that in our situation—Y is closed and X is an (FS) space—we have Y  X /Y ◦ .

By using the Euclidean Fourier transform in combination with the last lemma, we get the following corollary. Corollary 2.3 Let E be a nuclear Fréchet space, then ˆ E) (S∗ (R) ⊗

ˆ E  )/(P(R) ⊗ E  ) (S (R) ⊗

(66)

ˆ E. and P(R) ⊗ E  is closed in S (R) ⊗ ˆ E and Furthermore, this characterization for the dual spaces of S(R× ) ⊗ ˆ E by quotient spaces, enables us to find subspaces of S (R) ⊗ ˆ E  which S∗ (R) ⊗ are embedded into these dual spaces. Suppose F is a Banach space, such that there is a continuous embedding E → F with dense range. Then we may see, that ˆ E  and into the Lebesgue-Bochner spaces Lp (R; F  ) are embedded into S∗ (R) ⊗  × ˆ  S (R ) ⊗ E for p ∈ (1, ∞). Here we define the distribution corresponding to f ∈ Lp (R; F  ) by  Tf (ϕ) :=

R

f (x), ϕ(x) dx,

ϕ ∈ S(R; E),

(67)

where ·, · denotes the dual pairing on F  ×F . Notice, that Tf is indeed an injective map into S (R; E  ), since f = 0 almost everywhere, iff Tf (ϕ ⊗ e) = 0 for all ϕ ∈ S(R) and all e ∈ E.

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Though, we can make a much more general claim. For this purpose, we define ˆ E. the following subspaces of S (R) ⊗ |x|→∞ ˙  (R; E  ) := {φ ∈ S (R) ⊗ ˆ E  | ∀ϕ∈S(R) ⊗ˆ E φ(ϕ(· − x)) −−−−→ 0} B

(68)

λ→0  . ˆ E  | ∀ϕ∈S(R) ⊗ˆ E φ(ϕ(λ−1 ·)) −−− → 0} B (R; E  ) := {φ ∈ S (R) ⊗

Lemma 2.4 Let F be a Banach space as described above. The Lebesgue-Bochner ˙  (R; E  ) for p ∈ [1, ∞) and a subspace of space Lp (R; F  ) is a subspace of B   . B (R; E ) for p ∈ [1, ∞] with respect to the embedding f → Tf . ˆ E then also Proof Let f ∈ Lp (R; F  ) and let ϕ ∈ S(R) ⊗ [x → (1 + x 2 ) ϕ(x)] ∈ Lq (R; F )

(69)

for each 1 = 1/p+1/q. Suppose first p ∈ [1, ∞), then for some C > 0 independent of x ∈ R 

 |Tf (ϕ(· − x))| ≤

R

|f (y), ϕ(y − x)| dy ≤ C

p

R

f (y)F  dy (1 + (x − y)2 )p

1

p

. (70)

Now let ε > 0 be arbitrary and let R > 0 be big enough, such that 

p

f (y)F  dy ≤ ε,

(71)

|y|≥R

With this inequality, we get 

p f (y)F  R

(1 + (x − y)2 )p

1

p

dy

⎛ ⎜ ≤ ⎝ε +



|y|≤R

⎞1 p f (y)F 

(1 + (x − y)2 )p

p

⎟ x→±∞ 1 dy ⎠ −−−−→ ε p . (72)

˙  (R; E  ), because ε > 0 can be arbitrarily small. With the same Hence Tf ∈ B calculation as before, we get ⎛ ⎜ |Tf (ϕ(·/λ))| ≤ C ⎝ε +



|y|≤R

⎞1 p f (y)F  (1 + (y/λ)2 )p

p

1 ⎟ λ→0 dy ⎠ −−−→ Cε p .

(73)

Gelfand Triples for Homogeneous Lie Groups

71



Thus Tf ∈ . B (R; E  ). Now suppose p = ∞. Here we have  |Tf (ϕ(·/λ))| ≤ λ ess sup f (x)F  x∈R

λ→0

R

ϕ(y) dy −−− → 0.

(74)



B (R; E  ) for this case. Hence also Tf ∈ .





˙ (R; E  ) can have any form in a bounded region, Note, that the distributions in B  whereas distributions in . B (R; E  ) can have any form away from zero, as long as they are tempered. Proposition 2.1 The quotient maps ˆ E  → S (R× ) ⊗ ˆ E, S (R) ⊗ ˆ E  → S∗ (R) ⊗ ˆ E, S (R) ⊗

(75)

restrict to embeddings  . ˆ E, B (R; E  ) → S (R× ) ⊗

˙  (R; E  ) → S∗ (R) ⊗ ˆ E. B

(76)

Proof A short calculation yields  . ˙  (R; E  ) ∩ P(R) ⊗ E  . B (R; E  ) ∩ E0 (R) ⊗ E  = {0} = B

(77)

Together with the above lemma and corollary, this already concludes the proof.



2.2 Flat Orbits of Homogeneous Lie Groups Let Ad be the adjoint action of G on g. Denote by Cax ξ := ξ ◦ Adx −1 the coadjoint action of x ∈ G on linear functionals ξ ∈ g . A subalgebra m ⊂ g is called polarizing to  ∈ g , iff ([m, m]) = {0} and m is a maximal algebra fulfilling this condition. For any ξ ∈ g we can find at least one polarizing algebra. There is a bijection between the coadjoint Orbits and the irreducible unitary representations of G. It can be described by [π] ↔  = CaG ξ , where π is unitarily equivalent to the induced representation of χ(m) = e2πiξ(m) for m ∈ m ⊂ G for some maximal subordinate algebra m of  [9, Theorems 2.2.1–2.2.4]. This correspondence only depends on the orbit  and not on the choice of element ξ spanning  or the choice of polarizing algebra m. We will write π ∼ ξ or π ∼ , if the equivalence class  with the initial topology of π corresponds to the orbit  = CaG ξ . We equip G  to g /G. For any ξ the with respect to the bijection [π] →  for π ∼  from G

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orbit  = CaG ξ is an even dimensional polynomial manifold [6, page 521] and [9, Lemma 1.3.2]. A Jordan–Hölder basis of g, is a basis (ej )j , such that the linear hull gk = span{e1 , . . . ek }, is an ideal in g for each k ≤ dim G. Let qk be the quotient map g → g /g◦k . The set of jump indices J is the set of j > 1, such that dim qj () − dim qj −1 () = 1

(78)

Let us denote gJ := span{ej | j ∈ J }. From Corollary 3.1.5 of [9] follows, that a polynomial chart of  is given by σ :  → gJ : ξ → ξ gJ .

(79)

This equivalence between orbits and the corresponding subspaces gJ , leads to the definition of the orbital Fourier transform as the integral 

e−2πiξ(x)ϕ(ξ ) dθ (ξ ),

F ϕ(x) :=

x ∈ gJ , ϕ ∈ S(),

(80)



where θ ◦ σ−1 is a Haar measure on gJ . The Pedersen quantization [20] is the equivalent of the Weyl quantization for general simply connected nilpotent Lie groups. It is defined by the integral  opπ (ϕ) :=

 gJ

π(x)

e−2πiξ(x)ϕ(ξ ) dθ (ξ ) dν (x),

(81)



for some representation π ∼  and a fitting Haar measure ν on gJ . We can easily see, that the outermost integral converges in L(Hπ ). The following theorem fixes the choice of ν . Theorem 2.1 For each θ as above, there is a unique ν , such that the Pedersen quantization to π ∼  extends to a Gelfand triple isomorphism opπ : G(, θ ) → Gop (π).

(82)

Proof This is essentially stated in [20, Theorem 4.1.4]. Here Pedersen proves, that S() → B(Hπ )∞ : a → opπ (a)

(83)

is a homeomorphism, where B(Hπ )∞ is the space of smooth operators with respect to π. The spaces of smooth operators is defined to be B(Hπ )∞ = H∞ , where  is

Gelfand Triples for Homogeneous Lie Groups

73

the unitary representation of G × G on HS(Hπ ) defined by (x, y)T = π(x) ◦ T ◦ π(y)−1 . Furthermore Pedersen shows that  a b dθ = Tr[opπ (a) opπ (b)∗ ], for a, b ∈ S() (84) 

for a suitable choice of ν . In order to fit this result in our scheme we will make sure, that L(Hπ−∞ , Hπ∞ ) = B(Hπ )∞ as topological vector spaces. It is easy to see, that this identity holds in the sense that each T → T ◦ I is a bijection from the left-hand side to the righthand side, where I : Hπ → Hπ−∞ is the embedding defined by the real structure on G(π). It is left to check that the topologies on both sides coincide. For P ∈ u(gL ) denote by π(P )∗ ∈ L(Hπ−∞ ) the unique operator fulfilling Iπ(P )v = π(P )∗ Iv for v ∈ Hπ∞ . By Cartier [21] there is P ∈ u(gL ), such that π(P ) is invertible on Hπ∞ , π(P )−1 can be extended to a nuclear operator on Hπ and Hπ−∞ is the compact inductive limit of the Hilbert spaces Hπ−k := π(P )k∗ IHπ , equipped with the norm w−k := I−1 π(P )−k wHπ . But this means each bounded B ⊂ Hπ−∞ is in fact a bounded set in some Hπ−k , i.e. B = π(P )k∗ IB˜ for B˜ bounded in Hπ . The topology in L(Hπ−∞ , Hπ∞ ) is defined by the seminorms T → sup π(P )j T φ , φ∈B

for B ⊂ Hπ−∞ bounded and j ∈ N0 .

(85)

The above implies, that actually it is enough to consider the seminorms T → π(P )j T I π(P )k  ,

for j, k ∈ N0 .

(86)

Now the above seminorms are also continuous on B(Hπ )∞ , so by the open mapping theorem for Fréchet spaces, L(Hπ−∞ , Hπ∞ ) = B(Hπ )∞ as topological vector spaces and S() → L(Hπ−∞ , Hπ∞ )∞ : a → opπ (a)

(87)

is a homeomorphism. Finally by the dense and continuous embeddings S() → L2 (, θ )

and L(Hπ−∞ , Hπ∞ ) → HS(Hπ )

(88)

we may extend opπ to a unitary operator between L2 (, θ ) and HS(Hπ ), and as such, even to a Gelfand triple isomorphism opπ : G(, θ ) → Gop (π).

(89)

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J. Brinker and J. Wirth

Note, that Pedersen uses the convention ξ ↔ χ(·) = eiξ(·) for bijection between functionals and characters. Though adjusting the formulas just results in additional constants, that may be hidden away inside the measures ν and θ .

Though in our case, we can simplify this process by a lot, since we are only interested in representations derived from generic resp. flat orbits. An orbit is called generic, if for each k the dimension of qk () is maximal compared to all other orbits. Let us denote the set of equivalence classes derived gen ⊂ G.  Note, that the Plancherel measure  from generic orbits by G μ is  concentrated on Ggen . A representation π ∈ Irr(G) is square integrable modulo the center, if x → |(π(x)v, w)Hπ | is square integrable on g/z with respect to the Haar measure for all v, w ∈ Hπ . Let us denote the set of irreducible representations, that are square integrable modulo the center, by SI/Z(G) ⊂ Irr(G) and pairs of such representations together with some matching real structure by SI/ZR (G). Suppose π ∼  = CaG ξ , then π ∈ SI/Z(G), if and only if  = ξ + z◦ [22]. Furthermore, if SI/Z(G) = ∅, then the orbits to representations in SI/Z(G) are exactly those having the maximal possible dimension [9, Corollary 4.5.6]. Also, the jump indices for π ∈ SI/Z(G) are given by J = {k + 1, k + 2, . . . , dim G}, where k = dim z,  is uniquely determined by the central character and the equivalence class [π] ∈ G 2πiξ(·) π z = e idHπ , where ξ ∈ ω◦ z . For this fact see [9, Corollaries 4.5.3 and 4.5.4]. For all π ∈ SI/Z(G) the Pedersen quantization is simpler, for we can just take one Haar measure θ on z◦ and translate it to a measure θ on  ∼ π for each π ∈ SI/Z(G). The subspace ω := gJ complements z in g and is the same for each representation in SI/Z(G). We get a Gelfand triple isomorphisms G(z◦ , θ ) → G(, θ ) : φ → φ ◦ Pz◦ ,

(90)

where Pz◦ is the projection onto z◦ along ω◦ . Using this isomorphism, we adjust the Pedersen quantization. Definition 2.4 We will use the Pedersen quantization opπ on G(z◦ , θ ) with respect to π ∈ SI/Z(G), defined by opπ : G(z◦ , θ ) → Gop (π), φ → opπ (φ ◦ Pz◦ ).

(91)

This version of Pedersen quantization takes on the form 



opπ (ϕ) =

π(x) ω

z◦

e−2πiξ(x)ϕ(ξ ) dθ (ξ ) dν(x),

(92)

where ν = ν depends on θ . Of course opπ is a Gelfand triple isomorphism, as well.

Gelfand Triples for Homogeneous Lie Groups

75

Now we will discuss the concept of generic orbits and square integrable (modulo the center) representation in context with homogeneous groups. The Lie group G = g is called a homogeneous Lie group, if it is equipped with a group of dilations (0, ∞) → Hom(G) : λ → δλ ,

(93)

where δλ x = elog(λ)A x is also a Lie algebra isomorphism and A is a diagonalizable map with positive eigenvalues. The number Q := Tr[A] is the homogeneous dimension of G. We may always decompose g into eigenspaces Eκ of A to Eigenvalues κ > 0, i.e. g= Eκ , where [Eκ , Eκ  ] ⊂ Eκ+κ  . (94) κ>0

Notice that the center z of g is always an eigenspace to both δλ and A, since [δλ z, x] = δλ [z, δλ−1 x] = 0

for all λ > 0, z ∈ z and x ∈ g.

(95)

/ For every μ > 0 the space κ≥μ Eκ is an ideal in g. We may always choose a Jordan–Hölder basis (ej )j through these ideals [9, Theorem 1.1.13]. If dim z = 1, then the center fulfils z = Eμ for μ = max{κ > 0 | Eκ = {0}}. Hence, one vector of our chosen Jordan–Hölder basis of eigenvectors will always lie in the center z. We also have the unique decomposition g = z ⊕ ω,

ω is A-invariant.

(96)

Now for λ < 0 denote δλ x := −δ|λ| x for x ∈ z,

and δλ x := δ|λ| x for x ∈ ω.

(97)

Furthermore, let also δλ ξ := ξ ◦ δλ for λ ∈ R× and ξ ∈ g . The question arises whether generic orbits are mapped to generic orbits by δλ . The dilation δλ on g /g◦k is a well defined vector space isomorphism by δλ ◦ qj := qj ◦ δλ , since gk and thus also g◦k are δλ -invariant. Furthermore dim qj (δλ ) = dim δλ ◦ qj () = dim qj ().

(98)

Thus δλ  is generic for each λ ∈ R× . Now take any π ∈ IrrR (G) with real structure Cπ and define π := Cπ πCπ ∈ IrrR (G) equipped with the same real structure. The representation π is equivalent to the dual representation of π. Now denote πλ (x) := π(δλ x) for λ > 0,

and πλ (x) := π |λ| (x) := π(δ|λ| g) for λ < 0. (99)

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J. Brinker and J. Wirth

All the representations πλ are irreducible unitary representations acting on Hπ resp. acting smoothly on H∞ π . With these definitions and the discussion above, we get the equivalence of the three statements • π ∈ SI/Z(G), if and only if πλ ∈ SI/Z(G), gen , if and only if [πλ ] ∈ G gen , • [π] ∈ G • π ∼ ξ , if and only if πλ ∼ δλ ξ . Suppose that SI/Z(G) = ∅ and dim z = 1. Furthermore, suppose we chose a Jordan–Hölder basis of eigenvectors to A. Let π ∈ SI/Z(G). As every equivalence gen only depends on its central character, we get a class of representations in G ×  gen . bijection between R and Ggen . Hence, π ∈ SI/Z(G), if and only if [π] ∈ G  We can even go one step further. The dilations δλ help us to understand G as measure space. For this purpose we need the Pfaffian Pf (ξ ) to a coadjoint orbit  = CaG ξ , which is defined by Pf (ξ )2 = det Bξ up to a sign. Here Bξ := (ξ([ej , ei ]))j,i∈J where the (ej )j ∈J span ω. We define κ > 0 and B ∈ L(ω), by δλ η := sgn(λ)|λ|κ η, for η ∈ ω◦ and A|ω = B. Proposition 2.2 Suppose G is a homogeneous group, π ∈ SI/ZR (G) and dim z = 1, then gen ,  (R× , κ|λ|Q−1 |Pf ()| dλ) → (G μ) : λ → [πλ ],

(100)

where π ∼  ∈ ω◦ , is a homeomorphism resp. a strict isomorphism between the Borel measure spaces. Furthermore, if  is a fixed generic orbit, then λ → δλ  defines a bijection between R× and the generic orbits. Proof Let U be the Zariski open set of functionals ξ ∈ g , such that CaG ξ is a generic orbit with respect to our basis. For ξ ∈ U we have δλ ξ ∈ U for each λ ∈ R× by Eq. (98). Each orbit meets U ∩ ω◦ in exactly one point [9, Theorem 3.1.9 and Theorem 4.5.5]. Furthermore, for any ξ ∈ ω× := ω◦ \ {0}, we have that R× → ω× : λ → δλ ξ

(101)

is a homeomoprhism. Thus also U ∩ ω◦ = ω× = {δλ  | λ ∈ R× }. But ω× also induces all maximal flat orbits, so they coincide with the generic orbits. Since the  is a homeomorphism, we also have U/G G gen with correspondence g /G G respect to the subspace topologies. Let q : U → U/G be the quotient map. Now q|ω× is a continuous bijection. We show, that it is also open. By [9, Theorem 3.1.9], there is a well define map ψ : ω× × z◦ → U , such that ψ(u, v) = w

⇔

w ∈ CaG u and Pz◦ w = v,

(102)

where Pz◦ is the projection onto z◦ along ω◦ . The map ψ is a rational, non singular bijection with rational non singular inverse. Hence ψ is a homeomorphism. If

Gelfand Triples for Homogeneous Lie Groups

77

V ⊂ ω× is open in ω× , then CaG V is open in U , since ψ(V × z◦ ) = CaG V .

(103)

Now, since q is open and q(CaG V ) = q(V ), the restriction q|ω× is an open map and thus a homeomorphism. If we now denote gen : λ → [δλ π], σ : R× → G

(104)

 → [0, ∞) be Borel then σ is a homeomorphism by the discussion above. Let ϕ : G measurable. Then by Theorem 4.3.10 and the subsequent discussion in [9] 

  G

ϕ([π]) d μ([π]) =

U ∩ω◦

ϕ([πξ ])|Pf (ξ )| d. μ(ξ ),

(105)

where . μ is the measure on U ∩ ω◦ , such that {t | t ∈ [0, 1]} has measure equal to one and πξ ∼ CaG ξ . Also, since our chosen Jordan–Hölder basis is an eigenbasis to A resp. δλ , we have 1

|Pf (δλ )| = | det(δλ ([ej , ei ]))j,i | 2

= | det(|λ|ni +nj ([ej , ei ]))j,i | 2 = |λ|Tr B |Pf ()|, 1

(106)

where |λ|nj is the eigenvalue of ej to δλ for j ∈ J . Both σ and σ −1 are measurable and we have d(. μ ◦ σ )(λ) = κ|λ|κ−1 dλ. Hence   ϕ([πξ ])|Pf (ξ )| d. μ(ξ ) = ϕ([πλ ])|Pf (δλ )| d(. μ ◦ σ )(λ) U ∩ω◦

 =

R× R×

(107) ϕ([πλ ])κ|λ|−1+Tr A |Pf ()| dλ

and σ is a strict isomorphism of measure spaces.



We will denote the Euclidean Fourier transform on g by  Fg ϕ(ξ ) =

g

e2πiξ(x)ϕ(x) dμ(x),

ϕ ∈ S(g), ξ ∈ g .

(108)

Of course, there is exactly one Haar measure μ on g , such that the Fourier transform is a Gelfand triple isomorphism G(g, μ) → G(g , μ ). Suppose  ∈ ω× . The map ℘ f (λ, ξ ) := f (δλ ( + ξ ))

for ξ ∈ z◦ , λ ∈ R× and f : g → C,

(109)

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J. Brinker and J. Wirth

together with the Euclidean Fourier transform and the Pedersen quantization will enable us to describe the group Fourier transform on G (see also [3] for a similar statement).

2.3 The Group Fourier Transform on Homogeneous Groups Let us from now on always denote by G a homogeneous Lie group with dim z = 1 and SI/Z(G) = ∅. Trivially, the group Fourier transform is an isomorphism between  Also, the group Fourier transform is a unitary map from L2 (G, μ) S(G) and S(G). 2  to L (G,  μ). Of course, we may define a Gelfand triple       L2 (G, μ), S (G)), G(G, μ) := (S(G),

(110)

such that FG becomes a Gelfand triple isomorphism. Now we will use the isomorphism from Proposition 2.2 in order to find a new representation of the group Fourier transform on L2 (G). This will be the basis for the definition of our new Gelfand triples and a Gelfand triple isomorphism in the form of an equivalent Fourier transform. Proposition 2.3 Suppose ϕ ∈ S(G) and π ∈ SI/ZR (G) with π ∼  ∈ ω× , then  ) * opπ ℘ Fg ϕ(λ, ·) , FG ϕ(πλ ) = ) * opπ ℘ Fg ϕ(λ, ·) ,

λ > 0, λ < 0.

(111)

Proof First of all, for any ϕ ∈ S(G), we have  FG ϕ(πλ ) = G

λ− Tr A ϕ(δλ−1 x)π(x)∗ dμ(x) = λ− Tr A FG (ϕ ◦ δλ−1 )(π),

(112)

for λ > 0. Also Fg (ϕ ◦ δλ−1 ) = λTr A (Fg ϕ) ◦ δλ ,

(113)

for λ > 0. Notice, that for x ∈ g and z ∈ z, we have x · z = x + z and thus e2πi(z)π(x) = π(z)π(x) = π(z · x) = π(z + x).

(114)

Gelfand Triples for Homogeneous Lie Groups

79

Let μz resp. ν be Haar measures on z resp ω, such that μ = μz ⊗ ν, then by the above calculation   FG ϕ(π) = π(x) e−2πi(z)ϕ(z − x) dμz (z) dν(x) 

z

ω

=

 π(x)

ω

z◦

(115) e−2πiξ(X)Fg ϕ(ξ ) dθ (ξ ) dν(x).

Here θ is the measure associated to ν as described in Definition 2.4. This formula indeed holds pointwise. Hence FG ϕ(πλ ) = λ− Tr A opπ (Fg (ϕ ◦ δλ−1 )) = opπ ((Fg ϕ) ◦ δλ ) ) * = opπ ℘ Fg ϕ(λ, ·)

(116)

for all λ > 0. For λ < 0 we get ) * FG ϕ(πλ ) = FG (π −λ ) = opπ ℘− Fg ϕ(−λ, ·) ,

(117)

since π ∼ −. Now we can conclude the proof, by using δ−λ (− + ξ ) = δλ ( + ξ ), for any ξ ∈ z◦ .

The above proposition (c.f. [3, Theorem 3.3]) shows that the group Fourier transform splits into operators which are easy to handle in the L2 -setting, if dim z =   1. If we use the isomorphism (G, μ) (R× , κ|λ|Q−1 |Pf ()| dλ) then we can see FG as the composition of unitary operators Fg : L2 (G, μ) → L2 (g , μ ), ℘ : L2 (g , μ ) → L2 (R× × z◦ ; κ|λ|Q−1 |Pf ()| dλ dθ (ξ )), ) * Opπ : L2 R× , κ|λ|Q−1 |Pf ()| dλ; L2 (z◦ , θ ) ) * → L2 R× , κ|λ|Q−1 |Pf ()| dλ; HS(Hπ ) , (118) where Opπ = P+ ⊗opπ +P− ⊗opπ , for the projection P± of L2 (R× ) onto L2 (R± ). It is very convenient, that here the operator component emerges as a tensor product factor, which in turn enables us to understand multiplication operators more easily. This motivates us to define the following alternative spaces of test functions.

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J. Brinker and J. Wirth

2.4 The Fourier Transform on S∗ (G) In order to know which function space is a good choice, we will first take a look at the pull back ℘ . Here our earlier discussion of polynomial manifolds comes into play again. Remember that R× is equipped with a polynomial structure defined by R× = R+ ∪˙ R− , i.e. defined by the polynomial structures on R± . Similarly, we − × define g+  , g and g by ◦ g±  := {t + η | t ∈ R± , η ∈ z },

for  ∈ ω× = ω◦ \ {0}

(119)

± ˙ − and g× = g+  ∪ g and equip g with the polynomial structure analogously to the ± one on R , i.e. the polynomial structure induced by the map ◦ g±  → R × z : (t + η) → (t − 1/t, η).

(120)

Then δλ induces a tempered diffeomorphism, as written in the following Lemma. The polynomial structure on g× is just the one induced by its connected components. Notice that we just have g× = g \ z◦ as a set. Lemma 2.5 Let  ∈ ω× . The Map w : R± × z◦ → g±  : (λ, ξ ) → δλ ( + ξ ) is a tempered diffeomorphism. g+ Proof We prove that R+ × z◦  via w . The proof to the second statement is j analogous. Suppose (ξ )j is the dual basis to our Jordan–Hölder basis (ej )2n j =0 of 2n ◦ eigenvectors. Here (ξj )j =1 is the basis of z . Let κj be the positive number, such that δλ ξ j = λκj ξ j for λ > 0. We use the charts σ resp. σ1 , defined by (λ,

2n

j =1

cj ξ j ) resp. (λ +

2n

cj ξ j ) → (λ − 1/λ, c1 , . . . , c2n ).

(121)

j =1

Then σ1 ◦ w ◦σ −1 (t, c1 , . . . , c2n )  √ 2κ0 (t + t 2 + 4)κ0 − , = √ 2κ0 (t + t 2 + 4)κ0  √ √ (t + t 2 + 4)κ1 (t + t 2 + 4)κ2n c1 , . . . , c2n , 2κ1 2κ2n

(122)

Gelfand Triples for Homogeneous Lie Groups

81

which is a slowly increasing function. Similarly σ ◦ w−1 ◦σ1−1 (t, c1 , . . . , c2n ) ⎛ 1 1 √ 2 κ0 (t + t 2 + 4) κ0 ⎝ − , = 1 1 √ 2 κ0 (t + t 2 + 4) κ0 (t +

√ t 2 + 4) 2

−κ1 κ0

−κ1 κ0

c1 , . . . ,

(t +

√ t 2 + 4) 2

−κ2n κ0

−κ2n κ0

⎞

(123)

c2n ⎠



is slowly increasing. By Lemma 2.1, we can see S(g±  ) as the space ∓  S(g±  ) = {ϕ ∈ S(g ) | ϕ ≡ 0 on g },

(124)

equipped with the subspace topology in S(g ). The tempered diffeomorphism from the last lemma induces a Gelfand triple isomorphism. Lemma 2.6 The pullback ℘ f := f ◦ w defines a Gelfand triple isomorphism  ± Q−1 dλ) ⊗ G(z◦ , θ ), ℘ : G(g±  , μ ) → G(R , κ|Pf ()| |λ|

(125)

where κ, |Pf ()| are the constants introduced in Proposition 2.2 and the preceding remarks and Q is the homogenous dimension of G. ± ± Proof We take an arbitrary f ∈ Cc (g±  ). Define ω := R · , then



 R±

z◦

f (δλ ( + ξ ))κ|Pf ()| |λ|Q−1 dλ dθ (ξ ) 

 =  =  =

R±

ω±

g± 

z◦

f ((δλ ) + ξ )κ|Pf ()| |λ|κ0 −1 dλ dθ (ξ )



z◦

f (η + ξ )) |Pf ()| dμω◦ (η) dθ (ξ )

f (ξ ) dμ (ξ ). (126)

For the last two lines we used that the measure μω◦ on ω◦ is defined by the Lebesgue measure and  and that θ is defined by μ = |Pf ()| μω◦ ⊗ θ . The rest follows with the fact, that ℘ f = f ◦ w , where w is the tempered diffeomorphism from Lemma 2.5.



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We also proved that the restriction of the Haar measure μ to g±  is actually a tempered measure with respect to our chosen polynomial structure. Now we are ready to define Gelfand triples, with respect to which we get a convenient theory for the group Fourier transform. Definition 2.5 We define the following reduced Schwartz space ˆ S(ω)} S∗ (G) := {ϕ ∈ S(G) | [(λ, x) → ϕ(λz + x)] ∈ S∗ (R) ⊗

(127)

for any choice z ∈ z \ {0}, equipped with the subspace topology in S(G), and the corresponding Gelfand triple G∗ (G, μ) := (S∗ (G), L2 (G, μ), S∗ (G)),

(128)

equipped with the real structure given by the pointwise complex conjugation. Furthermore, we define the Gelfand triple ⎞ S(R× ; π) G(R× ; π) := ⎝L2 (R× ; π)⎠ := G(R× , κ |Pf ()| |λ|Q−1 dλ) ⊗ Gop (π). S (R× ; π) ⎛

(129)

for each SI/ZR (G) ( π ∼  ∈ ω◦ . That G∗ (G) is indeed a Gelfand triple can be seen by using Proposition 2.1. We ˙ ∗ (z; S (ω)), then we see, since use any linear isomorphism R z to define B ˙ ∗ (z; S (ω)) by Lemma 2.4, that the space L2 (G, μ) is embedded L2 (G, μ) ⊂ B  into S∗ (G) = S∗ (z; S (ω)). This embedding is continuous, since the embedding L2 (G, μ) → S (G) is continuous. Of course, the canonical map of S∗ (G) into L2 (G, μ) is a continuous embedding as well. Now the Hahn–Banach theorem implies that both embeddings are also dense, for they are dual to each other. To be more precise, if S∗ (G)⊥ is the polar of S∗ (G) in L2 (G, μ), then it is also the kernel of the dual map L2 (G, μ) → S∗ (G). But this map has a trivial kernel by Lemma 2.4. Hence S∗ (G)⊥ = {0} and S∗ (G) is dense in L2 (G, μ). Now denote by Y the image of L2 (G, μ) in S∗ (G). Since S∗ (G) is reflexive, Y ◦ can be identified with the kernel of the embedding S∗ (G) → L2 (G, μ), which is trivial. Hence Y ⊂ S∗ (G) is dense as well. Notice that G∗ (G, μ) does not depend either on the choice of some π ∈ SI/ZR (G) or some z ∈ z. The Gelfand triple G(R× ; π) does depend on π ∈ SI/ZR (G) but each different choice of π leads to an isomorphic Gelfand triple as the theorem below shows. Theorem 2.2 Let SI/ZR (G) ( π ∼  ∈ ω× . Let the Fourier transform in π-picture, Fπ be defined by Fπ := Opπ ◦ ℘ ◦ Fg ,

(130)

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83

where Opπ = P+ ⊗ opπ + P− ⊗ opπ and P+ = 1 − P− is the projection of S(R× ) onto S(R+ ) along S(R− ). Then Fπ is a Gelfand triple isomorphism Fπ : G∗ (G) → G(R× ; π).

(131)

Proof The proof essentially writes itself by now and is a summary of previous statements. The Euclidean Fourier transform Fg is a Gelfand triple isomorphism between G∗ (g, μ) and G(g, μ ) = G(ω× , |Pf ()| μω◦ ) ⊗ G(z◦ , θ ) by Lemma 2.2, where we choose the Haar measures μω◦ and θ , such that μ = |Pf ()| μω◦ ⊗ θ and μω◦ is induced by the Lebesgue measure dλ via the map R ( λ → λ ∈ ω◦ . By Lemma 2.6, the pull back ℘ is a Gelfand triple isomorphism between G(g× , μg ) and G(R× , κ |Pf ()| |λ|Q−1 dλ) ⊗ G(z◦ , θ ). For the last step we just need to use that Opπ = P+ ⊗opπ +P− ⊗opπ is a Gelfand triple isomorphism between G(R× , κ0 |Pf ()| dλ) ⊗ G(z◦ , μz◦ ) and G(R× ; π) by Theorem 2.1 and Definition 2.4.

Let us now discuss a few properties of S∗ (G) and S(R× ; π). Their duals can be identified with quotient spaces, in particular S∗ (G) S (R× ; π)

S (G)/(P(z) ⊗ S (ω))

and

ˆ L(Hπ−∞ , Hπ∞ )/(E0 (R) ⊗ L(Hπ−∞ , Hπ∞ )), S(R× ) ⊗

(132)

by Lemma 2.3 and Corollary 2.3. By employing Proposition 2.1, we can identify a large space of distributions on G resp. R that are embedded into S∗ (G) resp. ˙  (z; S (ω)) by using any isomorphism R z, then S (R× ; π). I.e. if we define B ˙  (z; S (ω)) → S∗ (G) B

 and . B (R; L(Hπ∞ , Hπ−∞ )) → S (R× ; π).

(133)

We may, for example, identify Lp (G, μ), for p ∈ [1, ∞), and also S(G) as a sub˙  (z; S (ω)) and the Bochner–Lebesgue spaces Lp (R, dλ; L(Hπ )), for spaces of B  p ∈ (1, ∞], and also S(R; L(Hπ−∞ , Hπ∞ )) as subspaces of . B (R; L(Hπ∞ , Hπ−∞ )).  × × The definition of S(R ; π) and S (R ; π) enables us to define a multiplication with a large class of smooth functions via Theorem 1.2. Proposition 2.4 For any π ∈ SI/Z(G), the multiplications OM (R× ; L(Hπ∞ )) × S(R× ; π) : (f, ϕ) → f ϕ OM (R× ; L(Hπ−∞ )) × S(R× ; π) : (f, ϕ) → ϕ f,

(134)

defined pointwise by composition of operators in L(Hπ−∞ , Hπ∞ ), L(Hπ∞ ) and L(Hπ−∞ ), are hypocontinuous bilinear maps.

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Proof We just need show that we may apply Theorem 1.2. The compositions of operators L(Hπ∞ ) × L(Hπ−∞ , Hπ∞ ) : (A, B) → A B L(Hπ−∞ ) × L(Hπ−∞ , Hπ∞ ) : (A, B) → B A

(135)

are hypocontinuous, since separately continuous maps on barrelled spaces are hypocontinuous. Since Hπ∞ S(Rn ) and Theorem 1.2, the above operator spaces are barrelled by [14, Corollaire 2 on page 128 of chapter 2] and [11, Corollary to 8.4 in chapter 2]. Also, the multiplication of slowly increasing functions and Schwartz functions is hypocontinuous. This follows directly from the definition and comments on pages 243 and 244 of [17]. Now we just need to remind ourselves, that S(R× ; π) is a tensor product of nuclear Fréchet spaces.

Now, we will prove the analogous result for the multiplication with the operator valued tempered distributions S (R× ; π). As we used in the proof above, Hπ∞ is reflexive for any π ∈ SI/Z(G). Thus, by using the transpose, we get the two isomorphisms of topological vector spaces L(Hπ∞ ) ( A → At ∈ L(Hπ−∞ )

and L(Hπ−∞ ) ( B → B t ∈ L(Hπ∞ ). (136)

Denote for f in OM (R× ; L(Hπ∞ )) or in OM (R× ; L(Hπ−∞ )) the operator valued function f t (λ) := f (λ)t . Then we may define multiplications on S (R× ; π) by (f φ)(ϕ) := φ(f t ϕ)

and (φ g)(ϕ) := φ(ϕ g t ),

(137)

for all φ ∈ S (R× ; π) and ϕ ∈ S(R× ; π), if we choose f ∈ OM (R× ; L(Hπ−∞ )) and g ∈ OM (R× ; L(Hπ∞ )). We get the following corollary. Corollary 2.4 For any π ∈ SI/Z(G), the multiplications OM (R× ; L(Hπ∞ )) × S (R× ; π) : (f, φ) → f φ, OM (R× ; L(Hπ−∞ )) × S (R× ; π) : (f, φ) → φ f

(138)

are hypocontinuous. Proof This follows directly from the definition of the multiplication and the fact that the dual pairing is hypocontinuous. Equivalently, we could also directly employ Theorem 1.2.

Let us now relate the Fourier transform in π picture with the group Fourier ∞ transform. Denote by jπ the map jπ (σ ) := [R× ( λ → σ (πλ ) ∈ L(H−∞ π , Hπ )],  defined on S(G).

Gelfand Triples for Homogeneous Lie Groups

85

Due to Proposition 2.2, we know that jπ has a unitary extension from gen ,  L2 (G μ) onto L2 (R× ; π). Because the Plancherel measure  μ is concentrated    on Ggen [9, Theorem 4.3.16], we can see jπ as a map defined on L2 (G, μ). 2 Proposition 2.3 implies the L -diagram below. L2 (G, μ)

FG Fπ

L2 (G, μ) jπ

L2 (R× ; π)

S(G)

FG

⊂ j0

j∗ ⊂

S∗ (G)

S (G)

S(G)

FG F−1 π

FG

S (G)

j∗

S∗ (G)

S(Ggen ) jπ−1

j0 FG

Fπ

S(R× ; π)

S (Ggen ) (jπ )−1

S

(R× ; π)

For the Schwartz spaces and spaces of tempered distributions we get a very gen ) the image of S(R× ; π) under jπ−1 . The similar diagram. Denote also by S(G 2 commutative diagram for the L -spaces implies the commutative diagram for the Schwartz spaces above. Then, by duality, we get the commutative diagram for the tempered distributions. However, the group Fourier transformations FG are defined by duality on S (G) resp. S∗ (G) and are not the same map, even though we use the same symbol. By Corollary 2.3 the map j∗ can be seen as the quotient map S (G) → S (G)/(P(z) ⊗ S (ω))

S∗ (G),

(139)

which is an open map. This also implies that j0 is surjective and open. Since w : R× × z◦ → g× , w (λ, ξ ) = δλ ( + ξ ) is a tempered diffeomorphism, we can also see ℘ as an isomorphism between OM (g× ) and OM (R× × z◦ ) resp. between S(g× ) and S(R× × z◦ ). However, in order to examine the Fourier image of S(G), it is even better to consider mixed spaces. We equip ω× = R× ·  with the ˆ S(z◦ ) can be seen polynomial structure transported from R× . The space OM (ω× ) ⊗ × ˆ S(z◦ ). as a subspace of OM (g ). In this manner we define ℘ on OM (ω× ) ⊗ Lemma 2.7 The Gelfand-Triple isomorphism ℘ restricts to an isomorphism ˆ S(z◦ ) → OM (R× ) ⊗ ˆ S(z◦ ). ℘ : OM (ω× ) ⊗

(140)

R× and z◦ R2n and z◦ R2n via our basis of Proof We identify ω× eigenvectors to the dilations. As usual, it is enough to consider the R+ -part, since OM (R× ) = OM (R+ ) ⊕ OM (R− ). With these adjustments, we need to exchange ℘ by the map ℘, where ℘g(λ, x) = g(λκ0 , (λκj xj )2n j =1 ).

(141)

First of all, we realize that λ → λκ0 is a tempered diffeomorphism. Hence T ∈ L(OM (R+ )), where T ψ(λ) := ψ(λκ0 ), is an isomorphism.

86

J. Brinker and J. Wirth 2n Now let us define linear isomorphisms fλ (x) = (λκj /κ0 xj )2n j =1 on R . Then it is

easy to see, that both λ → fλ and λ → fλ−1 define functions in OM (R+ ; L(R2n )) with values in Gl(R2n ). Now we denote by Fλ the corresponding operator Fλ ϕ := ϕ ◦ fλ and set F : λ → Fλ resp. F −1 : λ → Fλ−1 . A standard calculation shows that for any continuous seminorm p on L(S(R2n )) and any k ∈ N0 , there is a polynomial q on L(R2n )k+2 , such that p(∂λk Fλ ) ≤ q(fλ−1 , fλ , ∂λ fλ , . . . , ∂λk fλ ).

(142)

Of course, an analogous inequality is valid for F −1 . Hence, we may conclude F, F −1 ∈ OM (R+ ; L(S(R2n ))).

(143)

Here, F −1 is indeed the inverse of F in the algebra OM (R+ ; L(S(R2n ))). Due to Theorem 1.2, we know that the multiplication OM (R+ ; S(R2n )) ( g → F g ∈ OM (R+ ; S(R2n )), with (F g)(λ, x) = Hλ (g(λ, ·))(x), is continuous and in fact an isomorphism. Because ℘g = (T ⊗ 1)(F g), we can conclude that ℘ is an isomorphism.

(144)



Using the above lemma, we may now prove the following continuity property for the Fourier transform in π-picture on S(G). Proposition 2.5 The Fourier transform in π-picture restricts to a continuous map ∞ ˆ L(H−∞ Fπ : S(G) → OM (R× ) ⊗ π , Hπ ).

(145)

Proof This statement follows from the continuity of the maps Fg

℘

 ˆ S(z◦ ) −→ ˆ S(z◦ ), S(G) −→ S(g ) → OM (ω× ) ⊗ OM (R× ) ⊗

(146)

in which we use the continuous inclusion S(ω◦ ) ⊂ OM (ω× ), and also from the continuity of Opπ = P+ ⊗ opπ + P− ⊗ opπ as map ∞ ˆ S(z◦ ) → OM (R× ) ⊗ ˆ L(H−∞ Opπ : OM (R× ) ⊗ π , Hπ ).

(147)



Gelfand Triples for Homogeneous Lie Groups

87

3 Gelfand Triples for the Kohn–Nirenberg Quantization In [7] a pseudo-differential calculus resp. a Kohn–Nirenberg quantization for graded nilpotent Lie groups was developed. We will embed this definition into our context and derive an integral formulation for the Kohn–Nirenberg quantization for a general class of symbols. First, consider the map T : S(G × G) → S(G × G), Tf (x, y) := f (x, xy −1 ).

(148)

Then, it is easy to see that T extends to a Gelfand triple isomorphism T : G(G, μ) ⊗ G(G, μ) → G(G, μ) ⊗ G(G, μ).

(149)

Denote by K the kernel map K : L(G(G, μ), G(G, μ)) → G(G, μ) ⊗ G(G, μ)

(150)

from Lemma 1.2. We may define the Kohn–Nirenberg quantization as the Gelfand triple isomorphism  μ) → L(G(G, μ), G(G, μ)). Op := K−1 T−1 (1 ⊗ F−1 G ) : G(G, μ) ⊗ G(G,  (151) That means for a ∈ L2 (G × G, μ ⊗ μ), we have Op(a) ∈ HS(L2 (G)) and ( Op(a)f, g)L2 (G,μ)   = Tr[a(x, π) ((1 ⊗ FG inv)Tg ⊗ f )(x, π)∗ ] dμ(x) d μ([π])  G G

(152) for all f, g ∈ L2 (G, μ), where inv f (x) := f (−x) and (·, ·)L2 (G,μ) is the inner product in L2 (G, μ). We denote the right translation of functions f by Rx f (y) := f (yx). Because Tr[a(x, π) ((1 ⊗ FG inv)Tg ⊗ f )(x, π)∗ ] ) *∗ = Tr a(x, π) g(x) (FG Rx−1 inv f )(π)

= g(x) Tr a(x, π) (FG inv f )(π)∗ π(x) = g(x) Tr[a(x, π) FG f (π) π(x)]

(153)

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J. Brinker and J. Wirth

 we may write the operator Op(a) as for almost all (x, [π]) ∈ G × G,  Op(a)ϕ =

 G

Tr[π(·) a(·, π) FG f (π)] d μ([π]),

for f ∈ L2 (G),

(154)

where the integral converges in L2 (G, μ). We will now define the Kohn–Nirenberg quantization in the context of the Gelfand triples G∗ (G, μ) and G(R× ; π) for π ∈ SI/ZR (G). Subsequently, we will discuss an integral formula similar to the L2 -case above, but for a different class of symbols.

3.1 The Kohn–Nirenberg Quantization for Operators Defined on S∗ (G) We still take G to be a homogeneous Lie group with dim z = 1 and π ∈ SI/ZR (G).  by We already saw that FG is a Gelfand triple isomorphism from G(G, μ) to G(G),  the definition of G(G,  μ). Now we want to use the corresponding statement for the S∗ (G) test functions. Again we need to show that T is a Gelfand triple isomorphism in this context. Although the map T is not well defined on G∗ (G, μ) ⊗ G∗ (G, μ), it is well defined on G(G, μ) ⊗ G∗ (G, μ). Lemma 3.1 The map T S(G×G) extends to a Gelfand triple from G(G, μ) ⊗ G∗ (G, μ) onto itself, which we will also call T by a slight abuse of notation. ˆ S∗ (G) and q ∈ P(G), then for all x ∈ G and y ∈ ω Proof Suppose ϕ ∈ S(G) ⊗ 

 z

q(z)ϕ(x, x(−z − y)) dμz(Z) =

z

q((−x)(−z − y))ϕ(x, z) dμz(z) = 0 (155)

ˆ S∗ (G). Analogously Because [z → q((−x)(−z − y)] ∈ P(z). Hence Tϕ ∈ S(G) ⊗ ˆ S∗ (G) onto itself. Because S(G) ⊗ ˆ S∗ (G) we may prove that T−1 maps S(G) ⊗ ˆ S(G), the continuity of T and T−1 on carries the subspace topology in S(G) ⊗ ˆ S∗ (G) is evident. Since also S(G) ⊗   ψ Tϕ d(μ ⊗ μ) = ϕ T−1 ψ d(μ ⊗ μ), (156) G×G

G×G

for all ϕ, ψ ∈ S(G × G), we may extend T S(G×G) to a Gelfand triple isomorphism.

Now a direct conclusion is the formulation of the Kohn–Nirenberg quantization as a Gelfand triple isomorphism that incorporates the new Gelfand triples G∗ (G, μ) and G(R× ; π).

Gelfand Triples for Homogeneous Lie Groups

89

Proposition 3.1 The Kohn–Nirenberg quantization in π-picture × Opπ := K−1 T−1 (1 ⊗ F−1 π ) : G(G, μ) ⊗ G(R ; π) → L(G∗ (G, μ), G(G, μ)), (157)

where K is the kernel map between G(G, μ)⊗G∗ (G, μ) and L(G∗ (G, μ), G(G, μ)), is a Gelfand triple isomorphism. As for the Fourier transformation in π-picture, we may relate Opπ to the original Kohn–Nirenberg quantization Op via the diagrams on page 85.

3.2 The Integral Formula Representation in SI/ZR (π) can also be seen as slowly increasing functions. This is integral to our approach and will be proven in the proposition following the next lemma. Lemma 3.2 Suppose E is a complete locally convex space and f ∈ OM (G; E) and let F (λ, x) := f (δλ x). Then F ∈ OM (R± × G; E). Proof It is enough to show that for each continuous seminorm p on E, each k ∈ N0 each P ∈ DiffP (G) there is a polynomial q ∈ P(G) and l > 0, for which p(∂λk Px F (λ, x)) ≤ (1 + |λ|l + |λ|−l )q(x).

(158)

We realize that there are polynomial differential operators Pv , such that ∂λk Px F (λ, x) =



λv (Pv f )(δλ x),

(159)

v∈R

qv , as a finite linear combination. Since each p(Pv f ) is bounded by a polynomial . we may find polynomials qv such that p(∂λk Px F (λ, x)) ≤

v∈R

|λ|v. qv (δλ x) =



|λ|v qv (x).

(160)

v∈R

This concludes the proof.



Proposition 3.2 Suppose π ∈ SI/ZR (G), then the operator valued function −∞ × (x, λ) → πλ (x) is both in OM (R× × G; L(H∞ π )) and in OM (R × G; L(Hπ )) Proof By Lemma 3.2 it is enough to show, that x → π(x) is slowly increasing. For this purpose we choose an equivalent representation, that is more easily understood. n There is a representation σ ∼ π on Hσ = L2 (Rn ), such that H∞ σ = S(R ) and σ (x)f (t) = e2πiξ(a(x,t ))f (x −1 · t)

(161)

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J. Brinker and J. Wirth

where ξ is a linear functional on a subalgebra m of g, a : G×Rn → m is polynomial and G×Rn ( (x, t) → x ·t ∈ Rn is a polynomial action of G on Rn by Pedersen [6] and [9, Corollary 4.1.2]. Because (x, t) → x −1 · t is polynomial, we may represent the action of G on Rn by a linear combination x·t =



sk,j (x) uk,j (t) ej ,

(162)

j,k

where (ej )j is the standard basis on Rn and sk,j , uk,j are polynomials. Thus, we also have

sk,j (x) σ (x)(uk,j f )(t). (163) tj σ (x)f (t) = k

For the same reason, there are polynomials qj,k , . qj,k on G, rj,k , . rj,k on Rn such that ∂tj f (x −1 · t) =



. qj,k (x). rj,k (t) (∂k f )(x −1 · t)

k

=



qj,k (x) rj,k (x −1 · t) (∂k f )(x −1 · t).

(164)

k

Hence, for all α, β ∈ N0 we find operators Ak ∈ L(S(Rn )) and polynomials vk ∈ P(G), such that t β ∂tα σ (x)f (t) =



vk (x) σ (x)(Ak f )(t),

(165)

k

as a finite linear combination. The topology on L(S(Rn )) is induced by the seminorms p : A → sup sup |t β ∂tα Af (t)|, f ∈B t ∈Rn

B ⊂ S(Rn ) bounded, α, β ∈ Nn0 .

(166)

Now if L ∈ u(g)L is any left invariant differential operator on G and p is a seminorm as above, we get p(Lx σ (x)) ≤

k

=

k

vk (x) sup sup |σ (x)(Ak σ (L)f )(t)| f ∈B t ∈Rn

vk (x) sup sup |(Ak σ (L)f )(t)|.

(167)

f ∈B t ∈Rn

The right-hand side of the above inequality is a sum of continuous seminorms times polynomials, since σ (L) ∈ L(S(Rn )). Thus x → σ (x) is slowly increasing. Due to π ∼ σ the map x → π(x) is slowly increasing, too. Now (x, λ) → πλ is

Gelfand Triples for Homogeneous Lie Groups

91

slowly increasing with values in L(Hπ∞ ) due to Lemma 3.2. We finish the proof by remarking that L(Hπ∞ ) and L(Hπ−∞ ) are isomorphic by the transposition and πλ (x)t = Cπ πλ (−x)Cπ . This implies that π is also slowly increasing with values in L(Hπ−∞ ).

With the help of the above proposition, we want to write the inverse Fourier transform as an integral, which converges in OM (G). For this purpose, we need to explain a small fact about the dual space OM (G). Denote by ∂1 , ∂2 , . . . the directional derivative to any basis v1 , v2 , . . . of g. Each continuous linear functional on OM (g) can be represented by the set OM (G) = spanC {∂ α f | α ∈ Ndim(G) and f ∈ C(G) is rapidly decreasing}, 0 (168) where we used the standard multi-index notation, see [14, page 130 of chapter 2], if we use the dual pairing  ∂ f, g := α

f (−∂)α g dμ.

(169)

G

Here we say f : G → C is rapidly decreasing, iff qf is a bounded function for any q ∈ P(G). The differential operators ∂ α , α ∈ Ndim(G) span the P(G)-module 0 DiffP (G). Since the multiplication of Schwartz functions with polynomials is continuous, we may exchange ∂ α with arbitrary P ∈ DiffP (G) in the pairing above. By [9, Lemma A.2.2] the P(G)-span of the left invariant differential operators u(gL ) is equal to DiffP (G). Now let w1 , w2 , . . . be the dual basis to v1 , v2 , . . . and let X1 , X2 , . . . be the left invariant vector fields associated to v1 , v2 , . . . . A quick calculation shows that for all φ ∈ S (G) and all j, k there exists a polynomial q ∈ P(G) with wj Xk φ = q φ + Xk (wj φ).

(170)

Of course, the set of rapidly decreasing continuous functions is invariant under the multiplication with polynomials. In conclusion, we may represent the dual to OM (G) by OM (G) = spanC {Pf | P ∈ u(gL ) and f ∈ C(G) is rapidly decreasing}.

(171)

Lemma 3.3 If ϕ ∈ S(g) and ω× (  ∼ π ∈ SI/Z(G), then the integral  ϕ=

R×

Tr[πλ Fπ ϕ(λ)] dλπ

exists in OM (G), where dλπ := κ |Pf ()| |λ|Q−1 dλ.

(172)

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J. Brinker and J. Wirth

Proof Let f : G → C be continuous and rapidly decreasing, let P ∈ u(gL ) and let ϕ ∈ S(G). Then, f and P t ϕ are L2 functions and we may apply Plancherel for Fπ . Hence   P t f , ϕ = f P ϕ dμ = Tr[f(λ)∗ Fπ (P ϕ)(λ)] dλπ , (173) R×

G

in which we used the shorthand  g (λ) = Fπ g(λ) for functions g. Since f ∈ L1 (G, μ), we know that the integral that evaluates the Fourier transform in π-picture converges in L(Hπ ) with respect to the weak operator topology. That means for each pair u, v ∈ Hπ we have (f(λ)∗ u, v)Hπ =

 G

f (x) (πλ (x)u, v)Hπ dμ(x).

(174)

Because P ϕ ∈ S(G), we have Fπ (P ϕ)(λ) = πλ (P )  ϕ (λ) ∈ L(Hπ−∞ , Hπ∞ ), which is a nuclear operator for each λ ∈ R× . Hence for each orthonormal basis (ek )k∈N ⊂ Hπ 

|f (x) (πλ (x) πλ (P )  ϕ (λ)ek , ek )Hπ | dμ(x) G k∈N

≤ f L1 (G,μ) πλ (P )  ϕ (λ)N(Hπ ) < ∞, (175) where  · N(Hπ ) is the trace-norm on the space of nuclear operators on Hπ . Using Fubini with respect to the counting measure and μ results in Tr[f(λ)∗ Fπ (P ϕ)(λ)] =

 f (x) Tr[πλ (x) πλ (P )  ϕ (λ)] dμ(x),

(176)

G

since f ∈ L1 (G, μ). Naturally, we have πλ (x) πλ (P ) = Px πλ (x). By the embedding of L(Hπ−∞ , Hπ∞ ) into the nuclear operators N(Hπ ), we may see Tr as a continuous functional on L(Hπ−∞ , Hπ∞ ). Because the operator valued function πλ  ϕ (λ) is a slowly increasing map from G to L(Hπ−∞ , Hπ∞ ), we get Tr[πλ (x) πλ (P )  ϕ (λ)] = Px Tr[πλ (x)  ϕ (λ)], ϕ (λ)] ∈ OM (G). Tr[πλ 

(177)

Gelfand Triples for Homogeneous Lie Groups

93

Finally we get  P t f , ϕ =  =

 f (x) Px Tr[πλ (x)  ϕ (λ)] dμ(x)dλπ

R× R×

G

(178)

P t f , Tr[πλ (·)  ϕ (λ)] dλπ ,



which concludes the proof. ρ ∗ (λ, x)

:= πλ (−x) for some π ∈ SI/Z(G). Let us write ρ(x, λ) := πλ (x) and With Lemma 1.1, we already proved the continuity of the map L(OM (G)) → L(OM (G × R× ; L(Hπ∞ ))), A → A ⊗ 1.

(179)

Of course the evaluation map L(OM (G × R× ; L(Hπ∞ ))) → OM (G × R× ; L(Hπ∞ )),  → (ρ)

(180)

is continuous, as well. Finally, since the multiplication in OM (G×R× ) is continuous [17, page 248] and because of Theorem 1.2, the map S defined by S : L(OM (G)) → OM (G × R× ; L(Hπ∞ )), A → ρ ∗ · (A ⊗ 1)(ρ)

(181)

is continuous. Now this map looks exactly like the inverse Kohn–Nirenberg quantization on compact Lie groups H from [8]. Namely, for any B ∈ L(D(H )) the unique Kohn–Nirenberg symbol b with B = Op(b), evaluated at the irreducible unitary representation ξ , is given by ξ ∗ · (A ⊗ 1)(ξ ) ∈ D(H ; L(Hξ )). Lemma 3.4 The embedding S∗ (G) → OM (G) is continuous and has dense range. Proof The multiplication on S(G) is a continuous bilinear map. This implies the continuity of the canonical embedding τ : S∗ (G) → OM (G), since S∗ (G) carries the subspace topology in S(G). Now consider the dual map τ  : OM (G) → S∗ (G),

where τ  φ, ϕ = φ, ϕ,

for all ϕ ∈ S∗ (G). (182)

That this is indeed an embedding, can be seen from Proposition 2.1 and the representation (168) of the dual space OM (G). By the Hahn-Banach theorem, the operator τ has dense image.

In the Lemma above we saw that S∗ (G) → OM (G) has dense range. Naturally we also have OM (G) → S (G) and OM (R× ) → S (R× ), thus we get embeddings L(OM (G)) → L(S∗ (G), S (G)),

(183)

OM (G × R× ; L(Hπ∞ )) → S (G; S (R× ; π)).

(184)

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J. Brinker and J. Wirth

Notice that we can exchange L(Hπ∞ ) with L(Hπ−∞ ), in the paragraph above. By using the embeddings above, we will see that the map S does indeed reproduce the Kohn–Nirenberg symbol. We can even go one step further. Of course for A ∈ L(OM (G), S(G)), we can still define the map S, since S(G) → OM (G). However, ˆ OM (R× ; L(Hπ∞ )) or not, we are lacking tools to check whether S(A) ∈ S(G) ⊗ since we cannot apply Theorem 1.2. We run into the same problem if we try to define S for operators A ∈ L(OM (G); S (G)). Before we prove that the definition of S gives us Kohn–Nirenberg symbols, we need two final lemmata. ˆ S (R× ; π), then Lemma 3.5 Suppose a ∈ S (G) ⊗ ρ · a = (1 ⊗ Fπ inv)T−1 (1 ⊗ F−1 π )a,

(185)

where inv f (x) = f (−x), f ∈ S(G), continued to distributions. ˆ S(R× ; π). Then we just have Proof First we take a ∈ S(G) ⊗ −1 −1 −1 (1 ⊗ F−1 π )(ρ · a)(x, y) = (1 ⊗ Fπ )a(x, yx) = (1 ⊗ inv)T (1 ⊗ FG )a(x, y), (186)

by the integral formula for the inverse Fourier transform from Lemma 3.3. Now the rest simply follows due to the continuity of the involved maps.

Lemma 3.6 Define χx (ξ ) := e2πiξ(x) for x ∈ g and ξ ∈ g . Then Opπ ℘ (χx ) = πλ (x)

(187)

for any λ ∈ R× and SI/ZR (G) ( π ∼  ∈ ω× . Proof Let ψ ∈ D(z◦ ), such that ψ ≡ 1 on some neighbourhood of zero. If we define ψk (x) := ψ(x/k) for k ∈ N, then ψk χx → χx for k → ∞ in S (z◦ ). Due to the continuity of opπ , we may deduce for λ > 0 Opπ ℘ (χx ) = lim e2πi(δλ x) opπ (ψk · χδλ x z◦ ) k→∞



k (y − δλ. π(y)ψ x ) dν(y),

= lim e2πi(δλ x) k→∞

(188)

ω

k ∈ S(ω) is the Euclidean Fourier transform of ψk and . in which ψ x is the projection k (· − δλ. of x onto ω along z. If we consider the functions ψ x ) as distributions 

k (y − δλ. ψ x )ϕ(y) dν(y)

S(G) ( ϕ →

(189)

ω

k (· − δλ. x ) converges to the Dirac in S (G), then the sequence of functions ψ  distribution supported on δλ x in S (G). By Proposition 2.1 and the continuity of

Gelfand Triples for Homogeneous Lie Groups

95

Fπ , we arrive at x ) = πλ (x). Opπ ℘ (χx ) = e2πi(δλx) π(δλ.

(190)

For λ < 0 the calculation is analogous, we merely need to exchange π with π.



Theorem 3.1 For any A ∈ L(OM (G), E), E ∈ {S(G), OM (G)}, the equality a := S(A) = Op−1 π (A) is valid. Furthermore,  Aϕ =

R×

Tr[πλ a(·, λ) Fπ ϕ(λ)] dλπ ,

for ϕ ∈ S(G),

(191)

where the integral exists in E. Proof First we will prove the integral formula for A ∈ L(OM (G), E). From Lemma 3.3 we know, that for ϕ ∈ S(G)   ) * Tr[πλ  ϕ (λ)] dλπ = A Tr[πλ  ϕ (λ)] dλπ , (192) Aϕ = A G

G

where the integral converges in E and where we used the shorthand Fπ ϕ(λ) =  ϕ (λ). Due to Propositions 2.5 and 3.2, we know that ˆ L(Hπ−∞ , Hπ∞ ). ϕ (λ)) ∈ OM (G) ⊗ (πλ 

(193)

The trace operator Tr, restricted from the nuclear operators on Hπ , is a continuous functional on L(Hπ−∞ , Hπ∞ ), so we may use the tensor product structure of the above expression to get ) * ) * ) * A Tr[πλ  ϕ (λ)] = (A ⊗ Tr) πλ  ϕ (λ) = (1 ⊗ Tr)(A ⊗ 1) πλ  ϕ (λ) ,

(194)

for each λ ∈ R× . Furthermore, (A ⊗ 1)(πλ  ϕ (λ)) = πλ · πλ∗ · (A ⊗ 1)(πλ ) ·  ϕ (λ),

(195)

in which the multiplication is defined pointwise by the multiplication in L(Hπ∞ ). Hence, we can represent A ϕ by the integral  Aϕ =

R×

Tr[πλ a(·, λ)  ϕ (λ)] dλπ ,

(196)

with a := S(A). Now it is left to check that indeed A = Opπ (a). First of all, due to Lemma 3.5 −1 T−1 (1 ⊗ F−1 π )a = (1 ⊗ inv Fπ )(ρ · a).

(197)

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J. Brinker and J. Wirth

We define the function χ(x, ξ ) := e2πiξ(x) for ξ ∈ g , x ∈ g, then χ ∈ OM (g × g× ). Because (1 ⊗ Opπ ℘ )χ(x, λ) = πλ (x) = ρ(x, λ), due to Lemma 3.6, and Fπ = Opπ ℘ Fg , we know that −1 (1 ⊗ inv F−1 π )(A ⊗ 1)(ρ) = (A ⊗ inv Fg )(χ) = (A ⊗ Fg )(χ).

(198)

We choose arbitrary ϕ ∈ S(g) and ψ ∈ S∗ (G). The integral  ϕ=

g

χ(·, ξ ) Fg ϕ(ξ ) dμ (ξ )

(199)

converges in OM (g). Hence (A ⊗ Fg )χ, ψ ⊗ ϕ = (A ⊗ 1)χ, ψ ⊗ Fg ϕ  = A(χ(·, ξ )), ψ Fg ϕ(ξ ) dμ (ξ ) g

(200)

= Aϕ, ψ. Combining the calculations above implies KA = T−1 (1 ⊗ F−1 π )a, for the kernel map K for G(G) ⊗ G∗ (G). I.e. Op−1 π (A) = S(A).

(201)



Acknowledgments Jonas Brinker was supported by ISAAC as a young scientist for his participation at the 12th ISAAC Congress in Aveiro. We thank David Rottensteiner for discussing the possibility of our approach in this setting (homogeneous Lie groups with one-dimensional center and flat orbits) and we thank Christian Bargetz for a very helpful discussion and for pointing out the relevant literature to the multiplication of vector valued distributions with vector valued functions. We especially thank the reviewer for many helpful comments and suggestions, which allowed to improve the paper.

References 1. Geller, D.: Fourier analysis on the Heisenberg group. I. Schwartz space. J. Funct. Anal. 36(2), 205–254 (1980) 2. Astengo, F., Di Blasio, B., Ricci, F.: Fourier transform of Schwartz functions on the Heisenberg group. Stud. Math. 214, 201–222 (2013) 3. M˘antoiu, M., Ruzhansky, M.: Quantizations on nilpotent lie groups and algebras having flat coadjoint orbits. J. Geom. Anal. 29(2), 2823–2861 (2018) 4. Gröchenig, K., Rottensteiner, D.: Orthonormal bases in the orbit of square-integrable representations of nilpotent Lie groups. J. Funct. Anal. 275(12), 3338–3379 (2018)

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5. Bargetz, C., Ortner, N.: Characterization of L. Schwartz’ convolutor and multiplier spaces OC and OM by the short-time Fourier transform. RACSAM 108(2), 833–847 (2014) 6. Pedersen, N.V.: Geometric quantization and the universal enveloping algebra of a nilpotent Lie group. Trans. Am. Math. Soc. 315(2), 511–563 (1989) 7. Fischer, V., Ruzhansky M.: Quantization on Nilpotent Lie Groups. Birkäuser, Boston (2016) 8. Ruzhansky, M., Turunen, V.: Pseudo-Differential Operators and Symmetries. Birkäuser, Boston (2010) 9. Corwin, L., Greenleaf, F.P.: Representations of Nilpotent Lie Groups and Their Applications. Cambridge University Press, Cambridge (1990) 10. Treves, F.: Topological Vectors Spaces, Distributions and Kernels. Academic Press, San Diego (1967) 11. Schaefer, H.H.: Topological Vector Spaces, Springer, New York (1971) 12. Schwartz, L.: Théorie des distributions à valeurs vectorielles. II. Ann. Inst. Fourier 8, 1–209 (1958) 13. Kaballo, W.: Aufbaukurs Funktionalanalysis und Operatortheorie. Springer, Berlin (2014) 14. Grothendieck, A.: Produit tensoriels topologiques et espace nucléaire. Memoirs of the American Mathematical Society, vol. 16. American Mathematical Society, Providence (1955) 15. Bargetz, C: Explicit representations of spaces of smooth functions and distributions. J. Math. Anal. Appl. 424(2), 1491–1505 (2015) 16. Gel’fand, I.M., Vilenkin, N.Y.: Generalized Functions, vol. 4, Academic Press, New York (1964) 17. Schwartz, L.: Théorie des Distributions. Hermann, Paris (1978) 18. Grafakos, L.: Modern Fourier Analysis. Springer, New York (2009) 19. Komatsu, H.: Projective and injective limits of weakly compact sequences of locally convex spaces. J. Math. Soc. Jpn. 19(3), 366–383 (1967) 20. Pedersen, N.V.: Matrix coefficients and a Weyl correspondence for nilpotent Lie groups. Invent. Math. 118, 1–36 (1994) 21. Cartier, P.: Vecteurs différentiables dans les représentations unitaires des groupes de Lie. Sémin. Bourbaki, Exp. No. 454, 20–34 (1976) 22. Moore, C., Wolf, J.: Square integrable representations of nilpotent groups. Trans. Am. Math. Soc. 185, 445–445 (1973)

A Multiplicity Result for a Non-Homogeneous Subelliptic Problem with Sobolev Exponent Annunziata Loiudice

Abstract We prove a multiplicity result for the inhomogeneous subelliptic problem − G u = |u|2

∗ −2

u+f

in ,

u = 0 on ∂,

where G is a sub-Laplacian on a Carnot group G, 2∗ = 2Q/(Q − 2) is the critical Sobolev exponent in this context,  is a bounded domain of G and f is small in a suitable sense. Precisely, we prove the existence of two distinct solutions, that are positive if f is. We adapt to the present subelliptic setting the well-known technique developed by Tarantello (Ann Inst H Poincarè Anal Non Linéaire 9:281–309, 1992). Keywords Non-homogeneous problem · Critical Sobolev exponent · Carnot groups · Multiplicity result

1 Introduction In this paper we provide existence results for the following non-homogeneous subelliptic problem  ∗ − G u = |u|2 −2 u + f u=0

in , on ∂.

(1)

Here, G is a sub-Laplacian on a Carnot group G, 2∗ = 2Q/(Q − 2) is the critical Sobolev exponent in this context,  is a bounded domain of G and the inhomogeneous term f satisfies suitable summability assumptions.

A. Loiudice () Department of Mathematics, University of Bari, Bari, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 V. Georgiev et al. (eds.), Advances in Harmonic Analysis and Partial Differential Equations, Trends in Mathematics, https://doi.org/10.1007/978-3-030-58215-9_4

99

100

A. Loiudice

We recall that a great deal of interest has been paid in the literature to subelliptic equations with critical exponent on Stratified Lie groups, starting from the papers [19, 21] (See e.g. [2, 3, 6, 12, 13, 27, 29–34, 36, 38, 45]). The present results further explore the analysis performed by the author in [29, 31], where the homogeneous case f = λu, λ ∈ R, has been investigated for problem (1). We recall that, for f = 0, problem (1) does not admit any nonnegative non trivial solution, sufficiently regular up to the boundary, on starshaped domains of G (see [19, 21]; see also [31] for a Hardy–Sobolev version of the problem). In [29] the author proves that, when f is a homogeneous linear perturbation f = λu, problem (1) admits at least one positive solution for any dimension Q ≥ 4, when 0 < λ < λ1 , λ1 being the first eigenvalue of the sub-Laplacian G with Dirichlet boundary conditions, extending the celebrated results by Brezis–Nirenberg [7] to the subelliptic Carnot context. In the present paper, we investigate the problem with a non-homogeneous term f and, under suitable assumptions, we prove the existence of at least two solutions, which are positive if f is. We recall that multiplicity results for critical problems in the subelliptic context can be found e.g. in [10, 13, 18, 29, 35, 37]. In particular, Garagnani–Uguzzoni [18] and Maalaoui–Martino [35] obtain existence of multiple solutions depending on the topology of the domain when G is the Heisenberg group Hn . Our multiplicity result depends on the variational structure of the problem, it is obtained by using Ekeland’s variational principle and the Nehari manifold and generalizes to the present context a celebrated Euclidean result by Tarantello [44]. Let us introduce our results. We denote by S01 () the Sobolev–Stein space defined as the completion of C0∞ () with respect to the norm 

1/2

u := 

|∇G u|2

(2)

.

We are interested in weak solutions of problem (1), i.e. u ∈ S01 () such that 

 

∇G u∇G φ =

|u|2

∗ −2

 uφ +

f φ,





∀ φ ∈ S01 ().

(3)

We assume that the non-homogeneous term f belongs to S −1 (), that is the dual space of S01 (), and that its S −1 -norm is appropriately small. We look for solutions to (1) as critical points of the functional J (u) =

1 2

 

|∇G u|2 −

1 2∗



∗



|u|2 − 

f u, 

u ∈ S01 ().

(4)

A Multiplicity Result for a Non-Homogeneous Subelliptic Problem

101

Let us set CQ :=

4 Q−2



Q−2 Q+2

(Q+2)/4 .

(5)

f u ≤ CQ (∇G u2 )(Q+2)/2 ,

(6)

Our main results can be stated as follows. Theorem 1.1 Let f = 0 satisfy the following inequality  

 ∗ for any u ∈ S01 (),  |u|2 = 1, where CQ is the constant defined in (5). Then, problem (1) has at least one weak solution u0 ∈ S01 (). Moreover, u0 ≥ 0 if f ≥ 0. If, moreover, the estimate (6) holds with the strict sign, the existence of a second solution can be proved. We have the following result. Theorem 1.2 Let f = 0 satisfy the following strict inequality  

f u < CQ (∇G u2 )(Q+2)/2 ,

(7)

 ∗ for any u ∈ S01 (),  |u|2 = 1, where CQ is the constant defined in (5). Then, problem (1) has at least two weak solutions u0 , u1 ∈ S01 (). Moreover, u0 ≥ 0, u1 ≥ 0 if f ≥ 0. We also note that, for f ≥ 0, by means of Bony’s maximum principle [5], such solutions turn out to be strictly positive. Remark 1.1 We observe that condition (6) naturally holds if f S −1 ≤ CQ S Q/4 ,

(8)

where S is the best Sobolev constant on G. We point out that the technique introduced in [44] has been applied in many different contexts: see e.g. some applications to singular elliptic problems in Chen– Chen [9], Kang–Deng [26], a generalization to the fractional Laplacian in Shang– Zhang-Yang [43], an application to a critical problem on the Heisenberg group with indefinite nonlinearity in Chen–Huang–Rocha in [10]. The main difficulty one encounters when adapting the Euclidean technique to the general context of Carnot groups is the lack of knowledge of the related Sobolevtype extremals, whose explicit form is not known in the abstract Carnot setting. Indeed, the explicit form of Sobolev-type minimizers on Carnot groups is known only when G is a group of Iwasawa type (see Jerison–Lee [25] and Frank–Lieb [17] for the Sobolev minimizers in the Heisenberg group Hn , Ivanov–Minchev–Vassilev [23] and Christ–Liu–Zhang [11] for the remaining cases).

102

A. Loiudice

However, as observed by the author in [29], this difficulty can be overcome since some qualitative properties of the extremal functions, namely their existence and their exact asymptotic behavior at infinity, are sufficient to perform the required Brezis–Nirenberg type asymptotic expansions [7] and achieve the existence results by a mountain pass procedure (see e.g. [24, 29, 31]). We also recall that the sharp asymptotic behavior at infinity of Sobolev-type minimizers has been studied by the author in [29, 30, 32]. Finally, we note that obviously the preceding theorems hold for the subcritical case, i.e. when the power p = 2∗ is replaced by q ∈ (2, 2∗ ): in such a case standard compactness arguments apply (see [20] for compact Sobolev-type embeddings in the subelliptic setting). The paper is organized as follows: in Sect. 2 we recall the main notation and definitions related to the Carnot setting; in Sect. 3 we prove the existence of the first solution; in Sect. 4 we establish the existence of the second solution.

2 The Functional Setting of Carnot Groups We recall the main definitions and notations related to the Carnot groups setting. For a complete treatment, we refer to the monograph [4] and the classical papers [15, 16]. We also quote [14, 40] for an overview on general homogeneous Lie groups. A Carnot group (or Stratified group) (G, ◦) is a connected, simply connected nilpotent Lie /group, whose Lie algebra g admits a stratification, namely a decomposition g = rj =1 Vj , such that [V1 , Vj ] = Vj +1 for 1 ≤ j < r, and [V1 , Vr ] = {0}. The number r is called the step of the group G. The integer Q = ri=1 i dimVi is called the homogeneous dimension of G. We shall assume throughout that Q ≥ 3. By means of the natural identification of G with its Lie algebra via the exponential map (which we shall assume throughout), it is not restrictive to suppose that G is a homogeneous Lie group on RN = RN1 × RN2 × . . . × RNr , with Ni = dimVi , equipped with a family of group-automorphisms (called dilations) δλ of the form δλ (ξ ) = (λ ξ (1) , λ2 ξ (2) , · · · , λr ξ (r) ),

(9)

where ξ (j ) ∈ RNj for j = 1, . . . , r. Let m := N1 and let X1 , . . . , Xm be the set of left invariant vector fields of V1 that coincide at the origin with the first m partial derivatives. The second order differential operator

G =

m

Xi2

(10)

i=1

is called the canonical sub-Laplacian on G. We shall denote by ∇G = (X1 , . . . , Xm )

(11)

A Multiplicity Result for a Non-Homogeneous Subelliptic Problem

103

the related horizontal gradient. The operator G is left-translation invariant with respect to the group action τξ (η) = ξ ◦ η and δλ -homogeneous of degree two. In other words, G (u ◦ τξ ) = G u ◦ τξ , G (u ◦ δλ ) = λ2 G u ◦ δλ . Moreover, due to the stratification condition, the Lie algebra generated by X1 , . . . , Xm is the whole g, and therefore it is everywhere of rank N; therefore, the sub-Laplacian operator

G satisfies the well-known Hörmander’s hypoellipticity condition. The simplest example of Carnot group is the additive group G = (RN , +). In this case Q = N and the sub-Laplacians are exactly the constant coefficients elliptic operators on RN . Moreover, if G is a Carnot group of homogeneous dimension Q ≤ 3, then necessarily G is the ordinary Euclidean space. The simplest nonabelian Carnot group is the Heisenberg group Hn = (R2n+1 , ◦) with homogeneous dimension Q = 2n + 2 and composition law given by ξ ◦ ξ  = (x + x  , y + y  , t + t  + 2(x  , y − x, y  )), for every ξ = (x, y, t), ξ  = (x  , y  , t  ) ∈ R 2n+1 , where x, y ∈ Rn and t ∈ R. When Q ≥ 3, Carnot groups possess the following property: there exists a suitable homogeneous norm d on G such that the function (ξ ) =

C d(ξ )Q−2

(12)

is a fundamental solution of − G with pole at 0, for a suitable constant C > 0 (see [15]). By definition, a homogeneous norm on G is a continuous function d : G → [0, +∞), smooth away from the origin, such that d(δλ (ξ )) = λ d(ξ ), for every λ > 0 and ξ ∈ G, d(ξ −1 ) = d(ξ ) and d(ξ ) = 0 if and only if ξ = 0. Moreover, if we define d(ξ, η) := d(η−1 ◦ ξ ), then d is a pseudo-distance on G. Throughout the paper, we shall denote by d the homogeneous norm associated with the fundamental solution of the sub-Laplacian operator by (12). We shall denote by Br (ξ ) = Bd (ξ, r) the d-ball with center at ξ and radius r. A fundamental rôle in the functional analysis on Carnot groups is played by the following Sobolev-type inequality due to Folland and Stein [15]: if G is a Carnot group of homogeneous dimension Q ≥ 3, there exists a positive constant S = S(G) such that 

 |∇G u| dξ ≥ S

2∗

2

G

G

|u| dξ



2 2∗

∀ u ∈ C0∞ (G),

(13)

where 2∗ = 2Q/(Q − 2) denotes the critical exponent in this context. It is known that the best constant in (13) is attained (see [21, 45]), but the explicit form of the minimizers is known only for the class of Iwasawa-type groups (see [11, 17, 23, 25]). Qualitative properties of Sobolev minimizers on Carnot groups, such as their sharp summability in Lp -weak spaces and their asymptotic behavior at infinity, have been studied by the author in [29] for the pure Sobolev case, in [30] for the Hardy–Sobolev case, in [33] for the general Sobolev inequality with exponent 1 < p < Q. We also recall that a quantitative form of Sobolev inequality (13) on the Heisenberg group Hn has been obtained by the author in [28].

104

A. Loiudice

Finally, we point out that a large variety of functional inequalities can be obtained on Carnot groups, and more generally on graded and homogeneous Lie groups. We quote [39–42] and the references therein for recent developments on this topic.

3 Existence of the First Solution In this section, we prove the existence of the first solution for problem (1). Since J is bounded from below on the so-called Nehari manifold N := {u ∈ S01 () | < J  (u), u >= 0} + ,    1 2 2∗ = u ∈ S0 () | |∇G u| − |u| − fu = 0 , 



(14)



we shall consider the following minimization problem c0 := inf J.

(15)

N

Let us split N into the following sets + ,   ∗ |∇G u|2 − (2∗ − 1) |u|2 > 0 , N+ := u ∈ N |

(16)

+ ,   ∗ |∇G u|2 − (2∗ − 1) |u|2 = 0 , N0 := u ∈ N |

(17)

+ ,   2 ∗ 2∗ N := u ∈ N | |∇G u| − (2 − 1) |u| < 0 .

(18)













−

We shall obtain that, under the assumption (6), c0 is attained at a function u0 ∈ N and, if the more restrictive assumption (7) holds, u0 is a local minimum for J and belongs to N+ , i.e. c0 = inf J = inf J. N

(19)

N+

We begin with some preliminary lemmas. The first lemma clarifies the purpose of the assumption (7) on the non-homogeneous term f . Lemma 3.1 Let f = 0 satisfy condition (7). Then, for any u ∈ S01 (), u = 0, there exists a unique t + = t + (u) > 0 such that t + u ∈ N− ,  t+ >



u|2  |∇G  ∗ (2∗ − 1)  |u|2

01/(2∗−2) := tmax

(20)

A Multiplicity Result for a Non-Homogeneous Subelliptic Problem

105

and J (t + u) = max J (tu).

(21)

t ≥tmax

Moreover, if



 fu

> 0, there exists a unique t − = t − (u) > 0 such that t − u ∈ N+ ,  t

−


2,  is concave and achieves its maximum in  tmax =



u|2  |∇G  ∗ (2∗ − 1)  |u|2

01/(2∗−2) (25)

and )

(tmax ) =

* 2 (Q+2)/4  |∇G u| CQ ) * 2∗ (Q−2)/4  |u|

(26)

where CQ is the constant defined in (5).  Now, if  f u ≤ 0, there exists a unique t + > tmax such that (t + ) =  f u and  (t + ) < 0, that is, t + u ∈ N− and J (t + u) ≥ J (tu) for any t ≥ tmax . If, instead,  f u > 0, by the assumption (7) on f we get that  f u < (tmax ).

(27)



Hence, there exists a unique 0 < t − < tmax < t + such that (t + ) =



f u = (t − ) 

(28)

106

A. Loiudice

and  (t + ) < 0 <  (t − ),

(29)

that is, t + u ∈ N− and t − u ∈ N+ . Moreover, J (t + u) ≥ J (tu), ∀ t ≥ t − and J (t − u) ≤ J (tu), ∀ 0 ≤ t ≤ t + .



Lemma 3.2 For f = 0, the infimum  

inf



(Q+2)/4

CQ

2∗  |u| =1



|∇G u|



 −

2

f u := μ0

(30)



is attained. In particular, if f satisfies (7), then μ0 > 0. Proof The proof is rather technical and it straightforwardly follows the Euclidean outline. We omit it, referring to the Appendix in [44].

Lemma 3.3 Assume that condition (7) holds. Then, for any u ∈ N, u = 0, we have  



∗

2

∗

|u|2 = 0,

|∇G u| − (2 − 1)

(31)



i.e. N0 = {0}. Proof For any u ∈ N, we have 

 

|∇G u|2 =



∗

|u|2 +

f u.



(32)



By contradiction, assume that (31) does not hold. Then, there exists u0 ∈ N, u0 = 0 such that   ∗ |∇G u0 |2 − (2∗ − 1) |u0 |2 = 0. (33) 



From (32) and (33) we get  |u0 | 

2∗

1 = ∗ 2 −2

 f u0

(34)



and 

2∗ − 1 |∇G u0 | = ∗ 2 −2 



2

f u0 .

(35)



Set ∗

v := u0 /u0 22∗ .

(36)

A Multiplicity Result for a Non-Homogeneous Subelliptic Problem

107

From the preceding equalities, we easily get 1 ∗ ∗ (2∗ − 2)(2 −2)/(2 −1) ∗ 2 −1



 |∇G v| = 2



(2∗ −2)/(2∗−1) fv

,

(37)



that is 

4 fv = Q −2 



Q−2 Q+2

(Q+2)/4 

(Q+2)/4 

|∇G v|

2

,

(38)



which contradicts assumption (7). ±

Lemma 3.4 Assume that f = 0 satisfy (7). Then, for any u ∈ N , there exist ε > 0 and a differentiable function t = t (v) > 0, v ∈ S01 (), v < ε, such that t (v)(u − v) ∈ N± ,

t (0) = 1,

for v < ε,

(39)

and 

t (0), v =

2







∗ 2∗ −2 uv −  ∇G u∇G v − 2  |u|  fv . ∗ 2 ∗ 2 u − (2 − 1)  |u|

(40)

Proof Define the map F : R × S01 () → R as F (t, v) = tu − v2 − t 2

∗ −1



∗



|u − v|2 − 

f (u − v).

(41)



Since F (1, 0) = 0,

∂F (1, 0) = u2 − (2∗ − 1) ∂t



∗

|u|2 = 0,

(42)



by applying the implicit function Theorem at the point (1, 0) we get the result.



The following lemma provides an upper bound for c0 . Lemma 3.5 Let f ∈ S −1 () satisfy (7). Then, there exists a negative constant θ0 , depending on f , such that c0 < θ0 < 0. Proof Let v ∈ S01 () be the unique solution of  − G v = f v=0

in , on ∂.

(43)

108

A. Loiudice

Since f = 0, we have that v2 > 0 and v2 = f 2S −1 . Moreover, consider t − = t − (v) > 0 provided by Lemma 3.1. Hence t − v ∈ N+ and consequently c0 ≤ J (t − v) =− 0 for n large (otherwise we are done). By Lemma 3.1 applied to w = δJ  (un )/J  (un ) with δ > 0 small enough, we can get

A Multiplicity Result for a Non-Homogeneous Subelliptic Problem

109

tn (δ) = t (w) such that wδ := tn (δ)(un − w) ∈ N. From condition (45b) we have 1 wδ − un  n ≥J (un ) − J (wδ )

(50)

=(1 − tn (δ))J  (wδ ), un  + δtn (δ)J  (wδ ),

J  (u J  (u

n) n )

 + o(δ).

Dividing by δ > 0 and passing to the limit as δ → 0, we obtain 1 (1 + |tn (0)| un ) ≥ −tn (0)J  (un ), un  + J  (un ) = J  (un ), n

(51)

where we have set tn (0) := t  (0),

J  (un ) . J  (un )

(52)

Hence, taking into account (48), we get J  (un ) ≤

* C) 1 + |tn (0)| , n

(53)

for a suitable constant C > 0. The thesis then follows by showing that |tn (0)| is uniformly bounded with respect to n. Indeed, from (40) and (48), it follows that |tn (0)| ≤

C1 ∗ | un 2 − (2∗ − 1)un 22∗ |

(54)

for a suitable constant C1 > 0, and it can be verified, by using Lemma 3.2, that the denominator in the r.h.s. of (54) is bounded away from 0. We refer to [44] for further details. Now, let u0 ∈ S01 () be the weak limit, up to a subsequence, of un . From (47) we get that  f u0 > 0,

(55)



and from the fact that J  (un ) → 0 as n → ∞, we have that < J  (u0 ), w >= 0,

∀ w ∈ S01 (),

(56)

110

A. Loiudice

i.e. u0 is a weak solution for problem (1) and u0 ∈ N. Hence c0 ≤ J (u0 ) =

  1 1 u0 2 − 1 − ∗ f u0 ≤ lim J (un ) = c0 . n→∞ Q 2 

(57)

So we deduce that un → u0 strongly in S01 () and J (u0 ) = c0 . We can also deduce by Lemma 3.1 that u0 ∈ N+ (see e.g. [43, page 575]). Finally, if f ≥ 0, take t0 = t − (|u0 |) > 0 with t0 |u0 | ∈ N+ . Necessarily, t0 ≥ 1 and J (t0 |u0 |) ≤ J (|u0 |) ≤ J (u0 ).

(58)

So we can always take u0 ≥ 0. The proof for the case when f satisfies the weaker assumption (6) is achieved by an approximation argument, reasoning with fε = (1 − ε)f , 0 < ε < 1, for which the strict condition (7) holds and the preceding results apply. See the details in [44, page 292].



4 Existence of the Second Solution In this section, we prove that, under the strict assumption (7), the existence of a second solution to our problem (1) can be proved. Precisely, we show that, under the assumption (7), the following infimum c1 := inf J (u)

(59)

u∈N−

is achieved at a point u1 ∈ N− which defines a critical point of J . We need the following preliminary results. In what follows, S will denote the best Sobolev constant on G. Lemma 4.1 Assume that condition (7) holds. Then, the solution u0 obtained in Theorem 1.1 is a local minimum for J .  Proof From Lemma 3.1, for any u ∈ S01 () with  f u > 0 we know that  J (su) ≥ J (t − u),

for any 0 < s
0 and η ∈ G.

(69)

Now, recall that u0 = 0. We set ! ⊂  to be a set of positive measure such that u0 > 0 on ! (replace u0 with −u0 and f with −f if necessary). Let η ∈ ! and let R > 0 be such that Bd (η, R) ⊂ . Let ϕη ∈ C0∞ (Bd (η, R)) be a cut-off function, 0 ≤ ϕη ≤ 1, ϕη ≡ 1 in Bd (η, R/2). Define uε,η := ϕη Uε,η ,

ε > 0, η ∈ !.

(70)

We get the following crucial lemma. Lemma 4.3 For any t > 0 and for a.e. η ∈ !, there exists ε0 = ε0 (t, η) > 0 such that J (u0 + tuε,η ) < c0 +

1 Q/2 S , Q

(71)

for every 0 < ε < ε0 . Proof We have J (uo + tuε,η ) =





1 2



1 − ∗ 2

|∇G u0 |2 + t



|u0 + tuε,η |

 2∗

∇G u0 ∇G uε,η + 



−



t2 2

f u0 − t 

 

|∇G uε,η |2 (72)

f uε,η . 

We claim that, for a.e. η ∈ !,     ∗ 2∗ 2∗ 2∗ 2∗ ∗ |u0 + tuε,η | = |u0 | + t uε,η + 2 t |u0 |2 −2 u0 uε,η 



∗

+2 t

2∗ −1





2∗ −1



uε,η u0 + o(ε



(Q−2)/2

(73) ).

To verify (73), following Brezis–Nirenberg [8], we use the following formulas: for any 1 < q ≤ 3, there exists a positive constant C depending on q such that for α, β ∈ R we have ||α + β|q − |α|q − |β|q − qαβ(|α|q−2 + |β|q−2 )|  |α||β|q−1 if |α| ≥ |β|, ≤C |α|q−1 |β| if |α| ≤ |β|,

(74)

A Multiplicity Result for a Non-Homogeneous Subelliptic Problem

113

and for q ≥ 3, there exists a positive constant C depending on q such that for α, β ∈ R we have ||α+β|q −|α|q −|β|q −qαβ(|α|q−2 +|β|q−2 )| ≤ C(|α|q−2 β 2 +α 2 |β|q−2 ).

(75)

By the above formulas, we have that 



∗

∗

|u0 + tuε,η |2 = 

|u0 |2 + t 2 

+ 2∗ t 2

∗ −1

∗



∗





u2ε,η + 2∗ t

 |u0 |2

∗ −2

u0 uε,η



(76)

∗



u2ε,η−1 u0 + Rε(1) + Rε(2) ,

where Rε(1) and Rε(2) denote the terms corresponding, respectively, to the two terms in the right hand side of (74), if Q ≥ 6, and the two terms in the right hand side of (75) if 3 ≤ Q ≤ 5. Now, if Q ≥ 6, by (74) we have  |Rε(1) |

≤C

{|u0 |≥t uε,η }

|u0 |(tuε,η )2

∗ −1

.

(77)

We split the integration into d(ξ, η) < R/2 and d(ξ, η) ≥ R/2. Taking into account that the following decay estimate holds for Sobolev minimizers on Carnot groups U (ξ ) ≤ C(1 + d(ξ )Q−2 )−1 ,

∀ξ ∈ G,

(78)

∗

2 −1 ≤ Cε (Q+2)/2 on  \ B (η, R/2). Hence, letting (see [3, 29]), we have that Uη,ε d u0 to be zero outside , from (77) we get

|Rε(1) | ≤ Ct εγ2 (Q−2)/2

 G

|u0 (ξ )|1+γ1 dξ + Ct ε(Q+2)/2, d(η−1 ◦ ξ )γ2 (Q−2)

(79)

where γ1 , γ2 > 0 are chosen to satisfy γ1 + γ2 = 2∗ − 1 and 1 < γ2 < Q/(Q − 2). Note that |u0 |1+γ1 ∈ L2

∗ /(1+γ

1)

= L2

∗ /(2∗ −γ

2)

and d −γ2 (Q−2) ∈ LQ/[(Q−2)γ2],∞ .

(80)

Hence, their convolution on G belongs to Lr,∞ , where r = 2Q/[(Q − 2)γ2] (see [22, Theorem 1.2.13]). In particular  G

|u0 (ξ )|1+γ1 dξ < +∞, d(η−1 ◦ ξ )γ2 (Q−2)

for a.e. η ∈ !.

(81)

114

A. Loiudice (1)

It follows that Rε side of

= o(ε(Q−2)/2). In a similar fashion, estimating the right hand  |Rε(2) | ≤ C

{|u0 | 0 satisfy γ1 + γ2 = 2, with γ2 > 1. Then (ε2

1 1 1 ≤ 2γ , −1 2 2 −1 1 + d(η ◦ ξ ) ) ε d(η ◦ ξ )2γ2

(84)

which implies |Rε(1) |, |Rε(2) | ≤ Ct ε2−2γ1

 G

|u0 (ξ )|2

1 dξ, d(η−1 ◦ ξ )2γ2

(85)

where the convolution integral in the right hand side is finite for a.e. η ∈ !, being ∗ u0 ∈ L2 (see [22]). Therefore, Rε(1) = Rε(2) = O(ε2θ ), for any θ < 1, and so (73) follows for Q = 4. Finally, if Q = 5, by (75) we get |Rε(1) | ≤ Ct ≤

 G

Ct ε2

|u0 (ξ )|4/3 u2ε,η (ξ ) dξ

 G

|u0 (ξ )|

4/3

1 2 U (δ1/ε (η−1 ◦ ξ )) dξ εQ

(86)

and |R (2) | ≤ Ct ε2

 G

|u0 (ξ )|2

1 dξ. d(η−1 ◦ ξ )4

(87)

A Multiplicity Result for a Non-Homogeneous Subelliptic Problem

115

∗

Note that, since U ∈ L2 /2,∞ (G) ∩ L∞ (G) (see the regularity results in [29, 30]), we have in particular that U ∈ L2 (G) for Q > 4. Therefore, the kernel 1 2 U (δ1/ε (η−1 ◦ ξ )) εQ

K(ξ ) :=

(88)

appearing in (86), is a mollifier (up to a constant) and so  G

|u0 (ξ )|4/3



1 2 U (δ1/ε (η−1 ◦ ξ )) dξ → |u0 (η)|4/3 εQ

G

U 2 (ξ ) dξ

(89)

as ε → 0. Moreover, concerning (87), it holds that  G

|u0 (ξ )|2

1 d(η−1

◦ ξ )4

dξ < +∞,

(90)

for a.e. η ∈ !. Henceforth, we obtain that Rε(1) = Rε(2) = O(ε2 ), and so (73) holds also in this case. Now, from [29] we know that, as ε → 0  |∇G uε,η |2 = S Q/2 + O(εQ−2 ), (91a) 



∗



u2ε,η = S Q/2 + O(εQ ).

(91b)

Therefore, by (72), (73), (91a), and (91b), and taking into account that u0 is a weak solution of Eq. (1), we get ∗

t2 t2 ∗ J (u0 + tuε,η ) ≤J (u0 ) + S Q/2 − ∗ S Q/2 − t 2 −1 2 2 + o(ε

(Q−2)/2



∗



u2ε,η−1 u0

(92)

),

for a.e. η ∈ !. Setting u0 = 0 outside , we have 

∗



u2ε,η−1 u0 = ε−(Q+2)/2

 G

 = ε(Q−2)/2

G

u0 (ξ )ϕη2

u0 (ξ )ϕη2

∗ −1

∗ −1

(ξ )U 2

(ξ )

∗ −1

(δ1/ε (η−1 ◦ ξ )) dξ

1 2∗ −1 U (δ1/ε (η−1 ◦ ξ )) dξ. εQ

(93)

So, letting ε → 0, as before we get  G

1 ∗ ∗ u0 ϕη2 −1 Q U 2 −1 (δ1/ε (η−1 ε

 ◦ ξ )) dξ → u0 (η)

G

U2

∗ −1

dξ,

(94)

116

A. Loiudice

 ∗ for a.e. η ∈ !. In other words, setting D = G U 2 −1 , it holds  ∗ u2ε,η−1 u0 = ε(Q−2)/2u0 (η)D + o(ε(Q−2)/2).

(95)



Consequently, by (92) and (95) we get ∗

J (u0 + tuε,η ) ≤c0 + ( + o(ε

t2 t2 ∗ − ∗ )S Q/2 − t 2 −1 u0 (η)Dε(Q−2)/2 2 2 (Q−2)/2

(96)

).

Define ∗

h(t) := (

t2 t2 ∗ − ∗ )S Q/2 − t 2 −1 u0 (η)Dε(Q−2)/2, 2 2

for t ≥ 0.

(97)

Since h(t) goes to −∞ as t → ∞, supt ≥0 h(t) is achieved at some tε > 0. Since tε satisfies (tε − tε2

∗ −1

)S Q/2 = (2∗ − 1)tε2

∗ −2

u0 (η)Dε(Q−2)/2,

(98)

then necessarily tε < 1 and tε → 1 as ε → 0. Set tε = 1 − δε . From (98), we obtain (1 − δε − (1 − δε )2

∗ −1

)S Q/2 = (2∗ − 1)(1 − δε )2

∗ −2

u0 (η)Dε(Q−2)/2,

(99)

and so (2∗ − 2)S Q/2 δε = (2∗ − 1)u0 (η)Dε(Q−2)/2 + o(ε(Q−2)/2).

(100)

Therefore ∗

t2 t2 ∗ J (u0 + tuε,η ) ≤ c0 + ( ε − ε∗ )S Q/2 − tε2 −1 u0 (η)Dε(Q−2)/2 + o(ε(Q−2)/2) 2 2 1 Q/2 1 = c0 + ( − ∗ )S − u0 (η)Dε(Q−2)/2 + o(ε(Q−2)/2) 2 2 1 = c0 + S Q/2 − u0 (η)Dε(Q−2)/2 + o(ε(Q−2)/2). Q (101) Hence, for ε0 = ε0 (t, η) > 0 sufficiently small, we conclude J (u0 + tuε,η ) < c0 + for any 0 < ε < ε0 .

1 Q/2 S , Q

(102)



A Multiplicity Result for a Non-Homogeneous Subelliptic Problem

117

Now, by means of the preceding lemma, we are able to prove that the value c1 stands below the (PS)-threshold. Lemma 4.4 Let f = 0 satisfy (7). Then, c1 < c0 +

1 Q/2 S . Q

(103)

Proof Following [44], to prove estimate (103) we provide a mountain pass critical 1 Q/2 level that is below the threshold c0 + Q S and compares with c1 . To this aim, observe that, under the assumption (7), the manifold N− disconnects S01 () in two connected components E1 and E2 . Indeed, notice that, for every u ∈ S01 (), u = 1, by Lemma 3.1, there exists a unique t + (u) > 0 such that t + (u)u ∈ N−

and J (t + (u)u) = max J (tu). t ≥tmax

(104)

Set ,  + u E1 = {0} ∪ u : u < t + , u

,  + u E2 = u : u > t + . u

(105)

We have that S01 \ N− = E1 ∪ E2 and N+ ⊂ E1 . Moreover, observe that there exists a suitable constant C1 > 0 such that 0 < t + (u) < C1 , for all u, u = 1. Then, set T := (S −Q/2 |C12 − u0 2 |)1/2 + 1, and fix η ∈ ! such that Lemma 4.3 applies. It is possible to prove that u0 + T uε,η ∈ E2

(106)

for ε > 0 sufficiently small (see [44]). So, for such a choice of T and η ∈ !, fix ε > 0 such that both (71) and (106) hold, and define: := {h : [0, 1] → S01 () continuous, h(0) = u0 , h(1) = u0 + T uε,η }.

(107)

Clearly, . h : [0, 1] → S01 () defined by . h(t) := u0 + tT uε,η belongs to . So, by Lemma 4.3, we conclude that c := inf max J (h(t)) < c0 + h∈ t ∈[0,1]

Moreover, since the range of any h ∈

(108)

intersects N− , it holds that

c1 = inf J (u) ≤ c, u∈N−

which concludes the proof.

1 Q/2 S . Q

(109)



118

A. Loiudice

We are now able to complete the proof of Theorem 1.2, by providing the existence of the second solution. Proof (Theorem 1.2) Analogously to the proof of Theorem 1.1, by Ekeland’s variational principle, there exists a sequence {un } ⊂ N− such that J (un ) → c1 and J  (un ) → 0.

(110)

From Lemma 4.2 and (103), there exists u1 ∈ S01 () such that, up to a subsequence un → u1 strongly in S01 ().

(111)

Hence, u1 is a critical point for J , u1 ∈ N− (since N− is closed) and J (u1 ) = c1 . In particular, u1 is a solution to problem (1), u1 = u0 . Moreover, if f ≥ 0, let t + > 0 satisfy t + |u1 | ∈ N− . From Lemma 3.1, we get J (u1 ) = max J (tu1 ) ≥ J (t + u1 ) ≥ J (t + |u1 |). t ≥tmax

So, we can always take u1 ≥ 0.

(112)



Acknowledgments The author thanks Professors Georgiev, Ozawa, Ruzhansky and Wirth, organizers of the Session “Harmonic Analysis and Partial Differential Equations” at ISAAC Congress 2019 in Aveiro (Portugal), where some of the present results were presented.

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The Dixmier Trace and the Noncommutative Residue for Multipliers on Compact Manifolds Duván Cardona and César Del Corral

Abstract In this paper we investigate the Dixmier trace and the noncommutative residue (also called Wodzicki’s residue) of pseudo-differential operators by using the notion of a global symbol. We consider both cases, compact manifolds with or without boundary. Our analysis on the Dixmier trace of invariant pseudo-differential operators on closed manifolds will be based on the Fourier analysis associated with every elliptic and positive operator and the quantization process developed by Delgado and Ruzhansky. In particular, for compact Lie groups, this can be done by using the representation theory of the group in view of the Peter–Weyl Theorem and the Ruzhansky–Turunen symbolic calculus. The analysis of invariant pseudo-differential operators on compact manifolds with boundary will be based on the global calculus of pseudo-differential operators developed by Ruzhansky and Tokmagambetov. Keywords Dixmier trace · Non-commutative residue · Global operators · Representation theory · Closed manifolds · Manifolds with boundary · Hörmander classes · Boutet de Monvel’s algebra

1 Introduction The Dixmier trace arises in functional analysis, from the problem if there exists a non-trivial trace—which does not coincide with the spectral trace—for an ideal containing the set of trace class operators on a Hilbert space H . In 1966, J. Dixmier in [15] showed that there exists a trace which vanishes in the ideal of trace class operators, but it is non-trivial in the so-called Dixmier ideal L1,∞ (H ). As it was D. Cardona () Department of Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium e-mail: [email protected] C. Del Corral Department of Mathematics, Edificio Franco, Universidad de Los Andes, Bogota, Colombia e-mail: [email protected] © Springer Nature Switzerland AG 2020 V. Georgiev et al. (eds.), Advances in Harmonic Analysis and Partial Differential Equations, Trends in Mathematics, https://doi.org/10.1007/978-3-030-58215-9_5

121

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pointed out in [40] and references therein, the Dixmier trace is important due to its applications in fractal theory, foliation theory, spaces of non-commuting coordinates, perturbation theory, noncommutative geometry and quantum field theory. In this paper, we study the Dixmier trace of invariant pseudo-differential operators (multipliers) on compact manifolds with or without boundary. By applying the Connes trace theorem (see [4]) we derive a formula for the noncommutative residue of Fourier multipliers on compact Lie groups, and manifolds with boundary. This is possible based on the recent quantizations of pseudo-differential operators associated with global symbols introduced by M. Ruzhansky and V. Turunen [32] (for compact Lie groups), J. Delgado and M. Ruzhansky [10] (for compact manifolds without boundary), and finally, by M. Ruzhansky and N. Tokmagambetov [14] (for compact manifolds with boundary). One of the advantages of our approach is that we obtain results on the Dixmier trace and the noncommutative residue for Fourier multipliers in terms of global (full) symbols, where we give a new viewpoint, instead of the classical results where the problem was treated by using local symbols (see e.g. [4, 15, 16, 18, 21, 35, 41] and references therein). We summarize our research in the following results: • We provide necessary and sufficient conditions in order to guarantee that global invariant pseudo-differential operators on a manifold M with or without boundary lie in the Dixmier class L(1,∞) (H ), H = L2 (M), (see Eq. (2)). • We provide sufficient and necessary conditions in order to obtain Dixmier traceability for a type of pseudo-differential operators with global symbols in Hörmander classes on compact Lie groups and we express our results by using the representation theory of these groups. Also, we use Connes’ trace theorem in order to show formulae for the noncommutative residue of classical pseudodifferential operators on compact Lie groups. • On a compact Lie group, we provide formulae in terms of global symbols for the noncommutative residue of a type of classical pseudo-differential operators in terms of the representation theory of the group (see Proposition 3.1). • For a compact manifold M with or without boundary, we find criteria for global symbols in order that the corresponding operators belong to the Marcinkiewicz ideal L(p,∞) (H ), 1 < p < ∞, where H = L2 (M), (see Eq. (4)). In order to present our main results we precise some definitions. By following Connes [4], if H is a Hilbert space, the class L(1,∞) (H ) consists of those bounded linear operators A ∈ L(H ) satisfying

sn (A) = O(log(N)), N → ∞, (1) 1≤n≤N

where {sn (A)} denotes the sequence of singular values of A, i.e. the square roots of the eigenvalues of the non-negative self-adjoint operator A∗ A. So, L(1,∞) (H ) is endowed with the norm

1 AL(1,∞) (H ) = sup sn (A). (2) N≥2 log(N) 1≤n≤N

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The Dixmier trace Trω on positive operators in L(1,∞) (H ) can be formulated by using an average function ω of the following form lim ω



) 1 * 1 sn (A) := ω sn (A) , log(N) log(N) 1≤n≤N

(3)

1≤n≤N

where limω is a positive linear functional on l ∞ (N), the set of bounded sequences, satisfying: • limω (αn ) ≥ 0 if all αn ≥ 0; • limω (αn ) = lim(αn ) whenever the ordinary limit exists; • limω (α1 , α1 , α2 , α2 , α3 , α3 , . . . ) = limw (αn ). The functional Trω can be defined for positive operators, and later it can be extended to the whole ideal L(1,∞) (H ) by linearity (in this case Trω is not necessarily a positive functional). The functional Trω is trivial on the ideal L1 (H ) of trace class operators, (see [4] or [15]). The subset of L(1,∞) (H ) with Dixmier trace independent of the average function ω defines the class of Dixmier measurable operators. In noncommutative geometry, a remarkable result due to A. Connes shows that classical pseudo-differential operators acting on L2 (M) with order − dim(M), belong to the Dixmier class, and the Dixmier trace coincides with the noncommutative residue (see [41]), which is the unique trace (up to by factors) in the algebra of pseudo-differential operators with classical symbols on a closed manifold, (see [4, pag. 305] for this fact). R. Nest, E. Schrohe in [26], show that in general Connes’ result does not hold for the Boutet de Monvel algebra on a manifold with boundary (cf. (62) and (63)). More generally, the ideal L(p,∞) (H ) consists of those linear bounded operators A on H satisfying the condition:

sn (A) = O(N (1−1/p) ), N → ∞,

(4)

1≤n≤N

for 1 < p < ∞. On L(p,∞) (H ) the usual norm is given by AL(p,∞) (H ) := sup N N≥1

( p1 )−1



sn (A).

(5)

1≤n≤N

In order to present our results we introduce briefly the theory of pseudo-differential operators that we use here [9, 23, 32]. The notion of global symbol on Rn is natural because a pseudo-differential operator A is an integral operator defined by  Af (x) = Rn

ei2πxξ σA (x, ξ )fˆ(ξ ) dξ,

(6)

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for a suitable smooth function σA (x, ξ )—called the symbol of A—and satisfying some bounded conditions on its derivatives (see [23]). Here fˆ denotes the euclidean Fourier transform of the function f . If we consider a closed manifold M (i.e a compact manifold without boundary) and H = L2 (M), there exists a (global) Fourier analysis associated to every positive elliptic pseudo-differential operator E on M which gives for certain pseudo-differential operators A—called Einvariants—a discrete Fourier representation of the form Af (x) =

∞

σA,E (l)f(l), el (x)Cdl ,

(7)

l=0

where el (x) := (elm )1≤m≤dl and {elm : l ∈ N, 1 ≤ m ≤ dl } provides a basis of L2 (M) consisting of eigenfunctions elm , associated to certain eigenvalues λl , l ∈ N0 , of the operator E. The function σA,E is called the matrix valued symbol of A with respect to E. The process associating to every operator A acting in C ∞ (M) a such matrix valued symbol σA,E was introduced by J. Delgado and M. Ruzhansky, [13]. Also, If M = G is a compact Lie group and E = −LG is minus the Laplace– Beltrami operator on G, Ruzhansky and Turunen [32] give a discrete representation to every pseudo-differential operator A on C ∞ (G) in terms of the representation theory of the group G, in the following way Af (x) =



dξ Tr[ξ(x)σA (x, ξ )f(ξ )],

(8)

 [ξ ]∈G

 denotes the unitary dual of the group G. Ruzhansky–Turunen’s calculus here G m gives a characterization of the Hörmander classes ρ,δ (G) on a compact Lie group m G by using the notion of global symbols. In fact, A ∈ ρ,δ (G) if and only if its global symbol σA (x, ξ ) as in (8) satisfies   αξ ∂xβ σA (x, ξ )op ≤ Cα,β ξ m−ρ|α|+δ|β| , x ∈ G, [ξ ] ∈ G,

(9)

1  is the spectrum of −LG . In where ξ  := (1 + λ[ξ ] ) 2 and {λ[ξ ] : [ξ ] ∈ G} Remark 2.1 we will recall the relationship between the symbols σA,−L(G) and σA associated to the quantizations (7) and (8), respectively. Now, we recall that for a compact manifold M (without boundary) of dimension ", a classical pseudo-differential operator A on M of order m, can be defined by local symbols. This means that for any local chart U , the operator A has the form

 Au(x) =

eixξ σ A (x, ξ ) u(ξ ) dξ Tx∗ U

(10)

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where σ A (x, ξ ) is a smooth function on T ∗ U ∼ = U × Rn , Tx∗ U = Rn , admitting an asymptotic expansion σ A (x, ξ ) ∼

∞

σ Am−j (x, ξ )

(11)

j =0

where σm−j (x, ξ ) are homogeneous functions in ξ = 0, of degree m − j for ξ far from zero. The set of classical pseudo-differential operators of order m is  denoted by clm (M). If A ∈ cl (M), for all x ∈ M, |ξ |=1 σ−" (x, ξ ) dξ dx defines a local density which can be glued over M. So, the noncommutative residue, is the functional defined on classical operators by res (A) =

1 "(2π)"



 σ A−" (x, ξ ) dξ dx.

(12)

M |ξ |=1

An important feature of the noncommutative residue is that it vanishes on noninteger order classical operators. A complementary trace to the noncommutative residue is the canonical trace, A → TR(A), on the set of classical pseudodifferential operators of non-integer order. The term complementary is justified because the noncommutative residue is defined on the whole algebra of pseudodifferential operators, but does not extend the usual L2 -trace. However the canonical trace is not well defined on the whole algebra of classical pseudo-differential operators, but it extends the usual L2 -trace on a suitable set, see, e.g. [7, 22, 28, 29]. Now we present our main results. Our formulae for Dixmier traces will be presented for positive operators in L(1,∞) (L2 (M)) because such formulae can be extended to the whole ideal L(1,∞) (L2 (M)) by linearity. We start with the case of manifolds without boundary. Here, " denotes the dimension of a compact manifold  to be without boundary M. For every compact Lie group G, we define L(1,∞) (G) dξ ×dξ  the class of vector-valued matrices M : G → ∪[ξ ]∈G , such that, for every C  M([ξ ]) ≡ M(ξ ) ∈ Cdξ ×dξ , and [ξ ] ∈ G, 1 N→∞ log N

ML(1,∞) (G)  := lim



dξ Tr(|M(ξ )|) < ∞.1

(13)

[ξ ]:ξ ≤N

In a similar way, we define ML(p,∞)(G)  := sup N N≥1

( p1 −1) dim G



dξ Tr(|M(ξ )|) < ∞,

(14)

[ξ ]:ξ ≤N

for every 1 < p < ∞.

1 As

usual, |M(ξ )| is defined via the spectral calculus of matrices by |M(ξ )| =

√

M(ξ )∗ M(ξ ).

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Theorem 1.1 Let M be a "-dimensional compact manifold without boundary and ν (M) be a positive elliptic pseudo-differential operator on M. If A : let E ∈ +e 2 L (M) → L2 (M) is a E-invariant bounded operator with matrix-valued symbol (σA,E (l))l , then we have, • A is Dixmier traceable, i.e. A ∈ L(1,∞) (L2 (M)) if and only if τ (A) :=

1 1 lim dim(M) N→∞ log N



Tr(|σA,E (l)|) < ∞,

1 l:(1+λl ) ν

(15)

≤N

where σA,E are as in (7). Moreover, if A is positive, τ (A) = Trw (A). • The E-invariant operator A ∈ L(p,∞) (L2 (M)) if and only if γp (A) := sup N

dim M( p1 −1)

·

N≥1

1 l:(1+λl ) ν

Tr(|σA,E (l)|) < ∞,

(16)

≤N

for all 1 < p < ∞. In this case, the norms AL(p,∞) (L2 (M)) and γp (A) are equivalents, which we denote by AL(p,∞) (L2 (M)) ) γp (A). If M = G is a compact Lie group then AL(p,∞) (L2 (G)) ) σA L(p,∞) (G)  .

(17)

 be the unitary dual • Let M = G be a compact Lie group of dimension " and let G of G. If we denote by σA (x, ξ ) the matrix valued symbol associated to A, then under the condition  ∀α ∈ Rn ,  αξ ∂xβ σA (x, ξ )op ≤ Cξ −"−|α| , x ∈ G, [ξ ] ∈ G,

(18)

the operator A is Dixmier measurable. Moreover, if A is left-invariant, then A is Dixmier traceable if and only if τ (A) :=

1 × σA L(1,∞) (G)  < ∞. dim(G)

(19)

In this case, if A is positive, τ (A) = Trw (A). • If A ∈ cl−" (G) is a classical, positive and left-invariant pseudo-differential operator, the noncommutative residue of A (cf. (12)), is given in terms of representations on G by res(A) = τ (A) ≡

1 × σA L(1,∞) (G)  . dim(G)

(20)

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Moreover, if A is as above and its symbol admits an asymptotic expansion in homogeneous components of the form: σ A (x, ξ ) ∼

∞

am−j (x)σ Am−j (ξ ),

(21)

j =0

then res(A) =

1 dim(G)

 G

a−" (x)dx × σA−" L(1,∞) (G)  .

(22)

It is important to mention that our formula (15) for the Dixmier trace of a global operator A on a closed manifold M has two components: one is, the inverse of the dimension of M, which has a geometric nature. The other one has an analytic nature, which is defined by the global symbol σA,E (·) of A, determined by the Fourier analysis associated to E. We observe that the approach in the preceding result can be used in order to analyze the Dixmier traceability of Fourier multipliers on compact homogeneous manifolds; this analysis in terms of the Fourier analysis and the representation theory of such manifolds has been considered in Theorem 4.1. Equation (22) provides a formula of the noncommutative residue for a class of global pseudodifferential operators on compact Lie groups. In particular, our approach, provide formulae for the noncommutative residue of operators on the torus, in a different way to the work of Pietsch [30]. Now we present our result concerning to global operators on a manifold with boundary. In the formulation of our result we use the global quantization of pseudodifferential operators on compact manifolds with boundary due to J. Delgado, M. Ruzhansky and N. Tokmagambetov [31] which we briefly describe as follows. If M denotes a compact manifold M with boundary ∂M and L is a pseudodifferential operator on M satisfying some boundary conditions on ∂M and with a merely discrete spectrum {λξ | ξ ∈ I} (see Sect. 2.3), then every continuous linear operator A acting on a suitable domain CL∞ (M), has associated a function σA,L : M × I → C—called the L-global symbol of A—satisfying σA,L (x, ξ ) = uξ (x)−1 A(uξ )(x), ξ ∈ I, x ∈ M,

(23)

where uξ is the eigenvalue corresponding to λξ . In this case, the operator A can be written in terms of the global symbol σA,L (x, ξ ) as Af (x) =



uξ (x)σA,L (x, ξ )f(ξ ), for f ∈ CL∞ (M),

(24)

ξ ∈I

where f := FL (f ) denotes the L-Fourier transform of f introduced in [31], (see also (79)).

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An L-Fourier multiplier is a bounded linear operator A : CL∞ (M) → CL∞ (M) if it satisfies FL (Af )(ξ ) = σA,L (ξ )FL (f )(ξ ), for f ∈ CL∞ (M),

(25)

for some function σA,L : I → C which depends only on the Fourier variable ξ ∈ I, in this case σA,L (ξ ) corresponds with the L-symbol of A. With the notation above our result on the Dixmier traceability of operators on manifolds with boundary can be enunciated in the following way. We write I = {ξl : l ∈ N0 }. Theorem 1.2 Let M be a compact manifold with boundary ∂M. If A : L2 (M) → L2 (M) is a bounded Fourier multiplier and L is a self-adjoint operator on L2 (M) as in Sect. 2.3, then we have the following assertions, • A is Dixmier traceable if and only if 1

|σA,L (ξl )| < ∞. N→∞ log N

τ  (A) := lim

(26)

l≤N

In this case, if A is positive, τ  (A) = Trω (A). • Moreover, if L is an operator of order m satisfying the Weyl Counting Eigenvalue Formula, that is, 1

NL (λ) := #{l : |λξl | m ≤ λ} = C0 λdim M + O(λdim M−1 ), λ → ∞,

(27)

then A is Dixmier traceable if only if (cf. (26)) τ  (A) =

1 1 lim dim M N→∞ log N



|σA,L (ξl )| < ∞.

(28)

1 l:|λξl | m ≤N

In this case, if A is positive, τ  (A) = Trω (A). • A ∈ L(p,∞) (L2 (M)) if and only if γp (A) := sup N N≥1

( p1 −1)



|σA,L (ξl )| < ∞,

(29)

l≤N

for all 1 < p < ∞. In this case, AL(p,∞) (L2 (M)) ) γp (A). Moreover, if L satisfies (27), we have γp (A) ) sup N N≥1

dim M( p1 −1)

1 l : |λξl | m

|σA,L (ξl )|. ≤N

(30)

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 Moreover, if we assume the existence of L as before, and A =



P+

is an elliptic

T

operator and positive in the Boutet de Monvel algebra of order n, then the operator PT (realization P with respect to T ) can be regarded as an operator in Ruzhansky– Tokmagambetov’s calculus (see Remark (5.2)), and any parametrix R of PT is Dixmier’s traceable with Trω (R) = res (R) = lim

N→∞

1

|(σP ,L (ξl ))−1 |, ln N

(31)

l≤N

here (σP ,L (ξl ))l∈N0 denotes the global symbol of the operator P with respect to LM . The expression is the same for all parametrices and independent of the choice of the boundary condition and independent of the average function ω. As it was pointed out in [25, 39], the condition (27) holds true for several classes of elliptic operators on manifolds with boundary and without boundary, which are not necessarily compact manifolds. For our purposes, (27) is not restrictive because such condition implies that for some s0 ∈ R, τ (s0 , L) :=



ξ −s0 < ∞,

(32)

ξ ∈I

which is an important condition in the Ruzhansky–Tokmagambetov quantization (See Sect. 2.3) for manifolds with boundary. For an exhaustive historical perspective on this subject, we refer to [25]. Now, we present some references on the Dixmier traceability of pseudo-differential operators and related topics. The problem of classifying global pseudo-differential operators on certain ideals of bounded operators—as Schatten von Neumann classes and Gröthendieck nuclear operators—in terms of global symbols has been considered of a very satisfactory way in the references [9–13]. Nevertheless, the problem of finding sufficient conditions on the symbol σ of a pseudo-differential operator in order that the corresponding operator A will be trace class, Hilbert Schmidt or Dixmier traceable is classical. If H = L2 (Rn ) and we consider the pseudo-differential operator A defined by the Weyl quantization, 

  Af (x) =

e Rn

Rn

i2π(x,ξ )

σA

 1 (x + y), ξ f (y)dydξ 2

(33)

is well known that σ ∈ L2 (R2n ) implies that A is Hilbert–Schmidt. In the framework of the Weyl–Hörmander calculus of operators A associated to symbols σ in the S(m, g)-classes (see the definition of these classes in [23]), there exist two remarkable results. The first, due to Hörmander, which asserts that if σ ∈ S(m, g) and σ ∈ L1 (R2n ) then A is a trace class operator. The second, due to L. Rodino and

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F. Nicola expresses that if σ ∈ S(m, g) and m ∈ L1w , (the weak-L1 space), then A is Dixmier traceable [27]. Moreover, an open conjecture by Rodino and Nicola (see [27]) says that σ ∈ L1w (R2n ) gives an operator A with finite Dixmier trace. More general regularized traces for pseudo-differential operators (see S. Scoot [38] or S. Paycha [28] for complete a description) have been studied in different contexts, in first instance, G. Grubb, [19, 20] has obtained results about the zeta-regularized trace by using a resolvent approach; S. Paycha [29] has obtained important results about traces on meromorphic families of pseudo-differential operator; B. Fedosov, F. Golse, E. Leichtnam, E. Schrohe, [16] have defined the noncommutative residue and its properties on the algebra of pseudo-differential boundary value problems called Boutet de Monvel’s algebra, in the case of manifolds with boundary; G. Grubb and E. Schrohe in [22] have studied defect formulas for regularized traces on the Boutet de Monvel algebra, whose operators are choosing to satisfy the socalled Hörmander transmission property. Recently, one of the authors has studied in [8] the traceability property and uniqueness of the canonical trace for a class of operators not necessarily satisfying the transmission property. Finally, we explain the structure of our paper. • In Sect. 2, we present some preliminaries on the quantization of global pseudodifferential operators on compact Lie groups and general compact manifolds with or without boundary and we briefly describe the Boutet de Monvel calculus. Also, we present some notions and known results about the Dixmier trace as well as the noncommutative residue. • In Sect. 3 we prove our main results for operators acting on functions defined on compact manifolds without boundary. The case of compact manifolds with boundary will be addressed in Section 5. • In Sect. 4 we study the Dixmier traceability of multipliers and we investigate the noncommutative residue in this setting. • We end our paper with some examples in Sect. 6.

2 Preliminaries In this section, we present some basics about the Fourier analysis used in this paper. Also, we present some definitions on the noncommutative residue (sometimes called Wodzicki’s residue) and the Dixmier trace for pseudo-differential operators on compact manifolds with or without boundary. In this article, L(1,∞) (H ) denotes the Dixmier ideal of the algebra of bounded operators on an Hilbert space H, L(H ), Specp (A) denotes the pointwise spectrum of a (not necessarily bounded) linear operator A on H, and Trω (·) is the Dixmier trace functional on the ideal L(1,∞) (H ).

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2.1 Pseudo-Differential Operators on Compact Manifolds Without Boundary We recall that for every m ∈ R and for every open subset U ⊂ Rn , the Hörmander class S m (U × Rn ), (for a detailed description see [23]), is defined by functions satisfying the usual estimates β

|∂xα ∂ξ σ (x, ξ )| ≤ Cα,β ξ m−|β| ,

(34)

for all (x, ξ ) ∈ T ∗ U ∼ = U × Rn and α, β ∈ Nn . Then, a pseudo-differential operator associated to a function σ ∈ S m (U × Rn ) is a operator of the form  σ (x, D)f (x) =

ei2π(x,ξ )σ (x, ξ )f(ξ )dξ, for f ∈ C0∞ (U ),

(35)

Rn

where C0∞ (U ) denotes the set of smooth compact supported functions on U . The set of such operators is denoted by  m (U × Rn ). On a smooth manifold M without boundary of dimension n, for every m ∈ R, the Hörmander class of order m, S m (M) is defined by smooth functions on the cotangent T ∗ M which in local coordinates coincides with symbols in some open sets U of Rn satisfying inequalities as in (34). We observe that the notion of symbol on arbitrary manifolds is of local nature. However, it is possible to define a notion of global symbol if we restrict our attention to the case of compact Lie groups. Such a notion was developed by M. Ruzhansky and V. Turunen in [32]. In this theory, every operator A mapping C ∞ (G) itself, where G is a compact Lie group, can be described in terms of representations of G as follows.  be the unitary dual of G (i.e, the set of equivalence classes of continLet G uous irreducible unitary representations on G), the Ruzhansky–Turunen approach establishs that A has associated a matrix-valued global (or full) symbol σA (x, ξ ) ∈  on the noncommutative phase space G × G  satisfying Cdξ ×dξ , [ξ ] ∈ G, σA (x, ξ ) = ξ(x)∗ (Aξ )(x).

(36)

Then it can be shown that the operator A can be expressed in terms of such a symbol as [32], Af (x) =



dξ Tr(ξ(x)σA (x, ξ )f(ξ )).

(37)

 [ξ ]∈G

An important feature in this setting is that the Hörmander classes  m (G), m ∈ R where characterized in [32, 34] by the condition: A ∈  m (G) if only if its

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matrix-valued symbol σA (x, ξ ) as before satisfies ∂xα Dβ σA (x, ξ )op ≤ Cα,β ξ m−|β| ,

(38)

for every α, β ∈ Nn . For a rather comprehensive treatment of this quantization process we refer to [32]. The notion of global symbol on arbitrary compact manifolds is more delicate. This problem has been considered by J. Delgado and M. Ruzhansky in [11, 12]. Now, we explain this notion. If M is a compact manifold without boundary with a volume element dx, and E is a positive elliptic classical pseudo-differential operator of order ν > 0, we say that A : C ∞ (M) → C ∞ (M) is E-invariant if A and E commutes. In this case we can associate a full symbol to A in terms of the spectrum of E as follows: the eigenvalues of E form a positive sequence λj , which we label as 0 ≤ λ0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λn ≤ · · · .

(39)

For every eigenvalue λj the corresponding eigenspace Hj = Ker(E − λj I ). If λ0 = 0 then d0 = dim H0 = dim Ker(E). By the spectral theorem we can write L2 (M) =

∞ -

(40)

Hj .

j =0 d

j a basis of Hj for j ∈ N, where dj = dim Hj , and for So, if we denote by {ej k }k=1 every f ∈ L2 (M), we define the E-Fourier transform of f (relative to the operator dj ) at j ∈ N by E and the basis {ej k }k,j

⎡

f, ej 1 L2 (M) ⎢ f, ej 2 L2 (M) ⎢ (FE f )(j ) := f(j ) := ⎢ .. ⎣ . f, ej dj L2 (M)

⎤ ⎥ ⎥ ⎥ ⎦

.

(41)

dj ×1

If ej denotes the column vector with entries ej 1 , ej 2 , · · · ej dj then, for every f ∈ L2 (M), it follows from the Parseval Theorem that f has a Fourier series representation of the type, f (·) =

∞

f(l), el (·)dl .

(42)

l=0

where ·, ·dl is the usual inner product on Cdl . If {σ (l)}l∈N0 is a sequence of matrices such that for every j, σ (j ) is a square matrix of order dj , the E-invariant

The Dixmier Trace and the Noncommutative Residue on Compact Manifolds

133

operator A associated to {σ (l)}l∈N0 is the operator on C ∞ (M) defined by (Af )(·) =

∞

σ (l)f(l), el (·)dl .

(43)

l=0

 (l) = σ (l)f(l) we refer to E-invariant operators A as E-Fourier Since Af multipliers, or simply by Fourier multipliers, multipliers or E-multipliers. An important feature in the Delgado–Ruzhansky’s approach is that there exist formulae between the symbol σ (l) of a E-multiplier A from D  (M) into D  (M) (we use the notation D  (M) for the space of distributions in M) and its Schwartz kernel K(x, y) (see [23]). In fact we have   σ (l) = K(x, y)Ql (x, y)∗ dxdy (44) M

M

and K(x, y) =

∞

Tr(σ (l)Ql (x, y))

(45)

l=0

where Ql (x, y) = el (y)el (x)t . We end this section with the following remark of global symbols on compact Lie groups which will be useful in our analysis related with the Dixmier trace for Fourier multipliers. Remark 2.1 If A is left-invariant (with respect to the Laplacian) then we have two notions of global symbols for A. One is defined in terms of the representation theory of the group G and we will denote this symbol by (σA (ξ ))[ξ ]∈G  , and the other one is that defined when we consider the compact Lie group as a manifold, and in this case the symbol will be denoted by (σA (l))l∈N0 . The relation of this two symbols was discovered in [13, pag. 25]. Now, we describe this relation. In the setting of compact Lie groups the unitary dual being discrete, we can enumerate the unitary dual as [ξj ], for j ∈ N0 . In this way we fix the orthonormal basis d

1

dξ

j j {ej k }k=1 = {dξ2j (ξj )il }i,l=1

(46)

where dj = dξ2j . Then, we have the subspaces Hj = span{(ξj )i,l : i, l = 1, · · · , dξj }. With the notation above we have ⎡

σA (ξl ) 0dξl ×dξl 0dξl ×dξl ⎢ ⎢0dξl ×dξl σA (ξl ) 0dξl ×dξl σA (l) = ⎢ .. .. ⎢ .. ⎣ . . . 0dξl ×dξl 0dξl ×dξl 0dξl ×dξl

⎤ . . . 0dξl ×dξl ⎥ . . . 0dξl ×dξl ⎥ . .. ⎥ .. ⎥ . . ⎦ . . . σA (ξl ) d ×d l

l

(47)

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As a consequence of this discussion we obtain the following relation: Tr(|σA (l)|) = dξl Tr(|σA (ξl )|), where |A| :=

(48)

√ A∗ A denotes the absolute value of the matrix A.

2.2 Global Fourier Multipliers on Compact Homogeneous Manifolds In order to present our theorem for multipliers on compact homogeneous spaces, we recall some definitions on the subject. Compact homogeneous manifolds can be obtained if we consider the quotient of a compact Lie groups G with one of its closed subgroups K—there exists an unique differential structure for the quotient M := G/K–. Examples of compact homogeneous spaces are spheres Sn ∼ = SO(n + 1)/SO(n), real projective spaces RPn ∼ = SO(n + 1)/O(n), complex projective spaces CPn ∼ = SU(n + 1)/SU(1) × SU(n) and more generally Grassmannians Gr(r, n) ∼ = O(n)/O(n − r) × O(r). 0 the subset of G,  representations of G, that are class I with Let us denote by G 0 if there exists at least one non respect to the subgroup K. This means that π ∈ G trivial invariant vector a with respect to K, i.e., π(h)a = a for every h ∈ K. Let us denote by Bπ to the vector space of these invariant vectors and kπ = dim Bπ . Now we follow the notion of Multipliers as in [1]. Let us consider the class of symbols !(M), for M = G/K, consisting of those matrix-valued functions σ :

1

→ G

∞ 1

Cn×n such that σ (π)ij = 0 for all i, j > kπ .

(49)

n=1

Following [1], a Fourier multiplier A on M is a bounded operator on L2 (M) such that for some σA ∈ !(M) satisfies Af (x) =



dπ Tr(π(x)σA (π)f(π)), for f ∈ C ∞ (M),

(50)

0 π∈G

where f denotes the Fourier transform of the lifting f˙ ∈ C ∞ (G) of f to G, given by f˙(x) := f (xK), x ∈ G. Remark 2.2 For every symbols of a Fourier multipliers A on M, only the upper-left block in σA (π) of the size kπ × kπ can be not the trivial matrix zero.

The Dixmier Trace and the Noncommutative Residue on Compact Manifolds

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2.3 Global Pseudo-Differential Operators on Compact Manifolds with Boundary The natural analogy of the algebra of pseudo-differential operators on a closed manifold is the algebra of Boutet de Monvel (BdM), which will be described in this section (we will use a kernel integral description of such algebra, but also we give a local symbolic description), where we follow the references [2, 3, 36]. It was defined in [16], by B. Fedosov, F. Golsche, E. Leichtmann, and E. Schrohe a linear functional trace, agreeing with the noncommutative residue whether the boundary is empty, and it can be shown that such functional is the unique trace on the BdM algebra algebra. Moreover, R. Nest and E. Schrohe showed in [26] that such functional does not coincide with the Dixmier trace (with the exception of the closed case where the boundary is empty), however, they provide a particular set of operators in Boutet de Monvel’s algebra where these two traces agree. Let M be an n-dimensional compact smooth manifold with boundary ∂M which is embedded on a compact closed manifold , (we use n to denote the dimension of the manifolds underlying instead of ", as in the case of a closed manifold). Boutet de Monvel’s approach comprises operators acting on smooth functions over M and over ∂M as follows. A matrix operator of the form  A=

P+ +G K T

S



¯ ⊕ C ∞ (∂M) → C ∞ (M) ¯ ⊕ C ∞ (∂M), : C ∞ (M)

(51)

is said to be an operator in the Boutet de Monvel algebra of order m ∈ Z and of type r ∈ N, if • P+ := r + P e+ is an operator, called a truncated operator, obtained from a classical pseudo-differential operator P on  (here r + corresponds to the restriction map from functions on  to functions on M and the last one e+ refers to the extension by zero from functions on M to functions ), and the operator P satisfies the Hörmander transmission property (see [2]), namely, if σ P (x, ξ ) denotes the local symbol satisfying (11), (in local coordinates (x, ξ ) = (x  , xn , ξ  , ξn ) ∈ Rn−1 × R × Rn−1 × R on the boundary) of P , then for all α, β and any j = 0, 1, · · · , holds ∂ξα ∂xβn σjP (x  , 0, 0, +1) = (−1)j −|α| ∂ξα ∂xβn σjP (x  , 0, 0, −1).

(52)

• K is called a Poisson operator of order m, defined by  (Kv)(x) =

Rn−1

  ˜  , xn , ξ  )F[v](ξ  )dξ  , eix ξ k(x

(53)

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D. Cardona and C. Del Corral

with k˜ ∈ S m−1 (Rn−1 , S(R+ )), in the sense of (58). In this case, the symbol associated with the operator K is defined by ˜  , xn , ξ  )) k(x  , ξ  , ξn ) := Fxn →ξn (k(x

(54)

(here Fx→y (f ) means the Fourier transform of f which transform the variable x to y) and it is assumed an expansion in homogeneous functions km−1−j (x  , ξ  , ξn ) with respect to ξ (for |ξ | ≥ 1) of degree m − 1 − j . • T is called a Trace Operator of order m, a type r ∈ N, and it has the form r−1 SJ γj + T  , where Sj is pseudo-differential operator on Rn−1 of order T = j =0

m − j , and T  can be defined by the integral operator (T  u)(x  ) =

 Rn−1

t  (x  , xn , ξ  )Fx  →ξ  [u](ξ  , xn )dxn dξ  ,

(55)

with t˜ ∈ S m (Rn−1 × Rn−1 , S(R+ )), in the sense of (58), similarly the symbol of T is defined by t (x  , ξ  , ξn ) := Fxn →ξn (t˜(x  , xn , ξ  )).

(56)

• G is called a singular Green operator (s.G.o.) of order m and type r; it is defined  by an operator having the form G = r−1 K j =0 j γj + G , where Kj are Poisson operators; γj u(x  ) := Dxn u(x  , xn )|xn =0 (defines trace operators), and G can be defined by the integral operator j



(G u)(x) :=

 

Rn−1

eix ξ



∞ 0

 g(x ˜  , xn , yn , ξ  )Fx  →ξ  [u](ξ  , yn ) dξ  ,

(57)

with g˜ ∈ S m (Rn−1 × Rn−1 ; S(R+ × R+ )), i.e. g˜ ∈ C ∞ (Rn−1 × Rn−1 ; S(R+ × R+ )) such that for all α, β, l, l  , k, k  



xnk Dxkn ynl Dyl n Dξα Dx  g(x ˜  , xn , yn , ξ  )L2 (R2 β

++ )





≤ C(x  )ξ  m+1−k+k −l+l −|α| (58)

for a suitable positive constant C(x  ) depending of x  . The function g˜ has an asymptotic expansion g˜ ∼ j ≥0 g˜ m−j where g˜ m−j satisfy the following homogenety property: for λ ≥ 1 and |ξ  | ≥ 1,  1  1 g˜ d−j x  , xn , yn , ξ  = λm+2−j g˜d−j (x  , xn , yn , ξ  ). λ λ

(59)

The Dixmier Trace and the Noncommutative Residue on Compact Manifolds

137

In this case, the symbol associated with the operator G is defined by g(x  , ξ  , ξn , ηn ) := Fxn →ξn Fyn →ηn (ex+n ey+n g˜  (x  , xn , yn , ξ  )),

(60)

here (Fφ)(ξ ) := (Fφ)(−ξ ) for any φ ∈ L2 (R), the homogeneity property (59) corresponds to gd−j (x  , λξ  , λξn , ληn ) = λd−j g(x  , ξ  , ξn , ηn ), |ξ | ≥ 1, λ ≥ 1.

(61)

• Finally, S is a classical pseudo-differential operator on Rn−1 of order m. We have described Boutet de Monvel’s algebra acting on functions with values in R or C, which our setting of interest, however, such description can be done for a general setting as vector bundles. Now, let us mention an important result about the set of the boundary operators described above. Theorem 2.1 (L. Boutet de Monvel, [3]) Matrix operators as in (51) form an algebra, called the Boutet de Monvel algebra, more precisely, the composition AB of an operator A ∈ B with all entries of order m and type r with an operator B ∈ B with all entries of order m and type r  yields an operator in B of order m + m and type max{m + r, r  }. Let us mention that Boutet de Monvel’s algebra contains the parametrix operators of elements in such algebra (whether this exists). A parametrix here means that AB − I = S1 and BA − I = S2 are regularizing operators; their types are 0 and m, respectively. On the one hand, in [16] it is defined a linear functional on B, which extends the noncommutative residue to the case of a manifold with boundary, res (A) =

(2π)−n n

  M

(2π)−n+1 + n

|ξ |=1



∂M

p−n (x, ξ ) dξ dx  |ξ  |=1

trn (g−n )(x  , ξ  ) + s1−n (x  , ξ  ) dξ  dx  , (62)

∞ where trn (g−n )(x  , ξ  ) := 0 g˜−n (x  , xn , xn , ξ  ) dxn . On the other hand, R. Nest and E. Schrohe in [26] study the noncommutative trace and the Dixmier trace for suitable sets of the Boutet de Monvel algebra. The authors obtain a first result about the Dixmier class. Proposition 2.1 ([26], Proposition 2.1.) A bounded operator on the zero order Sobolev space L2 (M) with range in the n-order Sobolev space H n (M) is an element of L(1,∞) (M); moreover, if its range even is contained in H n+1 (M) then it is trace class.

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D. Cardona and C. Del Corral

If Dm denotes the set of matrix-operator A in B, as (51), for which P is an operator of order m ∈ Z; G is a. s.G.o. of order m and type zero, K is of order m + 1 and type zero, and S is an operator on the boundary of order m + 1. In [26] is showed that any operator A in D−n defines a bounded map A : L2 (M) ⊕ L2 (∂M) → H −n (M) ⊕ H 1−n (∂M). Therefore, D−n forms a subset of the Dixmier class L(1,∞) (H ) where H = L2 (M; E) ⊕ L2 (∂M; F ). Moreover, the authors show in the following result a explicit formula for the Dixmier trace in terms of the terms appearing in the noncommutative residue (cf. (62)), for operators in D−n . Theorem 2.2 For A ∈ D−n acting on H = L2 (M; E) ⊕ L2 (∂M; F ), we have (2π)−n Trω (A) = n

  M

(2π)−n+1 + n−1

|ξ |=1



∂M

p−n (x, ξ ) dξ dx (63)





|ξ  |=1







s−n+1 (x , ξ ) dξ dx .

In particular, Trω (A) is independent of the averaging procedure ω. Remark 2.3 The Theorem 2.2 shows that Conne’s theorem does not hold for the Boutet de Monvel algebra, see (62). Given P : C ∞ (M) → C ∞ (M) a differential operator of order m > 0. In general one is interested in solving either the in-homogeneous problem P u = f on M; T u = g at ∂M,

(64)

for f and g given, or else the homogeneous problem P u = f on M; T u = 0 at ∂M.

(65)

In order to treat problem (64), one consider operator of the form   A=

P T



: H m (M) → L2 (M) ⊕ H m−m −1/2 (∂M),

(66)

where m denotes de order of the trace operator T and it is such that m < m. For treat (65) one studies the realization PT which is defined as the unbounded operator PT on L2 (M), acting like P over D(PT ) = {u ∈ H m (M) | T u = 0}.

(67)

The ellipticity implies that there is a parametrix to PT in Boutet de Monvel’s calculus. It is of the form B = (Q+ + G K); the pseudo-differential part Q is a parametrix to P , while G is a singular Green operator of order −m and type zero and K is a potential operator of order −m.

The Dixmier Trace and the Noncommutative Residue on Compact Manifolds

139

Proposition 2.2 ([26], Theorem 3.2.) Let PT be the above elliptic boundary value problem and B = (Q+ +G K) its parametrix. Then there is a regularizing singular Green operator G0 of type zero, i.e. an integral operator with smooth kernel on M × M, such that R = (Q+ + G + G0 ) has the following properties: (i) R maps L2 (M) to D(PT ) and (ii) RPT − I and PT R − I are finite rank operators whose range consists of smooth functions. It shown in Corollary 3.2. in [26] that the parametrix R in the above theorem is unique up to a regularizing singular Green operator of type 0.   Proposition 2.3 ([26, Corollary 3.2.]) If m = n and A = P+ is elliptic and T

positive, the Dixmier trace for an arbitrary parametrix R to the operator PT , we have   1 Trω (R) = (σ P (x, ξ ))−1 dξ dx, (68) (2π)n n M |ξ |=1 n where σnP (x, ξ ) denotes the homogeneous component for σ P (x, ξ ) of degree n. The expression is the same for all parametrices and independent of the choice of the boundary condition. Moreover, it coincides with the noncommutative residue res (R) for R. By using non-harmonic analysis, M. Ruzhansky and N. Tokmagambetov in [31] give a different approach in terms of global symbols, which we summarize here. As above, M denotes a compact manifold of dimension n with boundary ∂M, and LM denotes the boundary value problem determined by a pseudo-differential operator L of order m on function in L2 (M), on the interior of M, satisfying a suitable boundary conditions (BC) on ∂M. It is assumed the following conditions: • The operator L associated with the (BC) has a discrete spectrum {λξ ∈ C | ξ ∈ I}, where I is a countable subset of Zk for some k ≥ 1, and the eigenvalues are ordered with the occurring multiplicities in the increasing order |λj | ≤ |λk | for |j | ≤ |k|. The authors introduce the weight ξ  := (1 + |λξ |2 )1/2m , and they assume that there exists a number s0 ∈ R such that

ξ −s0 < ∞. τ (s0 , L) :=

(69)

(70)

ξ ∈I

• If uξ denote the eigenfunction in L2 (M) of L associated with the (BC) corresponding to the eigenvalue λξ for ξ ∈ I, so Luξ = λξ uξ for ξ ∈ I.

(71)

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The adjoint spectrum problem is L∗ vξ = λξ vξ for ξ ∈ I,

(72)

which is equipped with the conjugate boundary conditions (BC)∗ . We can take biorthonormal system {uξ } and {vξ } for ξ ∈ I, i.e. uξ  = vξ  = 1 for ξ ∈ I and uξ , vη  = 0 for ξ = η and uξ , vη  = 1 for ξ = η,

(73)

 where f, g := M f x)g(x) dx is the usual product on the Hilbert space L2 (M). In this context, we assume that the system {uξ | ξ ∈ I} forms a basis of L2 (M), 2 i.e. for any f ∈ L (M) there exists a unique series ξ ∈I aξ uξ (x), so {vξ | ξ ∈ I} is a basis of L2 (M) too. • Also we assume the functions uξ and vξ do not have zeros in M for all ξ ∈ I and there exits C > 0 and N ≥ 0 such that as ξ  → ∞, inf |uξ (x)| ≥ Cξ −N , inf |uξ (x)| ≥ Cξ −N .

x∈M

x∈M

(74)

In this case the systems {uξ | ξ ∈ I} is called a WZ-system (without zeros system). As it was mentioned in [14, Remark 2.2], the condition given by WZ-system can be removed, however, under this situation the analysis underlying leads a matrixvalued version of the symbolic calculus, similar to Sect. 2.1, which consists in vectors of eigenfunctions so that its elements do not all vanish at the same timetypical examples of such situation is of operators on compact Lie groups, as it was described before. Now, we describe some elements involved in Ruzhansky–Tokmagambetov’s calculus which will be needed in this paper. The space CL∞ (M) :=

∞ 2

D(Lk )

(75)

k=1

where D(Lk ) := {f ∈ L2 (M) | Lj f ∈ D(L), j = 0, 1, · · · , k}, so that the boundary condition (BC) are satisfied by all the operators Lj . The Fréchet topology of CL∞ (M) is given by the family of norms f C k := max Lj f L2 (M) , k ∈ N0 , f ∈ CL∞ (M). L

j ≤k

(76)

Similarly, it is defined CL∞∗ (M) corresponding to the adjoint L∗ by CL∞∗ (M) :=

∞ 2 k=1

D((L∗ )k )

(77)

The Dixmier Trace and the Noncommutative Residue on Compact Manifolds

141

where D((L∗ )k ) := {f ∈ L2 (M) | (L∗ )j f ∈ D(L), j = 0, 1, · · · , k}, which also has to satisfy the adjoint boundary conditions corresponding to the operator L∗ . The Fréchet topology of CL∞∗ (M) is given by the family of norms f C k ∗ := max (L∗ )j f L2 (M) , k ∈ N0 , f ∈ CL∞ (M). L

j ≤k

(78)

Since {uξ } and {vξ } are dense in L2 (M) that CL∞ (M) and CL∞∗ (M) are dense in

L2 (M).

L-Fourier transform: Let S(I) be the space of rapidly decreasing functions φ : I → C, i.e. for any N < ∞, there exists a constant Cφ,N such that |φ(ξ )| ≤ Cφ,N ξ −N for all ξ ∈ I. The space S(I) forms a Fréchet space with the family of semi-norms pk (φ) := supξ ∈I ξ k |φ(ξ )|. The L-Fourier transform is a bijective homeomorphism FL : CL∞ (M) → S(I) defined by (FL f )(ξ ) := fˆ(ξ ) :=

 (79)

f (x)vξ (x) dx. M

∞ The inverse operator F−1 L : S(I) → CL (M) is given by

(F−1 h)(x) :=



(80)

h(ξ )uξ (x),

ξ ∈I

so that the Fourier inversion formula is given by f (x) =



fˆ(ξ )uξ (x), f ∈ CL∞ (M).

(81)

ξ ∈I ∞ (M) → Similarly, the L∗ -Fourier transform is a bijective homeomorphism FL : CL∗ S(I) defined by

(FL∗ f )(ξ ) := fˆ∗ (ξ ) :=



−1 ∞ Its inverse F−1 L∗ : S(I) → CL∗ (M) is given by (FL∗ h)(x) := that the conjugate Fourier inversion formula is given by

f (x) :=



(82)

f (x)uξ (x) dx. M

fˆ∗ (ξ )vξ (x), f ∈ CL∞∗ (M).



ξ ∈I h(ξ )vξ (x)

so

(83)

ξ ∈I

The space DL (M) := L(CL∞∗ (M, C)) of linear continuous functionals on CL∞∗ (M) is called the space of L-distribution. By dualizing the inverse L-Fourier transform

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D. Cardona and C. Del Corral

∞ F−1 L : S(I) → CL (M), the L-Fourier transform extends uniquely to the mapping

FL : DL (M) → S (I)

(84)

 by the formula FL w, φ := w, F−1 L∗ φ with w ∈ DL (M), φ ∈ S(I). 2 Moreover, ) 2 * the authors define Sobolev2 space adapted to LM . The space lL := FL L (M) is defined as the image of L (M) under the L-Fourier transform. Then the space of lL2 is a Hilbert space with the linear product

(a, b)l 2 := L



a(ξ )(FL∗ ◦ F−1 b(ξ )).

(85)

ξ ∈I

Then the space lL2 consists of the sequences of the Fourier coefficients of function in L2 (M), in which Plancherel identity holds, for a, b ∈ lL2 , −1 (a, b)l 2 = (F−1 L a, FL b)L2 .

(86)

L

For f ∈ DL (M) ∩ DL∗ (M) and s ∈ R, we say that f ∈ HsL (M) if and only if ξ s fˆ(ξ ) ∈ lL2 ,

(87)

provided with the norm f HsL :=

)

ξ 2s fˆ(ξ )fˆ∗ (ξ )

*1/2

.

(88)

ξ ∈I

Proposition 2.4 ([26], Proposition 6.3.) For every s ∈ R, the Sobolev space HsL (M) is a Hilbert space with the inner product (f, g)HsL (M) :=



ξ 2s fˆ(ξ )gˆ∗ (ξ ),

(89)

ξ ∈I

and the Sobolev space HsL (M) and Hs (M) are isometrically isomorphic. L-Quantization and L-symbol: It can be shown that, see Theorem 9.2. in [31], any continuous linear operator A : CL∞ (M) → CL∞ (M) can be expressed in terms the L-Fourier and its inverse transform as

uξ (x)σA,L (x, ξ )fˆ(ξ ) (90) Af (x) = ξ ∈I

The Dixmier Trace and the Noncommutative Residue on Compact Manifolds

143

for every f ∈ CL∞ (M) and x ∈ M, where σA,L (x, ξ ) is called the L-symbol of A and can be computed as (see [31, Theorem 9.2]) σA,L (x, ξ ) = uξ (x)−1 (Auξ )(x) for all x ∈ M and ξ ∈ I.

(91)

In particular, if A : CL∞ (M) → CL∞ (M) be a continuous linear operator such that FL (Af )(ξ ) = σ (ξ )FL (f )(ξ ), for f ∈ CL∞ (M) and ξ ∈ I,

(92)

for some σ : I → C; then A is called a L-Fourier multiplier. The adjoint operator A∗ is defined by the equation Auξ , vξ L2 = uξ , A∗ vξ L2 .

(93)

It follows from the Parceval identity, see Proposition 6.1. in [31] that A is an LFourier multiplier by σ (ξ ) if and only if A∗ is an L∗ -Fourier multiplier by σ (ξ ). Moreover, in [31], the authors provided several results in the same asymptotic spirit of Hörmander calculus for global symbols, e.g. asymptotic formulas for the symbol of the adjoint operator A∗ and for the symbol of any parametrix R of A, in terms of the symbol of the operator A; for the symbol of the product AB in terms of the symbol of A and B.

3 The Dixmier Trace and the Noncommutative Residue of Fourier Multipliers on Manifolds Without Boundary In this section we proof our main results concerning to the Dixmier trace and the noncommutative residue for invariant operators on a closed manifold M (a compact manifold without boundary). Remark 3.1 We are interested in bounded E-multipliers on L2 (M). We recall that the following inequality sup σA,E (l)op < ∞

(94)

l∈N0

is a necessary and sufficient condition for the boundedness of A on L2 (M). This is an immediate consequence of the Plancherel Formula. Now, we present our main theorem of this section. Theorem 3.1 Let M be a compact manifold without boundary and let E in the set ν (M) of positive elliptic pseudo-differential operators on M. If A : L2 (M) → +e 2 L (M) is a bounded E- invariant operator with matrix-valued symbol σA,E (l), then

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A is Dixmier traceable if and only if τ (A) :=

1 1 lim dim(M) N→∞ log N



Tr(|σA,E (l)|) < ∞.

(95)

1 l:(1+λl ) ν ≤N

Moreover, if A is positive, τ (A) = Trw (A). Proof We want to show that (95) is a necessary and sufficient condition for the Dixmier traceability of a E-invariant operator A with full symbol σA,E (l). As a consequence of the relation FE (Af )(l) = σA,E (l)(FE f )(l), l ∈ N0 ,

(96)

we obtain Specp (A) =

1

Specp (σA,E (l)).

(97)

l∈N0

In fact, if λ ∈ Specp (A) = ker(A − λI ), there exists u = 0, such that Au = λu. If we take the E-Fourier transform to both sides, we have σA,E (l) u(l) = λ u(l). There exists l0 = 0, such that  u(l0 ) = 0. Hence we obtain that λ is a eigenvector of the matrix σ (l0 ). For the proof of the converse, let us assume that for some l0 ∈ N0 , the complex number λ ∈ Specp (σ (l0 )) is an eigenvalue with eigenvector u(l0 ). By contradiction let us assume that λ is not a element of !p (A). It follows that A is a Fourier multiplier it have not residual or continuous spectrum, therefore λ is in the resolvent set of A, in particular A − λI is an injective operator. So we have that (A−λI )v = 0 implies that v = 0. Let us consider the sequence of matrices (vl )l∈N0 , defined by vl = 0 (zero matrix of dimension dl ) if l = l0 , and vl0 = u(l0 ). If u is the inverse E-Fourier transform of this sequence, we have  u(l) = vl . It is clear that for every l, σA,E (l) u(l) = λ u(l). By taking the inverse E-Fourier transform, we have Au = λu with u = 0 which is a contradiction. √ Applying (97) to the operator 2 A∗ A we obtain the following relation for the singular values of A : s(A) =

1

s(σA,E (l)).

(98)

l∈N0

Moreover, if λ1,dl , λ2,dl , · · · λkl ,dl are the singular values of σA,E (l) with corresponding multiplicities m1,dl , m2,dl , · · · mkl ,dl , then m1,dl + m2,dl + · · · mkl ,dl = dl

(99)

and m1,dl λ1,dl + m2,dl λ2,dl + · · · mkl ,dl λkl ,dl = Tr[|σA,E (l)|],

(100)

The Dixmier Trace and the Noncommutative Residue on Compact Manifolds

145

$ where |σA,E (l)| = 2 σA,E (l)∗ σA,E (l). With notations above, A is Dixmier traceable (by definition) if and only if τ (A) := lim

N→∞

kl Tr(|σA,E (l)|) l≤N j =1 mj,dl λj,dl = lim < ∞. kl N→∞ log( log( l≤N dl ) mj,dl )

l≤N

l≤N

j =1

(101) In this case τ (A) = Trw (A). Since lim

N→∞

l≤N Tr(|σA,E (l)|) = lim N→∞ log( l≤N dl )

1

(1+λl ) ν ≤N

log(



Tr(|σA,E (l)|) 1

(1+λl ) ν ≤N

dl )

,

(102)

by using the Weyl Eigenvalue Counting formula for the operator E we have

1 (1+λl ) ν

dl = C0 L" + O(L"−1 ),

(103)

≤L

where " = dim(M). So, we obtain Trw (A) =

1 1 lim dim(M) N→∞ log N



Tr(|σA,E (l)|)

(104)

1 (1+λl ) ν ≤N



which is the desired result.

An important fact in the formulation of our analysis is the following result proved by A. Connes in the 1980s, (see [4, pag. 307]). Theorem 3.2 (A. Connes [4]) Let M be a closed manifold of dimension ". Then every classical pseudo-differential operator A of order −" lies in L(1,∞) (L2 (M)) and Trw (A) = res(A).

(105)

Now, we apply the preceding theorem and some tools of representation theory in order to study the Dixmier trace and the noncommutative residue for the operator on a compact Lie group.  be the Theorem 3.3 Let M = G be a compact Lie group of dimension " and G unitary dual of G. If we denote by σA (x, ξ ) ≡ σA,−LG (x, ξ ) the matrix valued symbol associated to A, then under the condition   αξ ∂xβ σA (x, ξ )op ≤ Cξ −"−|α| , x ∈ G, [ξ ] ∈ G,

(106)

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D. Cardona and C. Del Corral

the operator A is Dixmier measurable. Moreover, if A is left-invariant and positive, its Dixmier trace is given by Trw (A) =

1 1 lim dim(G) N→∞ log N



dξ Tr(|σA (ξ )|).

(107)

ξ :ξ ≤N

Proof We observe that by condition (38), the operator A is a pseudo-differential operator of order −". It follows from Theorem 5.2 in [33] that A is bounded on L2 (G). Then, by Connes’ Theorem in the form of Theorem 3.2, A is Dixmier traceable. Now, if A is a Fourier multiplier on G, from Remark 2.1 and Eq. (48), we deduce Tr(|σA (l)|) = dξl Tr(|σA (ξl )|). Hence (95) becomes

Tr(|σA,E (l)|) log[ l≤N dl ]

(108)

l≤N

Trw (A) = lim

N→∞

= lim

N→∞

dξl Tr(|σA (ξl )|) = lim N→∞ log[ l≤N dξ2l ]

l≤N



dξ Tr(|σA (ξ )|) log[ ξ ≤N dξ2l ]

(109)

ξ ≤N

which proves (107). Finally, that (107) is a necessary and sufficient condition for the Dixmier traceability of A it follows from (95). By the Weyl Eigenvalue Counting Formula (see [9, pag. 539]), we have

dξ2 = C0 L" + O(L"−1 ),

(110)

ξ ≤N

where " = dim(G). Hence Trw (A) =

1 1 lim dξ Tr(|σA (ξ )|) dim(G) N→∞ log N

(111)

ξ ≤N

which completes the proof.



With an analogous analysis as in the previous result we obtain the following characterization of invariant pseudo-differential operators belonging to the Marcinkiewicz ideal L(p,∞) (L2 (M)).

The Dixmier Trace and the Noncommutative Residue on Compact Manifolds

147

Theorem 3.4 Let M be a compact manifold without boundary and let A be a bounded E-invariant operator on L2 (M). Then A ∈ L(p,∞) (L2 (M)) if only if γp (A) := sup N



dim M( p1 −1)

N≥1

1 l:(1+λl ) ν

Tr(|σA,E (l)|) < ∞,

(112)

≤N

for all 1 < p < ∞. In this case γp (A) ) AL(p,∞) (L2 (M)). Proof With the notation above we can give a short proof for this fact. Indeed, for N > 0 and SN := l≤N dl , A ∈ L(p,∞) (L2 (M)) if and only if

1− p1

Tr(|σA,E (l)|) = O(SN

(113)

).

l≤N

By the Weyl counting eigenvalue function, (113) becomes

1 l:(1+λl ) ν

Tr( |σA,E (l)| ) = O((C0 N " )

1− p1

) = O(N

"(1− p1 )

),

(114)

≤N

where " = dim(M). This observation completes the proof, indeed, 1

AL(p,∞) (H ) ) sup N p



−1

N≥1

1 l:(1+λl ) ν

Tr(|σA,E (l)|) =: γp (A) < ∞.

(115)

≤N

Corollary 3.1 If M = G is a compact Lie group, and A is a Fourier multiplier, (125) gives that σA L(p,∞)(G)  := sup N

( p1 −1) dim G

N≥1



dξ Tr(|σA (ξ )|) < ∞,

(116)

[ξ ]:ξ ≤N

is a sufficient and necessary condition in order that A ∈ L(p,∞) (L2 (G)) for all 1 < p < ∞. Remark 3.2 The classification of E-invariant operators A on Schatten-von Neumann classes Sp (L2 (M)) = L(p,p) (L2 (M)) and on the class of r-nuclear operators Nr (L2 (M)) have been considered in [9–13]. A remarkable result in [11] shows that the inequality

a(ξ )rSr dξ < ∞,

(117)

ˆ [ξ ]∈G 1

where σA (ξ )Sr := Tr(|σA (ξ )|r ) r , is a necessary and sufficient condition for the r-nuclearity of A. The r-nuclearity and boundedness of global pseudo-differential

148

D. Cardona and C. Del Corral

operators on compact Lie groups in the general setting of Besov spaces has been studied by one of the authors in [5, 6]. Our result on the noncommutative residue can be enunciated in the following way: Theorem 3.5 Let G be a compact Lie group of dimension " and A : C ∞ (G) → D  (G) be a positive left invariant continuous linear operator. Then if A ∈  −" (G) is a positive classical pseudo-differential operator, then the noncommutative residue of A, is given in terms of representations on G by res(A) =

1 1 lim dim(G) N→∞ log N



dξ Tr(|σA (ξ )|)

(118)

ξ :ξ ≤N

Proof From the Connes trace theorem in the form of Theorem 3.2 we observe that the Dixmier trace of A coincides with its noncommutative residue and (118) now follows from the Eq. (107).

Now, we use our results on multipliers in order to provide formulae for a class of non-invariant operators (operators with symbols depending on the spatial variable x ∈ G). Proposition 3.1 Let G be a compact Lie group and " = dim G. Let us assume that A ∈ clm (G) is a positive classic pseudo-differential operator with local symbol, admitting homogeneous components, of the form: σ A (x, ξ ) ∼

∞

am−j (x)σ Am−j (ξ ).

(119)

j =0

Then the noncommutative residue of A is given by 1 res(A) = dim(G)

 G

a−" (x)dx × σA−" L(1,∞) (G)  .

(120)

Proof By definition of the noncommutative residue, considering the measure of Haar of a compact Lie group is normalized, and the Connes theorem we have   1 a−" (x)σ A−" (ξ )dξ dx "(2π)" G |ξ |=1   = a−" (x)vol" (x) × res(A−" ) = a−" (x)dx × Trω (A−" )

res(A) :=



G

G

1 1 lim a−" (x)dx × = dim(G) N→∞ log N G



dξ Tr(|σA−" (ξ )|).

ξ :ξ ≤N

(121)

The Dixmier Trace and the Noncommutative Residue on Compact Manifolds

149

We finish the proof by observing that σA−" L(1,∞) (G)  =

1 log N



dξ Tr(|σA−" (ξ )|).

(122)

ξ :ξ ≤N

We end this section with the following remark about the noncommutative residue for invariant operators. In order to distinguish the local and global symbol, let us use an upper notation to denote the local symbol of A by σ A (ξ ), i.e, the symbol of A defined as a section of the cotangent bundle T ∗ G on G (see [23]), with asymptotic A (ξ ) in positive-homogeneous components σ A A expansion σ (ξ ) ∼ σm−j m−j of degree m − j then by combining of the Eqs. (118) and (12) we deduce the following relation between the global and the local symbol of A. Corollary 3.2 Let A be a classical operator satisfying the conditions of the previous result. So, res (A) :=

1 "(2π)"

 |ξ |=1

1 = lim N→∞ " log N

A σ−" (ξ ) dξ

(123)



dξ Tr(|σA (ξ )|),

ξ :ξ ≤N

A denotes the component of degree −" in the asymptotic expansion of σ A . where σ−"

4 The Dixmier Trace of Fourier Multipliers on Compact Homogeneous Manifolds In Theorem 3.1, on a closed manifold, we give a characterization of Dixmier traceable E-invariant operators through its global symbols. In this section, we consider the special case of multipliers on compact homogeneous manifolds and we describe such characterization in terms of the representation theory of such manifolds. We generalize the corresponding assertion on multipliers given in Theorem 3.3, but in contrast with such result we do not consider the case of Hörmander classes as in such theorem. Now we present our main result of this section. Theorem 4.1 Let us assume that A is a Fourier multiplier as (50) on M := G/K. Then A is Dixmier traceable on L2 (M) if and only if τ (A) :=

1 1 lim dim(M) N→∞ log N

0 :π≤N [π]∈G

In this case, if A is positive, τ (A) = Trw (A).

dπ Tr(|σA (π)|) < ∞.

(124)

150

D. Cardona and C. Del Corral

Proof If A is a Fourier multiplier on M, from Remark 2.1 and the Eq. (48), we obtain Tr(|σA,LG/K (l)|) = dπl Tr(|σA (πl )|),

(125)

0 . If we use (95) we obtain where {[πl ] : l ∈ N0 } is a enumeration of G l≤N Tr(|σA,E (l)|) Trw (A) = lim N→∞ log[ l≤N dl ] πl ≤N dπ Tr(|σA (π)|) l≤N dπl Tr(|σA (πl )|) = lim = lim N→∞ N→∞ log[ π≤N dπl kπl ] log[ l≤N dπl kπl ] (126) which proves (107). Finally, that (107) is a necessary and sufficient condition for the Dixmier traceability of A it follows from (95). By the Weyl Eigenvalue Counting Formula (see [1, pag. 7] or [37]), we have

dπl kπl = O(C0 L" ),

(127)

0 :πl ≤N [πl ]∈G

where " = dim(M), and C0 =

1 (2π)"

 dx dw,

(128)

σ1 (x,w)