Hardy-Littlewood and Ulyanov inequalities 9781470447588

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Number 1325

Hardy-Littlewood and Ulyanov Inequalities Yurii Kolomoitsev Sergey Tikhonov

May 2021 • Volume 271 • Number 1325 (second of 7 numbers)

Number 1325

Hardy-Littlewood and Ulyanov Inequalities Yurii Kolomoitsev Sergey Tikhonov

May 2021 • Volume 271 • Number 1325 (second of 7 numbers)

Library of Congress Cataloging-in-Publication Data Cataloging-in-Publication Data has been applied for by the AMS. See http://www.loc.gov/publish/cip/. DOI: https://doi.org/10.1090/memo/1325

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26 25 24 23 22 21

Contents Basic notation

vii

Chapter 1. Introduction 1.1. Background 1.2. Ulyanov inequalities for moduli of smoothness 1.3. Specifics of the case 0 < p < 1 1.4. Moduli of smoothness, K-functionals, and their realizations 1.5. Main tool: Hardy–Littlewood–Nikol’skii polynomial inequalities 1.6. Main goals and work organization

1 1 2 6 7 8 9

Chapter 2. Auxiliary results 2.1. Besov and Triebel-Lizorkin spaces and their embeddings 2.2. Fourier multipliers 2.3. Estimates of Lp -norm of an integrable function in terms of its Fourier coefficients 2.4. Estimates of Lp -norms of polynomials related to Lebesgue constants 2.5. Hardy–Littlewood type theorems for Fourier series with monotone coefficients 2.6. Weighted Hardy’s inequality for averages

11 11 12

Chapter 3. Polynomial inequalities of Nikol’skii–Stechkin–Boas–types

19

Chapter 4. Basic properties of fractional moduli of smoothness

25

Chapter 5. Hardy–Littlewood–Nikol’skii inequalities for trigonometric polynomials 5.1. Hardy–Littlewood–Nikol’skii (Lp , Lq ) inequalities for 0 < p ≤ 1 5.2. Hardy–Littlewood–Nikol’skii (Lp , Lq ) inequalities for 1 < p < q ≤ ∞ 5.3. Hardy–Littlewood–Nikol’skii (Lp , Lq ) inequalities for directional derivatives Chapter 6.

13 14 16 17

31 35 44 46

General form of the Ulyanov inequality for moduli of smoothness, K-functionals, and their realizations 51

Chapter 7. Sharp Ulyanov inequalities for K-functionals and realizations

55

Chapter 8. Sharp Ulyanov inequalities for moduli of smoothness 8.1. One-dimensional results 8.2. Comparison between the sharp and classical Ulyanov inequalities 8.3. Multidimensional results

67 67 68 70

iii

iv

CONTENTS

Chapter 9. Sharp Ulyanov inequalities for realizations of K-functionals related to Riesz derivatives and corresponding moduli of smoothness 75 Chapter 10. Sharp Ulyanov inequality via Marchaud inequality 10.1. Inequalities for realizations of K-functionals 10.2. Inequalities for moduli of smoothness

79 79 84

Chapter 11. Sharp Ulyanov and Kolyada inequalities in Hardy spaces 11.1. Sharp Ulyanov and Kolyada inequalities in Hp (Rd ) 11.2. Sharp Ulyanov and Kolyada inequalities in analytic Hardy spaces

87 87 91

Chapter 12. (Lp , Lq ) inequalities of Ulyanov-type involving derivatives

95

Chapter 13. Embedding theorems for function spaces 13.1. New embedding theorems for Lipschitz spaces 13.2. Embedding properties of Besov and Lipschitz-type spaces

101 102 107

Bibliography

111

List of symbols

117

Abstract We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness ωα (f, t)q and ωβ (f, t)p for 0 < p < q ≤ ∞. A similar problem for the generalized K-functionals and their realizations between the couples (Lp , Wpψ ) and (Lq , Wqϕ ) is also solved. The main tool is the new Hardy–Littlewood–Nikol’skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity D(ψ)(Tn )q , 0 < p < q ≤ ∞, sup Tn D(ϕ)(Tn )p where the supremum is taken over all nontrivial trigonometric polynomials Tn of degree at most n and D(ψ), D(ϕ) are the Weyl-type differentiation operators. We also prove the Ulyanov and Kolyada-type inequalities in the Hardy spaces. Finally, we apply the obtained estimates to derive new embedding theorems for the Lipschitz and Besov spaces.

Received by the editor June 22, 2017, and, in revised form, March 4, 2018. Article electronically published on August 18, 2021. DOI: https://doi.org/10.1090/memo/1325 2020 Mathematics Subject Classification. Primary 41A63, 42B15, 26D10; Secondary 46E35, 26A33, 41A17, 26C05, 41A10. Key words and phrases. Moduli of smoothness, K-functionals, Ulyanov’s inequalities, Hardy– Litlewood–Nikol’skii’s inequalities, embedding theorems, fractional derivatives. Yu. Kolomoitsev is the corresponding author. He is at the Universit¨ at zu L¨ ubeck, Institut f¨ ur Mathematik, Ratzeburger Allee 160, 23562 L¨ ubeck, Germany; Institute of Applied Mathematics and Mechanics of NAS of Ukraine, General Batyuk Str. 19, 84116 Slov’yans’k, Ukraine; [email protected]. S. Tikhonov is at the Centre de Recerca Matem` atica Campus de Bellaterra, Edifici C 08193 Bellaterra (Barcelona), Spain; ICREA, Pg. Llu`ıs Companys 23, 08010 Barcelona, Spain, and Universitat Aut` onoma de Barcelona; [email protected] . The first author was partially supported by the project AFFMA that has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 704030. The second author was partially supported by MTM 2017-87409-P, 2017 SGR 358, and by the CERCA Programme of the Generalitat de Catalunya. c 2021 American Mathematical Society

v

Basic notation Let Rd be the d-dimensional Euclidean space with elements x = (x1 , . . . , xd ), and (x, y) = x1 y1 + · · · + xd yd , |x| = (x, x)1/2 , |x|1 = |x1 | + · · · + |xd |, |x|∞ = max1≤j≤d |xj |, . Let N be the set of positive integers, Zd be the integer lattice in Rd , and Td = Rd /2πZd . In what follows, X = Rd or Td . As usual, the space Lp (X) consists of all measurable functions f such that for 0 < p < ∞   p1 f Lp (X) = |f (x)|p dx 0 of a function f ∈ Lp (Td ), 0 < p ≤ ∞, which appeared for the first time in 1970’s (see [13, p. 788], [100]): (1.4)

ωα (f, δ)Lp (Td ) := sup Δα h f Lp (Td ) , |h|≤δ

where (1.5)

Δα h f (x) =

∞ 

(−1)ν

ν=0

    α f x + (α − ν)h , ν

h ∈ Rd ,

1.2. ULYANOV INEQUALITIES FOR MODULI OF SMOOTHNESS

3

    and αν = α(α−1)...(α−ν+1) , α0 = 1. It is clear that for integer α we deal with the ν! classical definition (1.3). Also, in the case of 0 < p < 1, since ∞  p     α    < ∞ for α ∈ N ∪ (1/p − 1), ∞ , ν ν=0   it is natural to assume that α ∈ N ∪ (1/p − 1)+ , ∞ while defining the fractional modulus of smoothness in Lp . Note that the moduli of smoothness of fractional order satisfy all usual properties of the classical moduli of smoothness (see Chapter 4). The sharp Ulyanov inequality for the fractional moduli of smoothness for 1 < p < q < ∞ reads as follows: 1  δ

q1 dt q1 1 1 t−θ ωα+θ (f, t)Lp (Td ) − . , θ=d (1.6) ωα (f, δ)Lq (Td )  t p q 0 Moreover, this inequality is sharp over the class     Lip ω(·), α + θ, p = f ∈ Lp (T) : ωk+θ (f, δ)p = O (ω(δ)) (see [90]). More precisely, for any function ω ∈ Ωα+θ (i.e., such that ω(δ) is nondecreasing and δ −α−θ ω(δ) is non-increasing), there exists a function   f0 (x) = f0 (x, p, ω) ∈ Lip ω(·), α + θ, p such that for any q ∈ (p, ∞) and for any δ > 0 ωα (f0 , δ)Lq (T)

 1

q dt q δ ≥C , t−θ ω(t) t 0

where a constant C is independent of δ and ω. Nowadays the Ulyanov inequalities are used in approximation theory, function spaces, and interpolation theory (see, e.g., [7, 16, 20, 32, 33, 38, 50, 51, 53–55, 69, 79, 90, 102, 107]). Very recently [32], sharp Ulyanov inequalities were shown between the Lorentz-Zygmund spaces Lp,r (log L)α−γ , α, γ > 0, and Lq,s (log L)α over Td in the case 1 < p ≤ q < ∞, and the corresponding embedding results were established. In the case of quasi-normed Lebesgue spaces Lp , 0 < p < 1, an Ulyanov inequality was established only in its not-sharp form given by (1.2) (see [25, 96] and [16, Ch. 3.7]). It is known (see [102]) that in the limiting cases p = 1, q < ∞ or 1 < p, q = ∞, inequality (1.6) does not hold in general. The best possible inequality is given by (1.7)

ωα (f, δ)Lq (T)

1  δ

q1 dt q1  max(1/p ,1/q) −θ t ln 2/t  ωα+θ (f, t)Lp (T) , t 0

 max(1/p ,1/q)  max(1/p ,1/q) where ln 2/t cannot be replaced by o ln 2/t . Interestingly, in the case p = 1 and q = ∞, inequality (1.6) is valid in the one-dimensional case (see [102]). Note that in this case θ = 1 and one can use the classical (non fractional) moduli of smoothness.

4

1. INTRODUCTION

Another improvement of the Ulyanov inequality (1.2) was suggested by Kolyada [50]: 1   1 1 δ  θ−α p dt p q dt q α−θ −θ t t ωα (f, t)Lp (T) δ ωα (f, t)Lq (T)  , t t δ 0 where 1 < p < q < ∞ and θ = 1/p − 1/q. Unlike the one-dimensional case, in the multidimensional case (d ≥ 2, θ = d(1/p − 1/q)), the corresponding inequality also holds for p = 1 < q < ∞. Roughly speaking, the sharp Ulyanov inequality refines inequality (1.2) with respect to the right-hand side while the Kolyada inequality refines (1.2) with respect to the left-hand side. Note that the Kolyada inequality is not comparable to the sharp Ulyanov inequality. In this paper, one of our main goals is to study sharp (Lp , Lq ) inequalities of Ulyanov-type with variable parameters for moduli of smoothness, generalized K-functionals, and their realizations. More precisely, we solve the long-standing problem (see [51, 112]) of finding the sharp inequalities between the moduli of smoothness ωα (f, t)Lq (Td ) and ωβ (f, t)Lp (Td ) in the general case: 0 < p < q ≤ ∞ and d ≥ 1. For the recent results on Rd , see [49]. One of our main contributions is the full description of sharp Ulyanov inequalities given by the following result. Theorem 1.1. Let f ∈ Lp (Td ), 0 < p < q ≤ ∞, α ∈ N ∪ ((1 − 1/q)+ , ∞), and α + γ ∈ N ∪ ((1/p − 1)+ , ∞). Then, for any δ ∈ (0, 1), we have (1.8) 1    δ   ωα+γ (f, t)Lp (Td ) q1 dt q1 ωα+γ (f, δ)Lp (Td ) 1 ωα (f, δ)Lq (Td )  σ , + 1 1 δγ δ t 0 td( p − q ) where (1) if 0 < p ≤ 1 and p < q ≤ ∞, then

⎧ 1 ⎪ td( p −1) , γ > d 1 − 1q ; ⎪ ⎪

+ ⎪ ⎪ 1 ⎪ d( p −1) 1 ⎪ t , γ = d 1 − ≥ 1, d ≥ 2, and α + γ ∈ N; ⎪ q ⎪ ⎪

+ ⎪ 1 1 ⎪ ⎪ ⎪ td( p −1) ln q1 (t + 1), γ = d 1 − 1q ≥ 1, d ≥ 2, and α + γ ∈ N; ⎪ ⎨ + 1 1 σ(t) := td( p −1) ln q (t + 1), 0 < γ = d 1 − 1q = 1 and d = 1; ⎪ ⎪

+ ⎪ 1 1 ⎪ d( −1) ⎪ t p ln q (t + 1), 0 < γ = d 1 − 1q < 1; ⎪ ⎪ ⎪

+ ⎪ ⎪ d( 1 − 1 )−γ ⎪ 1 ⎪ t p q , 0 1 and p ≤ 1 are different: γ = d(1/p − 1/q) and γ = d(1 − 1/q), correspondingly. It is important to mention that the sharp form of the Ulyanov inequalities, that is, an optimal choice of the function σ, essentially depends on the method we choose to measure the smoothness or, in other words, how we define a modulus of smoothness. For simplicity, we explain this in the one dimensional case. On the one hand, it is well known that the classical modulus of smoothness is equivalent to the K-functional (for 1 ≤ p ≤ ∞) and the realization functional (for 0 < p ≤ ∞) defined by means of the classical Weyl derivative. In this case, the sharp Ulyanov inequality is given by Theorem 1.1. On the other hand, if instead of the Weyl derivative in the definition of the realization concept, we consider the Riesz derivative, then the corresponding modulus of smoothness of order α > 0 is given by (see [85]) (1.9)

ωα (f, δ)Lp (T)

       β|ν| (α)  = sup  , − f (· + νh) − f (·)  β0 (α) 0 0,

1 ≤ p < ∞,

changes in Lp (T), 0 < p < 1.  Recall that the fractional derivative of an integrable function f (x) ∼ k f k eikx in the sense of Weyl is given by  iπα f (α) (x) = (ik)α f k eikx , (ik)α = |k|α e 2 sign k . k

The Bernstein inequality in Lp (T), 0 < p < 1, reads as follows (see [3] and [48, 83] for the multidimensional case). Proposition 1.1. Let 0 < p < 1. Then ⎧ α α ∈ Z+ or α ∈ Z+ and α > ⎪ ⎨ n 1, −1 (α) α∈  Z+ and α < p1 − 1; np , Tn Lp (T)

sup ⎪ 1 1 Tn Lp (T) ≤1 ⎩ n p −1 log p n, α = 1 − 1 ∈ Z . + p

1 p

− 1;

It is worth mentioning that if the polynomial Tn is such that spec(Tn ) ⊂ [0, n], that is, ck = 0 for k < 0, then the Bernstein inequality changes drastically: for any 0 < p ≤ ∞, one has Tn(α) Lp (T)  nα Tn Lp (T) ,

α > 0;

see Belinskii’s paper [2]. Second, let us discuss the smoothness of functions in the sense of behaviour of their moduli of smoothness in Lp . As an example, we consider the splines of maximum smoothness. Denote by hm the following function sequence: h1 (x) = π 2 sgn(cos x), x ∈ T, and  x 

1 hm−1 (t) − hm (x) = hm−1 (z)dz dt, m = 2, 3, . . . . 2π T 0 Note that up to a constant hm (x) ∼

 ei(2k+1)x . (2k + 1)m k∈Z

It is well known (see [19, p. 359]) that, for 0 < p ≤ ∞ and l ∈ N, we have  m−1+ 1 p, l ≥ m; δ (1.11) ωl (hm , δ)Lp (T)

l δ, l < m. Let us compare the classical and sharp Ulyanov inequalities for such functions in the case 0 < p < 1. First, we have ωk (hr+k , δ)Lp (T) δ k , and the standard

1.4. MODULI OF SMOOTHNESS, K-FUNCTIONALS, AND THEIR REALIZATIONS

7

(not-sharp) Ulyanov inequality (1.2) implies ωk (hr+k , δ)Lq (T)  δ k−(1/p−1/q) . At the same time, the sharp Ulyanov inequality (1.8) with γ = r ∈ N yields ωk (hr+k , δ)Lq (T) 

ωk+r (hr+k , δ)Lp (T) 1− 1 δ p δr 1    δ ωr+k (hr+k , t)Lp (T) q dt q +

δk , 1 1 t 0 tp−q

which is the best possible estimate since ωk (hr+k , δ)Lq (T) δ k . Therefore, the sharp Ulyanov inequality implies better estimates for Lp -smooth functions when 0 < p < 1. It is interesting to note that, for any f ∈ C ∞ (T) and for all 0 < p ≤ ∞, one has ωγ (f, δ)Lp (T) δ γ . In other words, taking into account (1.11), we see that spline functions in the case 0 < p < 1 are smoother than C ∞ -functions in the sense of the behavior of their moduli of smoothness. This phenomena disappears if we deal with absolutely continuously functions. More precisely, if f (α−1) ∈ AC(T), α ∈ N, and ωα (f, δ)p = o(δ α ) as δ → 0, then f ≡ const (see [99] and Proposition 12.1). 1.4. Moduli of smoothness, K-functionals, and their realizations The key approach to obtain the sharp Ulyanov inequalities is to use the realizations of the K-functionals. For simplicity, we start with the one-dimensional case. It is known (see [5, p. 341]) that if 1 ≤ p ≤ ∞, then the classical modulus of smoothness is equivalent to the K-functional given by

inf Kα (f, δ)Lp (T) := f − gLp (T) + δ α g (α) Lp (T) , g (α) ∈Lp (T)

that is, (1.12)

ωα (f, δ)Lp (T) Kα (f, δ)Lp (T) ,

1 ≤ p ≤ ∞,

α > 0.

This equivalence fails for 0 < p < 1 since in this case Kα (f, δ)Lp (T) ≡ 0 (see [23]). A suitable substitute for the K-functional for p < 1 is the realization concept given by

f − T Lp (T) + δ α T (α) Lp (T) . (1.13) Rα (f, δ)Lp (T) = inf T ∈T[1/δ]

The next result was proved in [23] for α ∈ N. For any positive α, the proof follows from Theorem 3.1 below. Proposition 1.2. (Realization result) Let f ∈ Lp (T), 0 < p ≤ ∞, and  α ∈ N ∪ (1/p − 1)+ , ∞ . Then, for any δ ∈ (0, 1), we have ωα (f, δ)Lp (T) Rα (f, δ)Lp (T) .

(1.14)

In the multidimensional case, an analogue of equivalence (1.12) holds for the classical moduli of smoothness of integer order (see [5, p. 341]). More precisely, if 1 ≤ p ≤ ∞ and α ∈ N, we have for f ∈ Lp (Td )   ωα (f, δ)Lp (Td ) inf f − gLp (Td ) + δ α gW˙ α (Td ) , ˙ α g∈W p

where f W˙ pα (Td ) =

 |ν|1 =α

Dν f Lp (Td ) ,

p

Dν = Dν1 ,...,νd =

∂ ν1 +···+νd . ∂xν11 . . . ∂xνdd

8

1. INTRODUCTION

For the fractional moduli of smoothness (α > 0), Wilmes [114] proved that, for 1 < p < ∞,   (1.15) ωα (f, δ)Lp (Td )

f − gLp (Td ) + δ α (−Δ)α/2 gLp (Td ) , inf (−Δ)α/2 g∈Lp

where where (−Δ)α/2 g(x) =

(1.16)



|k|α g k ei(k,x) .

k∈Zd

The corresponding realization result (see, e.g., [32]) is given by   ωα (f, δ)Lp (Td ) inf f − T Lp (Td ) + δ α (−Δ)α/2 T Lp (Td ) . T ∈T[1/δ]

We will show (see Theorem 4.1 below) that for any 0 < p ≤ ∞ and for any α ∈ N ∪ ((1/p − 1)+ , ∞) the following realization result holds (1.17)



ωα (f, δ)Lp (Td )

where

∂ ∂ξ

inf

T ∈T[1/δ]

f − T Lp (Td ) + δ α

sup ξ∈Rd , |ξ|=1

  α   ∂   T  ∂ξ 

 ,

Lp (Td )

α T is the directional derivative of order α, that is, 

∂ ∂ξ

α T (x) =



α

(i(k, ξ)) ck ei(k,x) .

|k|∞ ≤[1/δ]

Equivalence (1.17) is a crucial relation to obtain sharp Ulyanov-type inequalities in the multidimensional case for all 0 < p ≤ q ≤ ∞. 1.5. Main tool: Hardy–Littlewood–Nikol’skii polynomial inequalities The Hardy–Littlewood inequality for fractional integrals states that for any  f ∈ Lp (Td ) such that Td f (x)dx = 0, we have (1.18)

(−Δ)

−θ/2

 f Lq (Td )  f Lp (Td ) ,

1 < p < q < ∞,

θ=d

1 1 − p q

 .

Remark that (1.18) does not hold outside the range 1 < p < q < ∞. On the other hand, the following Nikol’skii inequality (sometimes called the reverse H¨older inequality) holds for any trigonometric polynomial Tn of degree at most n:   1 1 θ (1.19) Tn Lq (Td )  n Tn Lp (Td ) , 0 < p < q ≤ ∞, θ = d − . p q Both inequalities are, in a way, the limiting cases of the following general estimate: (1.20)

(−Δ)−γ/2 Tn Lq (Td )  σ(n)Tn Lp (Td ) ,

0 < p < q ≤ ∞.

The key step in our proof of Ulyanov-type inequalities, which is of its own interest, is to obtain a sharp asymptotic behavior of σ(n) = σ(n, p, q, γ, d). Our result reads as follows (see Corollary 5.2).

1.6. MAIN GOALS AND WORK ORGANIZATION

If 0 < p ≤ 1 and p < q ≤ ∞, then ⎧ 1 ⎪ nd( p −1) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ nd( p1 −1) ln q11 (t + 1), (−Δ)−γ/2 Tn Lq (Td ) sup

1 1 ⎪ Tn Lp (Td ) Tn ∈Tn ⎪ nd( p − q )−γ , ⎪ ⎪ ⎪ ⎪ ⎩ d( p1 − 1q ) n , (1)

(2)

sup

If 1 < p < q ≤ ∞, then

Tn ∈Tn

⎧ 1, ⎪ ⎪ ⎪ ⎨ 1,

9

γ > d 1 − 1q ; + 0 < γ = d 1 − 1q ;

+ 0 < γ < d 1 − 1q ; +

γ = 0.

γ ≥ d( p1 − 1q ), q < ∞; γ > dp , q = ∞;

(−Δ)−γ/2 Tn Lq (Td ) 1

ln p (n + 1), γ = dp , q = ∞; ⎪ Tn Lp (Td ) ⎪ ⎪ ⎩ d( p1 − q1 )−γ , 0 ≤ γ < d( p1 − 1q ). n

In particular, this implies the following analogue of the Hardy–Littlewood inequal ity (1.18) for the limiting cases: If 1 ≤ p < q ≤ ∞, f ∈ Lp (Td ), and Td f (x)dx = 0, then (−Δ)−γ/2 f Lq (Td )  f Lp (Td ) holds provided γ > d(1/p − 1/q), p = 1 or/and q = ∞. In fact, we will study a more general problem than (1.20) by considering the Weyl-type derivatives defined by homogeneous functions in place of powers of Laplacians. 1.6. Main goals and work organization In this paper, our main goals are the following: 1. To prove Hardy–Littlewood–Nikol’skii polynomial inequalities for the generalized Weyl derivatives with the whole range of parameters 0 < p < q ≤ ∞. 2. To obtain sharp (Lp , Lq ) inequalities of Ulyanov-type with variable parameters for moduli of smoothness, generalized K-functionals, and their realizations. 3. To apply Ulyanov inequalities to obtain new optimal embedding theorems of Lipschitz-type and Besov spaces. The paper is organized as follows. In Chapter 2, we collect auxiliary results on embedding theorems for Besov and Triebel–Lizorkin spaces, Fourier multiplier theorems, and Hardy–Littlewood theorems on Fourier series with monotone coefficients. Moreover, we obtain a result which connects the Lp -norms of the Fourier transform of a continuous function and Lp -norm of polynomial generated by this function. Chapter 3 deals with Nikol’skii–Stechkin–Boas–type inequalities for the relationship between norms of derivatives and differences of trigonometric polynomials written as follows   α   ∂  Tn 

δ −α ωα (Tn , δ)Lp (Td ) , Tn ∈ Tn , sup    ∂ξ d d ξ∈R , |ξ|=1

Lp (T )

where 0 < δ ≤ π/n, 0 < p ≤ ∞, and α > 0. In particular, this relation plays a crucial role in obtaining the equivalence between moduli of smoothness and realization concepts, see (1.17).

10

1. INTRODUCTION

Basic properties of fractional multidimensional moduli of smoothness are given in Chapter 4. In Chapter 5, we obtain sharp Hardy–Littlewood–Nikol’skii inequalities for the generalized Weyl derivatives of trigonometric polynomials in the multidimensional case. In more detail, we find the asymptotic behavior of D(ψ)Tn Lq (Td ) , 0 < p < q ≤ ∞, sup  Tn ∈Tn D(ϕ)Tn Lp (Td ) where D(η) is the Weyl-type differentiation operators, i.e., D(η) :   generalized i(ν,x) i(ν,x) c e → η(ν)c e , and η is a homogeneous function of a certain ν ν ν ν =0 degree. In Chapter 6, we study properties of the generalized K-functionals and their realizations and prove a general form of sharp Ulyanov inequalities. Chapter 7 provides an explicit formula of the sharp Ulyanov inequality for the realization concepts. Chapter 8 deals with the sharp Ulyanov inequality for the moduli of smoothness in one-dimensional and multi-dimensional cases, where we use results of Chapters 6 and 7. Chapter 9 is devoted to the proof of sharp Ulyanov inequalities for moduli of smoothness and K-functional related to the Riesz derivatives. Chapter 10 deals with the sharp Ulyanov inequalities, which are obtained using Marchaud-type inequalities. In Chapter 11, we treat the Ulyanov and Kolyada-type inequalities in the real and analytic Hardy spaces. In Chapter 12, we prove the sharp Ulyanov inequalities involving derivatives. In particular, we improve the known estimate (see [26]) 1  δ

q1 dt q1 1 −r−( p − q1 ) (r) ωr+α (f, t)Lp (T) , 0 < p < q ≤ ∞, t ωα (f , δ)Lq (T)  t 0 where 1/p − 1/q < α. Finally, in Chapter 13, we discuss new embedding theorems for smooth function spaces, which follow from inequalities for moduli of smoothness.

CHAPTER 2

Auxiliary results 2.1. Besov and Triebel-Lizorkin spaces and their embeddings s Let us recall the definition of the Besov space Bp,q (Rd ) (see, e.g., [108]). We consider the Schwartz function ϕ ∈ S (Rd ) such that supp ϕ ⊂ {ξ ∈ Rd : 1/2 ≤ |ξ| ≤ 2}, ϕ(ξ) > 0 for 1/2 < |ξ| < 2 and ∞ 

ϕ(2−k ξ) = 1 if ξ = 0.

k=−∞

We also introduce the functions ϕk and ψ by means of the relations F ϕk (ξ) = ϕ(2−k ξ) and F ψ(ξ) = 1 −

∞ 

ϕ(2−k ξ).

k=1

We say that f ∈ S  (Rd ) belongs to the (non-homogeneous) Besov space s Bp,q (Rd ), s ∈ R, 0 < p, q ≤ ∞, if   1q ∞ q sqk s (Rd ) = ψ ∗ f L (Rd ) + 2 ϕk ∗ f Lp (Rd ) , θ= − . q r 2 s p 2 d( 1 − 12 )

If, in addition, f ∈ Lq (Rd ) and (−Δ)s/2 f ∈ Lr (Rd ), then f ∈ B˙ 2,pp

(Rd ).

The next result easily follows from the definition of the Besov spaces (see, e.g., [108, 3.4.1, p. 206]; mind the typo in [108, 5.3.1, Remark 4, p. 239]. Lemma 2.3. We have d

f (λ·)B˙ s

p,q (R

d)

 λs− p f (·)B˙ s

p,q (R

d)

.

2.2. Fourier multipliers In this paper, we will use Fourier multipliers only for periodic functions. Recall that a function m : Rd → C is called a Fourier multiplier (we will write m ∈ Mp ) if there exists a constant C = C(m, p, d) such that for any ε > 0 and f ∈ Lp (Td ), 1 ≤ p ≤ ∞, one has        

k ei(k,x)  ≤ C 

k ei(k,x)  ,  m(εk) f f     p

k∈Zd

where f k =

1 (2π)d

 Td

p

k∈Zd

f (x)e−i(k,x) dx,

k ∈ Zd ,

2.3. ESTIMATES OF Lp -NORM OF AN INTEGRABLE FUNCTION

13

are the Fourier coefficients of the function f . We need the following properties of Fourier multipliers, which can be found, for example, in [67], [76, p. 119] or [93, Ch. IV, 3.2]. Lemma 2.4. (i) (Mikhlin–H¨ormander’s theorem) Let 1 < p < ∞. If sup |ξ||ν|1 |Dν m(ξ)| < ∞

for all

0 ≤ |ν|1 ≤ [n/2] + 1,

ν ∈ Zd+ ,

ξ∈Rd

then m ∈ Mp . d/2 (ii) (Peetre’s theorem) If m ∈ B˙ 2,1 (Rd ), then m ∈ M1 or, equivalently, m ∈ M∞ . 2.3. Estimates of Lp -norm of an integrable function in terms of its Fourier coefficients The following result allows us to estimate a (quasi-)norm of an integrable function via its Fourier coefficients. In the case p = 1, this lemma was proved in [110]. Lemma 2.5. Let 0 < p ≤ 1, Φ ∈ L1 (Td ), and  ϕ(k)ei(k,x) . Φ(x) ∼ k∈Zd d( − ) 2 Let ϕc be such that ϕc (k) = ϕ(k) for any k ∈ Zd and ϕc ∈ B˙ 2,pp 2 ∩ B˙ 2,1 (Rd ). Then , ΦLp (Td ) ≤ Cϕc  d( p1 − 12 ) 1

B˙ 2,p

1

(Rd )

where a constant C is independent of Φ. d

2 (Rd ), there exists g ∈ L1 (Rd ) such that Proof. For any ϕc ∈ B˙ 2,1  −d/2 g(t)ei(t,x) dt ϕc (x) = (2π)

Rd

(see, for example, [76, p. 119]). For k ∈ Zd , we get  ϕ(k) = ϕc (k) = (2π)−d/2 g(t)e−i(k,t) dt Rd   −d/2 = (2π) g(t)e−i(k,t) dt = (2π)−d/2 = (2π)−d/2

μ∈Zd

Td +2πμ

μ∈Zd

Td

 



Td

where gT (t) ∼



g(t + 2πμ)e−i(k,t) dt

gT (t)e−i(k,t) dt,

g(t + 2πμ),

t ∈ Td .

μ∈Zd

Using Beppo Levi’s theorem, we have gT ∈ L1 (Td ). Since  gT (t) ∼ ϕ(k)ei(k,t) , k∈Zd

d

14

2. AUXILIARY RESULTS

then ΦLp (Td ) = gT Lp (Td ) ≤ gLp (Rd ) ≤ Cϕc 

d( 1 − 1 ) 2

B˙ 2,pp

, (Rd )

where in the last estimate we have used the Bernstein-type inequality f Lp (Rd ) ≤ Cf 

d( 1 − 1 ) 2

B˙ 2,pp

(Rd )



(see, e.g., [76, p. 119]).

2.4. Estimates of Lp -norms of polynomials related to Lebesgue constants We will widely use the following anisotropic result which connects the norms of the Fourier transform of a function and the polynomial generated by this function (see [111, p. 106]). Note that for p = 1, this question is closely related to a study of Lebesgue constants of approximation methods (see [1], [60], and [111, Ch. 4 and Ch. 9]). Theorem 2.1. Let 0 < p ≤ 1. Let ϕ ∈ C(Rd ) have a compact support and let ϕ

∈ Lp (Rd ). Then ⎛ sup

⎝

εj =0, j=1,...,d

where Φε (x) =

d 

⎞1− p1 |εj |⎠

Φε Lp (Td ) = (2π)d/2 ϕ

Lp (Rd ) ,

j=1



ϕ(ε1 k1 , . . . , εd kd )ei(k,x) ,

ε = (ε1 , . . . , εd ).

k∈Zd

Note that this theorem was proved in the case ε1 = · · · = εd in [1] for p = 1 and in [111, p. 106] for 0 < p ≤ 1. In applications below, the anisotropic case (εj = εi ) plays an essential role. Proof. We follow the proof of Theorem 4.4.1 from [111]. First, let us verify the estimate from above. By the Poisson summation formula (see [93, Ch. VII]), we get 



x1 + 2πk1 xd + 2πkd ϕ

,..., ε εd 1 d



k∈Z

= (2π)−d/2

d 

|εj |

j=1

= (2π)−d/2

d  j=1



ϕ (−ε1 k1 , . . . , −εd kd ) ei(k,x)

k∈Zd

|εj |

 k∈Zd

ϕ (ε1 k1 , . . . , εd kd ) e−i(k,x) .

2.4. ESTIMATES OF Lp -NORMS OF POLYNOMIALS

15

Further, simple calculations imply the following relations: ⎛ ⎞p−1 d  ⎝ |εj |⎠ Φε pLp (Td ) j=1

= (2π)pd/2

d 

|εj |−1



T

j=1

≤ (2π)pd/2

d 

|εj |

−1

= (2π)

d 

|εj |

−1

k∈Zd

 p    x1 + 2πk1 xd + 2πkd  ϕ

,...,   dx ε1 εd



Td k∈Zd

j=1 pd/2

   p  x1 + 2πk1 xd + 2πkd   ϕ

, . . . ,  dx  ε1 εd d

j=1



Td +2πk

k∈Zd

   p  x1 xd  ϕ  ε1 , . . . , εd  dx

pLp (Rd ) . = (2π)pd/2 ϕ To obtain the estimate from below, we use the substitution xj → εj xj , j = 1, . . . , d, which gives ⎛ ⎞p−1 d  ⎝ |εj |⎠ Φε pLp (Td ) j=1

⎛

=⎝

d 

⎞p |εj |⎠



2π ε1

 ···

0

j=1

0

2π εd

 p    −i( d εj kj xj )   j=1 ϕ (ε1 k1 , . . . , εd kd ) e   dx. k∈Zd

Assuming that the limit inferior of this value as |ε| = (ε21 + · · · + ε2d )1/2 → 0 is finite and equals M , we will show that

pLp (Rd ) ≤ M. (2π)pd/2 ϕ Given arbitrary N > 0 and δ > 0, we obtain for sufficiently small |ε|  (2.2) ⎛ ≤⎝

 p d     −i( d   j=1 εj kj xj ) ϕ (ε k , . . . , ε k ) e ε 1 1 d d j  dx  d

NT d 

j=1

|εj |⎠

j=1

k∈Zd

⎞p



2π ε1

 ···

0

0

2π εd

 p    −i( d ε k x ) j j j   dx ≤ M +δ. j=1 ϕ (ε1 k1 , . . . , εd kd ) e   k∈Zd

Noting that we deal with the Riemann as |ε| → 0 to obtain   p (2π)pd/2 |ϕ(x)|

dx = N Td

integral sum in (2.2), we pass to the limit

    d

NT

Rd

p  ϕ(y)e−i(x,y) dy  dx ≤ M + δ.

Letting N → ∞ and, consequently, δ → 0, we get the desired estimate from below. 

16

2. AUXILIARY RESULTS

Corollary 2.1. Let 0 < p Then ⎛ ⎞1− p1 d  ⎝ |εj |⎠ sup εj =0, j=1,...,d

j=1

≤ 1 and ϕ ∈ C ∞ (Rd ) have a compact support.     i(k,x)   ϕ(ε k , . . . , ε k )e 1 1 d d   k∈Zd

< ∞. Lp (Td )

2.5. Hardy–Littlewood type theorems for Fourier series with monotone coefficients It is well known that trigonometric series with monotone coefficients have several important properties. In particular, Hardy and Littlewood [117] studied the series  f (x) = an cos nx, an ≥ an+1 , n∈Z

and proved that a necessary and sufficient condition for f ∈ Lp (T), 1 < p < ∞, is  p p−2 < ∞. Moreover, n∈Z an n  f p

(2.3)



1/p apn np−2

.

n∈Z

It turns out that in applications one needs a more general condition on Fourier coefficients than just monotonicity. We will give a higher-dimensional generalization of the Hardy-Littlewood theorem for trigonometric series with general monotone coefficients. A non-negative sequence a = {am }, m = (m1 , . . . , md ) ∈ Nd , d ≥ 1, satisfies the GM d (β)-condition, written a ∈ GM d (β), β = {βk }, k = (k1 , . . . , kd ) ∈ Nd ([27]), if am −→ 0 and

∞ 

···

m1 =k1

(d) ≡

d 

∞ 

as

|m|1 → ∞

|(d) am | ≤ Cβk ,

k = (k1 , . . . , kd ) ∈ Nd ,

md =kd

j

and j am = am − am1 ,...,mj−1 ,mj +1,mj+1 ,...,md .

j=1

Lemma 2.6. (See [27, 28].) Let 1 < p < ∞, d ≥ 1, and let the Fourier series of f be of the following type (2.4)

f (x) ∼

∞  m1 =1

···

∞ 

am

md =1



cos mj xj

j∈B



sin mj xj ,

j∈N \B

where N = {1, 2, · · · , d} and B ⊆ N . (i) Let a ∈ GM d (β). Then ⎞ ⎛  p−2 1/p ∞ ∞ d   p ⎠ . ··· βm mj f Lp (Td )  ⎝ m1 =1

md =1

j=1

2.6. WEIGHTED HARDY’S INEQUALITY FOR AVERAGES

(ii) Let a ∈ GM d (β) with ∞ ∞   βk = ··· m1 =k1 /c

md =kd /c

Then (2.5)

⎛ f Lp (Td ) ⎝

∞ 

m1 =1

am1 ,...,md , m1 · · · md

∞ 

···

apm

c > 1,

 d

md =1

p−2 mj

17

am1 ,...,md ≥ 0. ⎞1/p ⎠

.

j=1

In particular (see also [68]), equivalence ( 2.5) holds for the series ( 2.4) satisfying the condition (d) am ≥ 0 for any m ∈ Nd . 2.6. Weighted Hardy’s inequality for averages We will frequently use the following weighted Hardy inequality (see, e.g., [58]). Lemma 2.7. Let 1 < λ < ∞. Suppose u(x), v(x) ≥ 0 on (0, δ). Then 

!

δ

u(x) 0

"λ

x

ψ(t) dt

 dx 

0

s

ψ λ (x)v(x)dx 0

holds for each ψ(x) ≥ 0 if and only if 1/λ   δ sup (2.6) u(x)dx 0 0, n ∈ N, and ξ ∈ Rd , 0 < |ξ| < π/n. Then, for any  ck ei(k,x) ∈ Tn , Tn (x) = |k|∞ ≤n

we have (3.1)

  α   ∂  α  Tn   ∂ξ  Δξ Tn p , p

where the constants in this equivalence depend only on p, α, and d. As we have already mentioned, inequalities of this type have a long and rich history starting from the 1940s. When d = 1 and 1 ≤ p ≤ ∞, this is the result of Nikol’skii [70], Stechkin [91], and Boas [8] (for integer α, see also [104, p. 214 and p. 251]), and Taberski [100] and Trigub [109] (for any positive α). When d = 1 and 0 < p < 1, (3.1) follows from [23] for α ∈ N and from [44] for α > 0. When d ∈ N and 1 ≤ p ≤ ∞, the part ”  ” was proved in [114, Theorem 3] for any α > 0. 19

20

3. POLYNOMIAL INEQUALITIES OF NIKOL’SKII–STECHKIN–BOAS–TYPES

Proof. First, let us prove the estimate from above in (3.1). Let the function v ∈ C ∞ (R), v(s) = 1 for |s| ≤ π and v(s) = 0 for |s| ≥ 3π/2. We define α

#  (i(k, ξ))

α v (k1 ξ1 )2 + · · · + (kd ξd )2 ei(k,x) Kξ (x) = (k,ξ) k∈Zd 2i sin 2 (in what follows we assume that 00 = 1). Note that Kξ is a trigonometric polynomial of degree at most C/|ξj | in variable xj . We can assume that ξj = 0 for any j = 1, . . . , d, otherwise we consider the same problem in the space Rd−k with some k > 0. Taking into account that α   ξα (k, ξ) α ck ei(k,x+ 2 ) Δξ Tn (x) = 2i sin 2 |k|∞ ≤n



12

and (k1 ξ1 )2 + · · · + (kd ξd )2 ≤ |k||ξ| ≤ π, one has that  α   ∂ ξα α (3.2) Tn (x) = (Kξ ∗ Δξ Tn ) x − , ∂ξ 2 where ∗ stands for the convolution of periodic functions. Let first 0 < p < 1. We will show that ⎛ ⎞1− p1   α  d    ∂  ⎝ Tn  |ξj |⎠ Kξ p Δα (3.3) ξ Tn p .    ∂ξ p

j=1

For this we need the following Nikol’skii’s inequality ([71], see also [73]) ⎞ p1 −1 ⎛ d  nj ⎠ Un1 ,...,nd p , Un1 ,...,nd 1  ⎝ j=1

where



Un1 ,...,nd (x) =

|k1 |≤n1

···



ak ei(k,x) ,

k ∈ Zd .

|kd |≤nd

By using (3.2) and the above Nikol’skii inequality for the polynomial Kξ (x)Δα ξ Tn (x − ξα/2 − t) of degree at most C/|ξj | in xj , we obtain   α p p   ∂  1 α   Tn (x) ≤ |Kξ (t)Δξ Tn (x − ξα/2 − t)|dt ≤  ∂ξ (2π)pd Td ⎞p−1 ⎛  d  p   Kξ (t)Δα ⎠ ⎝ |ξj |  ξ Tn (x − ξα/2 − t) dt. j=1

Td

Integrating the above inequality over x, we get ⎛ ⎞p−1 p      α d  p   ∂   Kξ (t)Δα   ⎝ ⎠ Tn (x) dx  |ξj | ξ Tn (x − ξα/2 − t) dtdx.  ∂ξ d d d T

j=1

T

T

Thus, applying Fubini’s theorem derives (3.3).

3. POLYNOMIAL INEQUALITIES OF NIKOL’SKII–STECHKIN–BOAS–TYPES

We now show that gξ (t) =

$ d

1−1/p



α

j=1 |ξj |

(t, ξ)

v

2 sin (t,ξ) 2

21

Kξ p is bounded from above. Set

#

(t1 ξ1 )2 + · · · + (td ξd )2 .

Since g1 ∈ C ∞ (R), 1 = (1, . . . , 1), and g1 has a compact support, we obtain that, for any r ≥ 1, the following holds   1 g 1 (y) = O as |y| → ∞. |y|r The latter together with |g 1 (y)| ≤ C for |y| ≤ 1 gives that g 1 ∈ Lp (Rd ). Now, by using Theorem 2.1, we obtain ⎛ ⎞1− p1 ⎛ ⎞1− p1 ⎛ ⎞ p1 p   d d      ⎝ ⎝ |ξj |⎠ Kξ p = ⎝ |ξj |⎠ gξ (k)ei(k,x)  dx⎠  d j=1

T

j=1

⎞1− p1 ⎛  d  ⎝ ⎝ ⎠ = |ξj | ⎛

⎞ p1  p    g1 (ξ1 k1 , . . . , ξd kd )ei(k,x)  dx⎠  d

T

j=1

k∈Zd

k∈Zd

≤ (2π)d/2 g 1 Lp (Rd ) . Thus, from the last inequality and (3.3) we obtain the part ”  ” in (3.1). To prove the estimate ”  ”, we consider the polynomial

α (k,ξ) #

 2i sin 2 % ξ (x) = v (k1 ξ1 )2 + · · · + (kd ξd )2 ei(k,x) , K α (i(k, ξ)) d k∈Z

the equality

 Δα ξ Tn (x)

=

%ξ ∗ K



∂ ∂ξ

α

  ξα Tn x+ 2

and repeat the above proof. Let us consider the case 1 ≤ p ≤ ∞. By (3.2) and Young’s inequality, we have   α    ∂ α  Tn    Kξ 1 Δξ Tn p .  ∂ξ p

As above, we derive that Kξ 1  1, which implies the inequality ”  ” in (3.1). The converse inequality can be proved similarly.  Using now the fact that ωα (Tn , δ)p

sup ξ∈Rd , |ξ|=1

 α  Δξδ Tn  , p

we obtain the following result. Corollary 3.1. Let 0 < p ≤ ∞, α > 0, n ∈ N, and 0 < δ ≤ π/n. Then, for any Tn ∈ Tn , we have   α   ∂  −α Tn  (3.4) sup   ∂ξ  δ ωα (Tn , δ)p . d ξ∈R , |ξ|=1

p

Further, using (3.4) we prove the following Bernstein type inequality for fractional directional derivative of trigonometric polynomials.

22

3. POLYNOMIAL INEQUALITIES OF NIKOL’SKII–STECHKIN–BOAS–TYPES

Corollary 3.2. Let 0 < p ≤ ∞, α ∈ N ∪ ((1/p − 1)+ , ∞), and n ∈ N. Then, for any Tn ∈ Tn , we have   α   ∂  α Tn  (3.5) sup     n Tn p . ∂ξ d ξ∈R , |ξ|=1

p

Inequality (3.5) is the classical Bernstein inequality for d = 1 (see, e.g., [20, Ch. 4]). For the fractional α and d = 1 see [61] and [3], cf. [83]. For 1 ≤ p ≤ ∞, α > 0, and d ∈ N, inequality (3.5) can be obtained from [114, Theorem 3]. Note also that (3.5) does not hold for α ∈ N ∪ ((1/p − 1)+ , ∞) (see Proposition 1.1). Proof. To prove (3.5), one should take δ = 1/n in (3.4) and use the simple inequality ∞   1/min(1,p)   α min(1,p) α Δh f p ≤ f p ≤ C(α, p) f p   ν ν=0 

(cf. property (c) in Chapter 4).

For moduli of smoothness of integer order, we have the following Nikol’skii– Stechkin–Boas result, which is new in the case 0 < p < 1, d ≥ 2. Theorem 3.2. Let 0 < p ≤ ∞, r ∈ N, and 0 < δ ≤ π/n. Then, for any T ∈ Tn , n ∈ N, one has Tn W˙ pr δ −r ωr (Tn , δ)p ,

(3.6)

where the constants in this equivalence are independent of Tn and δ. Proof. Let us first show that

π . Tn W˙ pr  nr ωr Tn , n p

(3.7) Since

Tn W˙ pr

     ∂ r Tn   =  ∂xk1 . . . ∂xkd  , |k|1 =r

1

d

p

|k|1 = r, one has we have to verify that for any k ∈  

  ∂ r Tn    nr ωr Tn , π .  ∂xk1 . . . ∂xkd  n p Zd+ ,

1

Indeed, denoting Un (x) =

d

∂ r−k1 k k ∂x2 2 ...∂xdd

p

Tn (x) and applying Theorem 3.1 to Un in the

one-dimensional case, we obtain     k     1 ∂ r Tn   =  ∂ Un   nk1 Δkπ1 Un p  ∂xk1 . . . ∂xkd   ∂xk1  n ;x1 p p 1 1 d     ∂ r−k1 k1  = nk1  Δ T π  ∂xk2 . . . ∂xkd n ;x1 n  , p 2 d where Δkπ1;x1 Un is the k1 -th difference with respect to x1 . Repeating this procedure n d − 1 times, we get     π π ∂ r Tn kd k1 r r    ∂xk1 . . . ∂xkd   n Δ πn ;x1 . . . Δ nπ ;xd Tn p  n ωk1 ,...,kd Tn , n , . . . , n p , 1

d

p

3. POLYNOMIAL INEQUALITIES OF NIKOL’SKII–STECHKIN–BOAS–TYPES

23

where ωk1 ,...,kd (Tn , π/n, . . . , π/n)p is the mixed modulus of smoothness (see, e.g., [105]). It remains only to apply the following equivalence result  π π π ωk1 ,...,kd Tn , , . . . ,

ωr Tn , . n n p n p |k|1 =r

In the case 1 ≤ p ≤ ∞, the proof of this equivalence is given in [105] and is based on the representation of the total difference of a function via the sum of the mixed differences. Hence, repeating the proof of Theorem 7 from [105], one can easily verify that the above equivalence holds in the case 0 < p < 1 too. Thus, we have proved (3.7). Noting that Corollary 3.1 implies the equivalence nr ωr (Tn , π/n)p δ −r ωr (Tn , δ)p , 0 < δ < π/n, we conclude the proof of the part ”  ” in (3.6). The reverse part follows from Corollary 3.1.  We will also need the following result on equivalence between the Riesz derivatives (see (1.16)) and the directional derivatives. Corollary 3.3. Let 1 < p < ∞ and α > 0. Then, for any T ∈ T , one has   α   ∂  α/2 (3.8) sup  T  ∂ξ  (−Δ) T p . d ξ∈R , |ξ|=1 p Moreover, if 0 < p ≤ ∞ and r ∈ N, then   r   ∂  (3.9) sup  T   T W˙ pr . ∂ξ ξ∈Rd , |ξ|=1 p Proof. The first part follows from (3.4) and the equivalence δ −α ωα (T, δ)p

(−Δ)α/2 T p , where 0 < δ ≤ 1/ deg T (see [114]). Equivalence (3.9) follows from Theorem 3.2 and Corollary 3.1.  We conclude this chapter by the following new description of the classical Sobolev seminorm in terms of the norm defined by the directional derivatives. In the case 1 ≤ p ≤ ∞, this extends (3.9) for any function f ∈ Wpr . Theorem 3.3. Let 1 ≤ p ≤ ∞ and r ∈ N. Then, for any f ∈ Wpr , one has   r   ∂  f W˙ pr

sup  f   . ∂ξ ξ∈Rd , |ξ|=1 p Proof. The estimate (3.10)

sup ξ∈Rd , |ξ|=1

  r   ∂   f  ∂ξ   f W˙ pr p

easily follows from the multinomial theorem given by  r! xk1 xk2 . . . xkdd (x1 + x2 + · · · + xd )r = k1 !k2 ! . . . kd ! 1 2 |k|1 =r

and the triangle inequality. Let us prove the upper estimate for f W˙ r . Let Tn ∈ Tn be such that p

f − Tn W˙ pr → 0 as

n → ∞.

24

3. POLYNOMIAL INEQUALITIES OF NIKOL’SKII–STECHKIN–BOAS–TYPES

Then, by (3.9) and (3.10), we obtain f W˙ pr

  r   ∂  ≤ Tn W˙ pr + f − Tn W˙ pr  sup  Tn    + f − Tn W˙ pr ∂ξ ξ∈Rd , |ξ|=1 p   r    r   ∂  ∂       sup  f + sup  (f − Tn )  + f − Tn W˙ pr ∂ξ ∂ξ d d ξ∈R , |ξ|=1 ξ∈R , |ξ|=1 p p   r   ∂   sup  f  ∂ξ  + f − Tn W˙ pr . ξ∈Rd , |ξ|=1 p

It remains only to pass to the limit as n → ∞.



CHAPTER 4

Basic properties of fractional moduli of smoothness Let us recall several basic properties of moduli of smoothness ([13,85,88,100]). For f, f1 , f2 ∈ Lp (Td ), 0 < p ≤ ∞, and α, β ∈ N ∪ ((1/p − 1)+ , ∞), we have (a) ωα (f, δ)p is a non-negative non-decreasing function of δ such that lim ωα (f, δ)p = 0; δ→0+   1 (b) ωα (f1 + f2 , δ)p ≤ 2( p −1)+ ωα (f1 , δ)p + ωα (f2 , δ)p ; ∞     β min(1,p) 1/min(1,p) (c) ωα+β (f, δ)p ≤ ωα (f, δ)p ;  ν  ν=0

(d) for λ ≥ 1,

ωα (f, λδ)p ≤ C(α, p, d)λα+d( p −1)+ ωα (f, δ)p ; 1

(e) for 0 < t ≤ δ, δ

ωα (f, δ)p 1 α+d( p −1)+

≤ C(α, p, d)

t

ωα (f, t)p 1 α+d( p −1)+

.

Statement (a) follows from |ωα (f1 , δ)p − ωα (f2 , δ)p |  f1 − f2 p , inequalities in (b) and (c) are clear, and (d) and (e) will be proved later in this Chapter. Now, we prove several basic results in approximation theory. In what follows, En (f )p is the best approximation of f ∈ Lp (Td ) by trigonometric polynomials T ∈ Tn , i.e., En (f )p = inf f − T p . T ∈Tn

We start with the direct inequality. Proposition 4.1. Let 0 < p ≤ ∞, α ∈ N ∪ ((1/p − 1)+ , ∞), and n ∈ N. Then   1 (4.1) En (f )p  ωα f, . n p Inequality (4.1) follows immediately from the Jackson-type inequality for the moduli of smoothness of integer order [98]. Indeed, for α+r ∈ N and r > (1/p−1)+ , property (c) implies     1 1 En (f )p  ωα+r f,  ωα f, . n p n p 25

26

4. BASIC PROPERTIES OF FRACTIONAL MODULI OF SMOOTHNESS

Note that for 1 < p < ∞ the technique from [18] allows us to obtain a sharper version of (4.1) for the fractional moduli of smoothness:  n  ρ1   1  1 αρ−1 ρ (k + 1) Ek (f )p  ωα f, , nα n p k=0

where ρ = ρ(p) = max(p, 2). The inverse inequality is given as follows. Proposition 4.2. Let 0 < p ≤ ∞, α ∈ N ∪ ((1/p − 1)+ , ∞), and n ∈ N. Then we have

(4.2)

  n  τ1  1  1  α (k + 1)ατ −1 Ek (f )τp , ωα f, n p n k=0

where (4.3)

 τ = τ (p) =

min(p, 2), p < ∞; 1, p = ∞.

In the case 1 ≤ p ≤ ∞, α ∈ N, this result is well known (see, e.g., [19, Ch. 7], [17], [104] and the reference therein). To the best of our knowledge, the case p = 1, ∞ and α > 0 does not seem to have been considered before for this moduli of smoothness (cf. [22]). In the case 0 < p < 1, d > 1 this result is new, while the case 0 < p < 1 and d = 1 was recently considered in [85]. Moreover, the corresponding results in terms of different realizations of the K-functional (in particular, using the Laplace operator) were obtained in the papers [82, 85]. Proof. The proof follows the standard argument. We give the proof only for the case 0 < p ≤ 1 or p = ∞. The case 1 < p < ∞ can be handled using [17]. Let Tn ∈ Tn be such that En (f )p = f − Tn p . Then for any m ∈ N we have with p1 = min(p, 1)    p p p 1 1 1 1 1 1 ωα f, ≤ ωα f − T2m+1 , + ωα T2m+1 , n p n p n p (4.4)  p1 1  E2m+1 (f )pp1 + ωα T2m+1 , . n p Next, by (3.4) and (3.5), choosing m such that 2m < n ≤ 2m+1 , we obtain    p p p m 1 1 1 1  1 1 ≤ ωα T1 − T0 , + ωα T2ν+1 − T2ν , ωα T2m+1 , n p n p n p ν=0   m  1  αp1 E0 (f )pp1 + 2(ν+1)αp1 E2ν (f )pp1 (4.5) n ν=0 m



1 nαp1

2 

(k + 1)αp1 −1 Ek (f )pp1 .

k=0

Combining (4.4) and (4.5), we proved the desired result.



4. BASIC PROPERTIES OF FRACTIONAL MODULI OF SMOOTHNESS

27

Now, we will prove a realization result, which will be crucial in our further study.   Theorem 4.1. Let f ∈ Lp (Td ), 0 < p ≤ ∞, and α ∈ N ∪ (1/p − 1)+ , ∞ . Then, for any δ ∈ (0, 1), we have ωα (f, δ)p Rα (f, δ)p ,

(4.6)

where Rα (f, δ)p is the realization of the K-functional, i.e., &   α  '   ∂ α Rα (f, δ)p := inf sup  T f − T p + δ  .  T ∈T[1/δ] ∂ξ d ξ∈R , |ξ|=1

p

Moreover, if T ∈ Tn , n = [1/δ], is such that f − T p  En (f )p , then   α   ∂  (4.7) ωα (f, δ)p f − T p + δ α sup  T   . ∂ξ d ξ∈R , |ξ|=1

p

This theorem is known for d = 1, 0 < p ≤ ∞, and α ∈ N (see [23]); for d = 1, 0 < p < 1, and α > 0 see [46]; for d = 1, 1 ≤ p ≤ ∞, and α > 0 see [89]. In the case d ≥ 1, 0 < p ≤ ∞, and α ∈ N, (4.7) was stated in [25] without the proof. Proof. First, let us prove the estimate from above in (4.6). Let n = [1/δ] and Tn ∈ Tn . By Corollary 3.1, we estimate ωα (f, δ)p  f − Tn p + ωα (Tn , δ)p   α   ∂  α  f − Tn p + δ sup  Tn   ∂ξ  . d ξ∈R , |ξ|=1

p

It remains to take infimum over all Tn ∈ Tn . Let us prove the estimate from below. Let T ∈ Tn be such that f − T p  En (f )p . Then by (4.1) and (3.4), we obtain   α     ∂  1 Rα f,  f − T p + n−α sup  T   n p ∂ξ ξ∈Rd , |ξ|=1 p       1 1 1  ωα f, + ωα T,  ωα f, , n p n p n p that is, (4.6) follows. In fact, we have proved that     1 1 ωα f,  Rα f, n p n p  f − T p + n

−α

sup ξ∈Rd , |ξ|=1

  α     ∂  1   T   ωα f, ,  ∂ξ n p

p



which implies (4.7).

Similarly, taking into account Theorem 3.2, we can prove the following result. Theorem 4.2. Let f ∈ Lp (Td ), 0 < p ≤ ∞, and r ∈ N. Then, for any δ ∈ (0, 1), we have ωr (f, δ)p Rr (f, δ)p , where   f − T p + δ r T W˙ pr . Rr (f, δ)p := inf T ∈T[1/δ]

28

4. BASIC PROPERTIES OF FRACTIONAL MODULI OF SMOOTHNESS

Moreover, if T ∈ Tn , n = [1/δ], is such that f − T p  En (f )p , then ωr (f, δ)p f − T p + δ r T W˙ r . p

In the case d = 1 and 0 < p ≤ ∞, this was proved in [23]; for 1 < p < ∞ (see [22]). The result is new in the case d > 1, 0 < p ≤ 1, and p = ∞. Next we prove the following important property of the moduli of smoothness. Theorem 4.3. Let f ∈ Lp (Td ), 0 < p ≤ ∞, α ∈ N ∪ ((1/p − 1)+ , ∞), λ > 0, and t > 0. Then (4.8)

ωα (f, λt)p  (1 + λ)α+d( p −1)+ ωα (f, t)p . 1

In particular, for any 0 < h < t, one has ωα (f, t)p 1 α+d( p −1)+

(4.9)



ωα (f, h)p

. 1 t hα+d( p −1)+ When d = 1 and 0 < p ≤ 1, this was obtained in [74, 80] (for integer α) and [85] (for positive α). For the case d ∈ N, 1 < p < ∞, and α > 0 see [114, Theorem 7]. Proof. We follow the idea from [82, p. 194]. We will consider only the case 0 < p ≤ 1. The case p = ∞ can be proved by the same way. The case 1 < p < ∞ follows from (4.6) and (1.15). Denote δ = (δ1 , . . . , δd ) ∈ Rd , δj > 0, j = 1, . . . , d, and  Kδ (x) = ϕδ (k)ei(k,x) , k∈Zd ∞

where ϕδ (t) = v ((δ, t)), v ∈ C (R), v(s) = 1 for |s| ≤ 1/2 and v(s) = 0 for |s| > 1. We can assume that λ > 1. Suppose also that |δ| ≤ h and T ∈ T1/h . Using Theorem 3.1, we obtain

(4.10)

p p α p Δα λδ f p  f − Kλδ ∗ T p + Δλδ (Kλδ ∗ T )p  p α   ∂ p p   f − T p + T − Kλδ ∗ T p +  (Kλδ ∗ T )  ∂(λδ) p   α p   ∂ = f − T pp + T − Kλδ ∗ T pp + λαp  T Kλδ ∗ ∂δ  . p

Now, we will show that (4.11)

T − Kλδ ∗ T p  λ

  α   ∂  T  ∂δ 

1 α+d( p −1) 

p

and (4.12)

   α   α    ∂   1 d( p −1)  Kλδ ∗ ∂  T  λ T    . ∂δ ∂δ p p

We will prove only the first inequality, the second one can be obtained similarly. It is easy to see that (4.11) is equivalent to the following inequality Aλδ ∗ T p  λα+d( p −1) T p , 1

where Aδ (x) =

 k∈Zd

ψδ (k)ei(k,x)

4. BASIC PROPERTIES OF FRACTIONAL MODULI OF SMOOTHNESS

and (1 − ϕδ (t)) ψδ (t) = v (it, δ)α

29

 # (t1 δ1 )2 + · · · + (td δd )2 . 2

The same argument as in the proof of (3.3) implies ⎛ ⎞1− p1 d  ( Aλδ ∗ T p  ⎝ δj ⎠ Aλδ p T p  ψ λ1 Lp (Rd ) T p j=1 1 )1 L (Rd ) T p  λd( p1 −1) T p ,  λd( p −1) ψ p

that is, (4.11) is verified. Now, combining (4.10)–(4.12), we obtain   α   ∂  1 α+d( p −1)  α T Δλδ f p  f − T p + λ  ∂δ  p  λ

1 α+d( p −1)

f − T p + |δ|

α

  α    ∂  sup  T   . ∂ξ d

|ξ|=1, ξ∈R

p

Finally, taking infimum over all T ∈ T1/h and using (4.6), we get α+d( p −1) Rα (f, h)p  λα+d( p −1) ωα (f, h)p , Δα λδ f p  λ 1

which implies (4.8).

1



We finish this chapter by the Marchaud inequality for the fractional moduli of smoothness. Theorem 4.4. Let f ∈ Lp (Td ), 0 < p ≤ ∞, α ∈ N ∪ ((1/p − 1)+ , ∞), and γ > 0 be such that α + γ ∈ N ∪ ((1/p − 1)+ , ∞). Then, for any δ ∈ (0, 1), one has  1  τ  τ1 ωγ+α (f, t)p dt α (4.13) ωα (f, δ)p  δ , α t t δ where τ is given by ( 4.3). The Marchaud inequality for the moduli of smoothness of integer order can be found, e.g., in [19, p. 48] and [21]. The case 1 < p < ∞ for fractional moduli was handled in [107, Theorem 2.1]. Proof. The proof is a combination of the direct and inverse estimates (4.1) and (4.2) as well as the monotonicity property of modulus of smoothness ωα (f, δ)p

 ωα (f, 2δ)p (see (4.8)).

CHAPTER 5

Hardy–Littlewood–Nikol’skii inequalities for trigonometric polynomials We say that a continuous on Rd \ {0} function ψ(ξ) belongs to the class Hα , α ∈ R, if ψ is a homogeneous function of degree α, i.e., ψ(τ ξ) = τ α ψ(ξ),

τ > 0,

ξ ∈ Rd .

In addition, if α ≤ 0, we assume that ψ ∈ C ∞ (Rd \ {0}). Any function ψ ∈ Hα generates the Weyl-type differentiation operator as follows:   D(ψ) : cν ei(ν,x) → ψ(ν)cν ei(ν,x) , ν∈Zd



where ν aν means Weyl-type operators:

 |ν|>0

ν∈Zd

aν . Let us give several important examples of the

(1) the linear differential operator 

Pm (D)f =

ak Dk f,

Dk = Dk1 ,··· ,kd =

k1 +···+kd =m k∈Zd +

with ψ(ξ) =



∂ k1 +···+kd ∂xk11 · · · ∂xkdd

,

ak (iξ1 )k1 . . . (iξd )kd ;

k1 +···+kd =m k∈Zd +

(2) the fractional Laplacian (−Δ)α/2 f (here ψ(ξ) = |ξ|α , ξ ∈ Rd ); (3) the classical Weyl derivative f (α) (here ψ(ξ) = (iξ)α , ξ ∈ R). In this chapter, we study the sharp Hardy–Littlewood–Nikol’skii inequality given by D(ψ)T q  η(n)D(ϕ)T p ,

T ∈ Tn ,

where 0 < p ≤ q ≤ ∞, ψ ∈ Hα , and ϕ ∈ Hα+γ . Note that if ψ(ξ) = |ξ|α , ϕ ≡ 1, 1 < p < q < ∞, and α = d(1/p − 1/q), then η(n) = 1 by the Hardy–Littlewood theorem on fractional integration. On the other hand, if ψ = ϕ ≡ 1, then η(n) = nd(1/p−1/q) , which corresponds to the Nikol’skii inequality for trigonometric polynomials (see Subsection 1.5). The main result of this chapter is the following theorem. 31

32

5. HARDY–LITTLEWOOD–NIKOL’SKII INEQUALITIES

Theorem 5.1. Let 0 < p < q ≤ ∞, α > 0, γ ≥ 0, ψ ∈ Hα , and ϕ ∈ Hα+γ . ϕ ∞ d ∞ d Let also ψ ϕ ∈ C (R \ {0}) and ψ ∈ C (R \ {0}). We have (5.1)

sup

T ∈Tn

D(ψ)T q

σ(n), D(ϕ)T p

where σ(·) is given as follows: (1) if 0 < p ≤ 1 and p < q < ∞, then

⎧ d( 1 −1) 1 p ⎪ t , γ > d 1 − ; ⎪ ⎪ q + ⎪ ⎪ 1 1 ⎪ ⎪ td( p −1) ln q (t + 1), 0 < γ = d 1 − 1q ; ⎪ ⎪ ⎪

+ ⎪ 1 ⎪ d( p − q1 )−γ 1 ⎪ , 0 dp , q = ∞; γ = dp ,

q = ∞;

0 ≤ γ < d( p1 − 1q ).

Proof. The proof of Theorem 5.1 can be obtained by combining Lemmas 5.1– 5.11 from the following two Subsections 5.1 and 5.2. In more detail, in the case (1) the proof follows from Lemmas 5.9 and 5.10 (the case γ = 0) and from Lemmas 5.1, 5.2, 5.6, and 5.7 (the case γ > 0). In the case (2), the proof follows from Lemmas 5.8 and 5.9. Concerning the case (3), see Lemma 5.11.  In what follows, the de la Vall´ee Poussin type kernel is defined by ⎞ ⎛   d   k j ⎠ i(k,x) ⎝ e v , (5.2) Vn (x) := n d j=1 k∈Z

∞

where v ∈ C (R) is monotonic for t ≥ 0, v(t) = v(−t), v(t) = 1 for |t| ≤ 1 and v(t) = 0 for |t| ≥ 2.

5. HARDY–LITTLEWOOD–NIKOL’SKII INEQUALITIES

33

Remark 5.1. We do not know the sharp growth behavior of supT ∈Tn

D(ψ)T q D(ϕ)T p

in the case 0 < p < q ≤ 1, γ = 0, d ≥ 2, and ψ(ξ) ϕ(ξ) ≡ const. However, we can show that 1 1 D(ψ)T q nd( p −1) D(ψ/ϕ)Vn q  sup  nd( p −1) D(ψ/ϕ)Vn q ,  D(ϕ)T  T ∈Tn p where Vn (x) = eix Vn/4 (2x) (see Lemma 5.1 and the proof of Lemma 5.10). Remark 5.2. Let again 0 < p < q ≤ 1, γ = 0, d ≥ 2, and ψ(ξ) ϕ(ξ) ≡ const. Let either ψ be a polynomial or d/(d + α) < q ≤ 1 and let either ϕ be a polynomial or d/(d + α) < p < 1. Then, we can show the following estimate from above  1 D(ψ)T q 1, 0 < q < 1; d( p −1) (5.3) sup n ln(n + 1), q = 1. T ∈Tn D(ϕ)T p Since the proof of (5.3) is based on Theorem 10.1 , we will give it in Chapter 10. Regarding the estimate from below, by Theorem 2.1 and Lemma 5.5 (ii), we 1 have that supn nd( q −1) D(ψ/ϕ)Vn q = ∞. Thus, by (5.3), there exists a sequence {εn } such that εn → ∞ and nd( p − q ) εn  sup 1

1

T ∈Tn

D(ψ)T q , D(ϕ)T p

which provides optimality of (5.3) in some sense. Remark 5.3. Without the assumption result remains true if (5.1) is replaced by sup

T ∈Tn

ϕ ψ

∈ C ∞ (Rd \ {0}) in Theorem 5.1, the

D(ψ)T q  σ(n). D(ϕ)T p

Remark 5.4. To illustrate a different behavior of σ(·) in (2) for γ = d = 1, we consider the following example. Taking ϕ(ξ) = (iξ) and ψ(ξ) = |ξ|2 , we get           1 1 eikx  iξ   p −1  σ(n) n p −1 

n V D n   |ξ|2 ik  ∞

0 0, γ ≥ 0, ψ ∈ Hα , and ϕ ∈ Hα+γ . (i) We have (5.4)

sup

T ∈Tn

1 D(ψ)T q  nd( p −1) D(ψ/ϕ)Vn q . D(ϕ)T p

(ii) Assuming that γ > 0 and ϕ(x) = 0 if and only if x = 0, we have nd( p −1) D(ψ/ϕ)Vn/4 q  sup 1

T ∈Tn

(iii) Assuming that γ > 0 and (5.5)

sup

T ∈Tn

ψ ϕ ϕ, ψ

D(ψ)T q . D(ϕ)T p

∈ C ∞ (Rd \ {0}), we have

1 D(ψ)T q

nd( p −1) D(ψ/ϕ)Vn q . D(ϕ)T p

Proof. (i) First, we consider the case 0 < q ≤ 1. We have     1 D(ψ/ϕ)Vn (t) D(ϕ)Tn (x − t) dt. D(ψ)Tn (x) = (2π)d Td    Considering the product D(ψ/ϕ)Vn (t) D(ϕ)Tn (x−t) as a trigonometric polynomial of degree 3n in each variable tj , j = 1, . . . , d, and applying (L1 , Lq ) Nikol’skii’s inequality, we obtain      D(ψ/ϕ)Vn (t) D(ϕ)Tn (x − t) q dt. |D(ψ)Tn (x)|q  nd(1−q) Td

Integrating this inequality with respect to x and applying the Fubini theorem and the (Lq , Lp ) Nikol’skii inequality, we get D(ψ)Tn q  nd( q −1) D(ψ/ϕ)Vn q D(ϕ)Tn q 1

 nd( p −1) D(ψ/ϕ)Vn q D(ϕ)Tn p . 1

Now, let 1 < q ≤ ∞. Then the Young convolution inequality and the Nikol’skii inequality between (Lp , L1 ) imply D(ψ)Tn q  D(ψ/ϕ)Vn q D(ϕ)Tn 1  nd(1/p−1) D(ψ/ϕ)Vn q D(ϕ)Tn p . Thus, the proof of (5.4) is complete. (ii) Denote   Vnϕ (x) := D(1/ϕ) Vn (x) − Vn (2x) .  Note that Vnϕ ∈ T4n . Then

(5.6)

D(ϕ)Vnϕ p = Vn (·) − Vn (2·)p ≤ 21/p Vn p .

Set B(ξ) :=

d  j=1

v (ξj )

36

5. HARDY–LITTLEWOOD–NIKOL’SKII INEQUALITIES

Thus, taking into account that B ∈ C ∞ (Rd ) and it has a compact support, by Corollary 2.1 and (5.6), we obtain

L (Rd )  n−d(1/p−1) . (5.7) D(ϕ)Vnϕ L (Td )  Vn L (Td )  n−d(1/p−1) B p

Note also that D(ψ)Vnϕ (x)

=

p

p

   ψ(2k)  k  k i(k,x) B B − e ei(2k,x) ϕ(k) n ϕ(2k) n

 ψ(k) k =0

k =0

1 = D(ψ/ϕ)Vn (x) − γ (D(ψ/ϕ)Vn )(2x) 2 and, therefore, D(ψ)Vnϕ q ≥ CD(ψ/ϕ)Vn q ,

(5.8)

where C = (1 − 2−γ q˜)1/˜q , q˜ = min(1, q). Thus, combining inequalities (5.7) and (5.8), we derive sup

 T ∈T4n

1 D(ψ)T q D(ψ)Vnϕ q ≥  nd( p −1) D(ψ/ϕ)Vn q . ϕ D(ϕ)T p D(ϕ)Vn p

(iii) Equivalence (5.5) follows from Lemma 5.2 in the case 0 < q ≤ 1, Lemmas 5.6 and 5.7 in the case 1 < q < ∞, and Lemma 5.8 in the case q = ∞.  Lemma 5.2. Let 0 < q ≤ 1, γ > 0, ψ ∈ Hα , ϕ ∈ Hα+γ , and then D(ψ/ϕ)Vn q 1.

ψ ϕ

∈ C ∞ (Rd \ {0}),

Remark 5.5. If γ = 0, the statement of Lemma 5.2 does not hold in the case ϕ(ξ) = Cψ(ξ), C = 0, since in this case one has D(ψ/ϕ)Vn q nd(1−1/q) .

(5.9)

Here, the part ”  ” follows from Corollary 2.1 and the part ”  ” follows from Nikol’skii’s inequality and the fact that Vn 1 1. Proof. Let us first show that D(ψ/ϕ)Vn q  1. We set  ψ(k) fψ,ϕ (x) = ei(k,x) . ϕ(k) |k| =0

Then (5.10)

D(ψ/ϕ)Vn q  D(ψ/ϕ)Vn 1 = Vn ∗ fψ,ϕ 1  Vn 1 fψ,ϕ 1 .

We have by Remark 5.6 below that fψ,ϕ 1  fψ,ϕ r  fγ r where fγ (x) =

for some

r > 1,

 ei(k,x) . |k|γ

|k| =0

The fact that fγ ∈ Lr (T ) follows from d

(5.11)

fγ (x) |x|γ−d

as x → 0,

0 0, we have   ∞  ∞      ikx iu x ∞ f (εk)e − f (u)e ε du ≤ 2εV−∞ (f ), sup ε  0 1, θ > 0, and u0 ∈ Sd−1 such that (5.12)

−β−d |f( , β η(ξ)| ≥ C2 |ξ|

ξ ∈ Ω,

where Ω ≡ Ω(ρ, θ, u0 ) = {ξ = ru : r ≥ ρ, u ∈ Sd−1 , cos θ ≤ (u, u0 ) ≤ 1}, Sd−1 is the unit sphere in Rd and C1 and C2 are some positive constants.

38

5. HARDY–LITTLEWOOD–NIKOL’SKII INEQUALITIES

Denote f (y) =

ψ(y) ϕ(y) v(y).

By Lemmas 5.4 and 5.5, we obtain

q  q1     iynx ∞  D(f )Vn q  n dy  dx + V−∞ (f )  f (y)e T R   q1 dx ∞ n + V−∞ (f ) q T (1 + |nx|)  1, 0 < q < 1;  ln(n + 1), q = 1. Similarly, one can prove the estimate from below. The proof of Lemma 5.3 is now complete.  The next two lemmas deal with the case 1 < q < ∞. Lemma 5.6. Let 1 < q < ∞ and γ ∈ R. Let also μ ∈ H−γ be such that μ, 1/μ ∈ C ∞ (Rd \ {0}). Then for any f ∈ Lq (Td ) such that (−Δ)−γ/2 f ∈ Lq (Td ) the following two-sided inequalities hold (5.13) and (5.14)

D(μ)f q (−Δ)−γ/2 f q ,

γ = 0,

     1  , f − f (x)dx D(μ)f q    d (2π) Td q

γ = 0.

Remark 5.6. If we do not assume μ ∈ C ∞ (Rd \ {0}) in Lemma 5.6, then we only have D(μ)f q  (−Δ)−γ/2 f q , γ = 0. Proof. To obtain (5.13), we consider the function u(ξ) = |ξ|γ μ(ξ). By properties of homogeneous functions, we have that the function Dν u is a homogeneous function of order −|ν|1 and it belongs to C ∞ (Rd \{0}) for any multi-index ν ∈ Zd+ . Hence, sup |ξ||ν|1 |Dν u(ξ)| < ∞. ξ∈Rd

Thus, by Lemma 2.4, we get that for any function f ∈ Lq (Td ), 1 < q < ∞, D(u)f q  f q . The proof of the reverse inequality is similar using 1/μ ∈ C ∞ (Rd \ {0}). Inequality (5.14) can be obtained similarly. Lemma 5.7. Let 1 < q < ∞ and γ ≥ 0. We ⎧ ⎪ ⎪ ⎪ 1, ⎨ 1 −γ/2 ln q (n + 1), Vn q

(5.15) (−Δ) ⎪ ⎪ ⎪ ⎩ nd(1− q1 )−γ ,

have, for any n ∈ N,

γ > d 1 − 1q ;

γ = d 1 − 1q ;

0 ≤ γ < d 1 − 1q .



5.1. HARDY–LITTLEWOOD–NIKOL’SKII (Lp , Lq ) INEQUALITIES FOR 0 < p ≤ 1

39

Proof. Let us prove (5.15) only for d = 2. For d = 1 or d > 2 the proof is similar. In what follows, ξ, η ∈ R, ⎧ 1 ⎨ , |ξ| ≤ 2n, |η| ≤ 2n, |ξ| + |η| = 0; gn (ξ, η) := (ξ 2 + η 2 )γ/2 ⎩ 0, otherwise, and

 (n)

ak,l = ak,l = gn (k, l)v We have (−Δ)−γ/2 Vn (x, y) =



|k| n

   |l| v , n

a0,0 = 0.

ak,l ei(kx+ly)

k∈Z l∈Z

=2

2n 

ak,0 cos kx + 2

k=1

2n 

a0,l cos ly + 4

l=1

2n 2n  

ak,l cos kx cos ly.

k=1 l=1

First, we estimate (−Δ)−γ/2 Vn q from below as follows (−Δ)−γ/2 Vn Lq (T2 )   2n  2n    ≥ 4 ak,l cos kx cos ly   k=1 l=1

− 4(2π)

   2n  ak,0 cos kx  

1/q 

Lq (T2 )

k=1

. Lq (T1 )

Using the Hardy–Littlewood theorem for series with monotone coefficients (see (2.3)), we get  2n 1/q    kq−2  k    2n ikx   (5.16) ak,0 e 

v

σ1 (n).  kγq n Lq (T1 ) k=1

Here we set

k=1

⎧

1 ⎪ 1, γ > d 1 − ⎪ q ; ⎪ ⎨ 1 1 σd (n) := ln q (n + 1), γ = d 1 − q ; ⎪

⎪ ⎪ d(1− q1 )−γ ⎩ n , γ < d 1 − 1q ,

where d stands for the dimension. To estimate the norm of the double sum, we use Lemma 2.6, which is a multidimensional analog of the Hardy-Littlewood theorem for series with monotone coefficients an,m in the sense of Hardy, i.e., Δ(2) an,m ≥ 0. Recall that for a sequence {an,m }n,m∈N the differences Δ(2) an,m is defined as follows: Δ1,0 an,m = an,m − an+1,m ,

Δ0,1 an,m = an,m − an,m+1 ,

Δ(2) an,m = Δ0,1 (Δ1,0 an,m ). Since for 1 ≤ k ≤ 2n − 1 and 1 ≤ l ≤ 2n − 1, we have     k+1  l+1  2  ξ ∂ η (2) (5.17) Δ gn (k, l) = gn (ξ, η)v v dξdη ≥ 0, ∂ξ∂η n n k l

40

5. HARDY–LITTLEWOOD–NIKOL’SKII INEQUALITIES

then Lemma 2.6 implies (5.18)   2n  2n   (n)  ak,l cos kx cos ly    k=1 l=1



Lq (T2 )

 2n 2n  



k=1 l=1 n n   k=1 l=1



n



n

1

1

kq−2 lq−2 (k2 + l2 )qγ/2 kq−2 lq−2 (k2 + l2 )qγ/2

    q 1/q k l v v n n 1/q

(ξη)q−2 dξdη (ξ 2 + η 2 )qγ/2

1/q

σ2 (n).

Assume first that γ ≤ d(1 − 1/q). To complete the proof in this case, it only remains to use the fact that for sufficiently large n and γ ≤ d(1 − 1/q) one has σ2 (n) − (2π)1/q σ1 (n)  σ2 (n), which follows from (5.16) and (5.18). Let now γ > d(1 − 1/q). To estimate (−Δ)−γ/2 Vn q from above, we use the inequality   2n  2n    a cos kx cos ly (−Δ)−γ/2 Vn Lq (T2 )   k,l   k=1 l=1

Lq (T2 )

   2n   + a cos kx k,0   k=1

 1, Lq (T1 )

where we again used (5.16) and (5.18). The estimate from below is given as follows (−Δ)−γ/2 Vn q  (−Δ)−γ/2 Vn 1  

 ak,l ei(k−1)x+i(l−1)y dxdy T2

k,l∈Z

 a1,1 > 0.  For the case q = ∞, the quantity D(ψ/ϕ)Vn q may have different growth properties depending on given ψ and ϕ (see also Remark 5.4). However, let us show that under some natural conditions it may happen only in the one-dimensional case. Lemma 5.8. Let α > 0, γ ≥ 0, ψ ∈ Hα , ϕ ∈ Hα+γ , and (i) If d = 1, then

(5.19)

D (ψ/ϕ) Vn ∞

⎧ 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ln(n + 1), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1−γ , n

ψ ϕ

∈ C ∞ (Rd \ {0}).

γ > 1; ψ(ξ) = A|ξ|−γ sign ξ ϕ(ξ) for some A ∈ C \ {0}; ψ(ξ) = A|ξ|−γ sign ξ γ = 1 and ϕ(ξ) for any A ∈ C \ {0}; 0 ≤ γ < 1,

γ = 1 and

5.1. HARDY–LITTLEWOOD–NIKOL’SKII (Lp , Lq ) INEQUALITIES FOR 0 < p ≤ 1

(ii) if d ≥ 2 and (5.20)

∈ C ∞ (Rd \ {0}), then ⎧ ⎨ 1, ln n, D (ψ/ϕ) Vn ∞

⎩ d−γ n ,

41

ϕ ψ

γ > d; γ = d; 0 ≤ γ < d.

Remark 5.7. Note that without the condition hand side estimate in (5.20), i.e. ”  ”, still holds.

ϕ ψ

∈ C ∞ (Rd \ {0}), the right-

Proof. (i) For d = 1, we write 1 ψ(ξ) = γ Φ (u) , ϕ(ξ) |ξ| ξ where u = |ξ| ∈ {−1; 1} and the function Φ(u) = 0 is such that one of the following conditions hold:

(1) Φ(1) = Φ(−1) = A; (2) Φ(1) = A, Φ(−1) = −A; (3) Φ(1) = A, Φ(−1) = B, and |A| = |B|, where A, B ∈ C. In the first case, we have D(ψ/ϕ)Vn ∞

     2n  k 1  = 2|A| v cos kx  γ n k

∞

k=1

⎧ γ > 1; ⎨ 1, ln n, γ = 1;

⎩ 1−γ , 0 ≤ γ < 1. n

In the second case, if γ ≤ 1, by Bernstein’s inequality, we get      2n  k 1  D(ψ/ϕ)Vn ∞ = 2|A| v sin kx   γ n k ∞ k=1     2n  k 1 2|A|   ≥ v cos kx   γ−1 n n k ∞ k=1

 n1−γ . If γ > 1, then D(ψ/ϕ)Vn ∞

    2n  k 1 |A|   ≥ v sin kx   γ π n k 1 k=1      2π  2n k 1 |A| ≥ v sin kx e−ix dx  1. π 0 n kγ k=1

At the same time, by using the boundedness of Vn 1 , we obtain 2n 2n       sin νx     sin νx     D(ψ/ϕ)Vn ∞ = 2Vn ∗ ≤ 2 V   n 1 γ γ ν ν ∞ ∞ ν=1 ν=1  1, γ ≥ 1;  n1−γ , 0 ≤ γ < 1

(see, e.g., [117, Ch. 5, §2]).

42

5. HARDY–LITTLEWOOD–NIKOL’SKII INEQUALITIES

In the third case, assuming for definiteness that |A| > |B|, we obtain (5.21) D(ψ/ϕ)Vn ∞

       2n −1    k 1 ikx k 1 ikx   = A v e + (B − A) v e   n kγ n kγ k=−2n

k=−2n

    2n −1   k k 1 1  2|A| v v − |B − A| |k|γ n |k|γ n k=1 k=−2n ⎧ γ ≥ 1; ⎨ 1, ln(n + 1), γ = 1;  ⎩ 1−γ n , 0 ≤ γ < 1.

∞

Estimate from above is similar to (5.21): ⎧ γ ≥ 1; ⎨ 1, ln(n + 1), γ = 1; D(ψ/ϕ)Vn ∞  ⎩ 1−γ , 0 ≤ γ < 1. n Thus, summarizing the above estimates and noting that the case (2) corre−γ sign ξ, we get (5.19) for d = 1. sponds to the case ψ(ξ) ϕ(ξ) = A|ξ| ξ 1 (ii) Let now d ≥ 2. As above, we have ψ(ξ) = Φ γ ϕ(ξ) |ξ| |ξ| , where Φ ∈ ϕ ∞ d ∈ C ∞ (Rd \ {0}) and ψ C ∞ (Sd−1 ). Since ψ ϕ ∈ C (R \ {0}), we obtain that Φ(u) > 0 or Φ(u) < 0 for all u ∈ Sd−1 . Moreover, one can find a constant C > 0 such that |Φ(u)| > C for all u ∈ Sd−1 . Thus,

D(ψ/ϕ)Vn ∞ ≥ |D(ψ/ϕ)Vn (0)|

⎧ γ > d; ⎨ 1, 1 ln(n + 1), γ = d; ...  ≥C ⎩ d−γ |k|γ , 0 ≤ γ < d. n k1 =−n kd =−n n  

On the other hand, D(ψ/ϕ)Vn ∞

n  

⎧ γ > d; ⎨ 1, 1 ln(n + 1), γ = d; ≤C ...  ⎩ d−γ |k|γ , 0 ≤ γ < d, n k1 =−2n kd =−2n 2n  

2n  



which implies (5.20).

We are now in a position to obtain an explicit form of the Hardy-LittlewoodNikol’skii inequality in the case γ = 0: for p ≤ 1 < q, see Lemma 5.9 and for p < q ≤ 1, Lemma 5.10. ψ ϕ

Lemma 5.9. Let 0 < p ≤ 1 < q ≤ ∞, α > 0, and ψ, ϕ ∈ Hα . Let also ϕ ∈ C ∞ (Rd \ {0}) and ψ ∈ C ∞ (Rd \ {0}). Then

(5.22)

sup

T ∈Tn

1 1 D(ψ)T q

nd( p − q ) . D(ϕ)T p

Proof. The estimate from above in (5.22) follows from Lemmas 5.1 and 5.6 and Lemma 5.7 (the case 1 < q < ∞) and Lemma 5.8 (the case q = ∞).

5.1. HARDY–LITTLEWOOD–NIKOL’SKII (Lp , Lq ) INEQUALITIES FOR 0 < p ≤ 1

43

Let us prove the estimate from below. For this, we denote   1 ϕ Vn (x) = D (VN n (x) − Vn (x)), ϕ where N ∈ N will be chosen later. By (5.7), we have D(ϕ)Vnϕ p  n−d( p −1) . 1

(5.23)

At the same time, using Lemma 5.6 and Lemma 5.7 (the case 1 < q < ∞) and Lemma 5.8 (the case q = ∞), we derive that (5.24)

D(ψ)Vnϕ q ≥ D(ψ/ϕ)VNϕn q − D(ψ/ϕ)Vnϕ q 1

1

1

≥ C1 (N n)d(1− q ) − C2 nd(1− q ) ≥ nd(1− q ) ,

where N d(1−1/q) ≥ (C2 + 1)/C1 . Thus, combining the inequality sup

 T ∈T2N n

D(ψ)T q D(ψ)Vnϕ q ≥ D(ϕ)T p D(ϕ)Vnϕ p 

with (5.23) and (5.24), we obtain the estimate from below in (5.22). Lemma 5.10. Let 0 < p < q ≤ 1, α > 0, and ψ, ϕ ∈ Hα . Let also ϕ C (Rd \ {0}) and ψ ∈ C ∞ (Rd \ {0}). Then ∞

ψ ϕ

∈

(5.25)

⎧ 1 ⎪ nd( p −1) , d = 1, 0 < q < 1, and ψ(ξ) ⎪ ϕ(ξ) ≡ const; ⎨ 1 D(ψ)T q ψ(ξ) d( p −1) sup

n ln(n + 1), d = 1, q = 1, and ϕ(ξ) ≡ const; ⎪ T ∈Tn D(ϕ)T p ⎪ ⎩ nd( p1 − 1q ) , d ≥ 1 and ψ(ξ) ϕ(ξ) ≡ const.

Proof. The estimate from above in all cases follows from Lemmas 5.1 and 5.3. The estimate from below in (5.25) in the case ψ(ξ) ϕ(ξ) ≡ const can be proved the same way as in the proof of the corresponding estimate in Lemma 5.9. We only note that we use Lemma 5.3 instead of Lemmas 5.6, 5.7, and 5.8. To prove the estimate from below in the case d = 1 and ψ(ξ) ϕ(ξ) ≡ const, we put        1 k v Vnϕ (x) = D ei(2k+1)x . ϕ n k∈Z

Then, by Corollary 2.1, we have D(ϕ)Vnϕ p = Vn (2·)p  n−( p −1) . 1

(5.26)

Next, using the fact that ψ/ϕ is a homogeneous function of order zero, we obtain       k ψ(2k + 1) ϕ i(2k+1)x   % (5.27) D(ψ)Vn q =  v e  = Vn q , ϕ(2k + 1) n q

k∈Z

where V%n (x) =

 k∈Z

    k k fn v eikx , n n

By properties of the Fourier transform, i ξ  f( n v(ξ) = e 2n f∞ vn (ξ) ,

fn (y) = 

ψ(2y + n1 ) . ϕ(2y + n1 )

1 vn (y) = v y − 2n

 .

44

5. HARDY–LITTLEWOOD–NIKOL’SKII INEQUALITIES

Thus, from (5.12) (see also the proof of (4.7) in [83]), it is easy to see that for sufficiently large n, |f( n v(ξ)| ≥

(5.28)

C , |ξ|

|ξ| > ρ,

where C and ρ do not depend on n. Now, using Lemma 5.4 and (5.28), by analogy with the proof of Lemma 5.3, we derive  1, 0 < q < 1; (5.29) V%n q  ln n, q = 1. It remains to combine inequalities (5.26), (5.27), and (5.29) with the inequality sup

 T ∈T2n

D(ψ)T q D(ψ)Vnϕ q ≥ . D(ϕ)T p D(ϕ)Vnϕ p 

5.2. Hardy–Littlewood–Nikol’skii (Lp , Lq ) inequalities for 1 < p < q ≤ ∞ Lemma 5.11. Let 1 < p < q ≤ ∞, α > 0, γ ≥ 0, ψ ∈ Hα , and ϕ ∈ Hα+γ . Let ψ ϕ ∞ d ∞ d ϕ ∈ C (R \ {0}) and ψ ∈ C (R \ {0}). We have ⎧ 1, γ ≥ d( p1 − 1q ), q < ∞; ⎪ ⎪ ⎪ ⎨ 1, γ > dp , q = ∞; D(ψ)T q 1

(5.30) η(n) := sup ln p (n + 1), γ = dp , q = ∞; ⎪ T ∈Tn D(ϕ)T p ⎪ ⎪ ⎩ d( p1 − 1q )−γ , 0 ≤ γ < d( p1 − 1q ). n

also

Proof. To estimate η(n) from above in the case q < ∞, we use the fact that η(n) sup

T ∈Tn

(−Δ)−γ/2 T q T p

(see Lemma 5.6). Let γ ≥ d(1/p − 1/q) and q < ∞. Using the Hardy-Littlewood inequality for fractional integrals, see (1.18), and taking into account the fact that by Lemma 2.4 the function 1 − v(2|ξ|) u(ξ) = γ−d( 1 − 1 ) p q |ξ| is a Fourier multiplier in Lp (Td ), we arrive at (−Δ)−γ/2 Tn q  D(u)Tn p  Tn p . If 0 < γ < d(1/p − 1/q) and q < ∞, the required estimate follows from the Hardy–Littlewood inequality for fractional integrals and the Bernstein inequality (see, e.g., [83]): (−Δ)−γ/2 Tn q

= (−Δ)−d( p − q )/2 (−Δ)(d( p − q )−γ)/2 Tn q 1

1

 (−Δ)(d( p − q )−γ)/2 Tn p 1

1

 nd( p − q )−γ Tn p . 1

1

1

1

5.2. HARDY–LITTLEWOOD–NIKOL’SKII (Lp , Lq ) INEQUALITIES

45

If γ = 0 and q < ∞, the part ”  ” in (5.30) follows from D(ψ)T p  D(ϕ)T p (see Lemma 5.6) and the classical Nikol’skii inequality: η(n)  sup

T ∈Tn

1 1 D(ψ)T q  nd( p − q ) . D(ψ)T p

Let now γ ≥ 0 and q = ∞. In this case, the proof follows from the inequality   

  −d  D(ψ/ϕ)Tn ∞ =  (2π) (t) T (x − t)dt D(ψ/ϕ)V n n   d T

∞

 D(ψ/ϕ)Vn p Tn p

and Lemmas 5.6 and 5.7. Now, we study the estimate of η(n) from below, i.e., the part ”  ” in equivalence (5.30). T If γ = 0 and q < ∞, then η(n) supT ∈Tn T pq and, therefore, (5.30) is the classical Nikol’skii’s inequality. The sharpness follows from the Jackson-type kernel example, see [104, § 4.9]:  2r sin nt 2 T (x) = , r ∈ N. n sin 2t If 0 < γ < d(1/p − 1/q) and q < ∞, using Lemmas 5.6 and 5.7, we estimate η(n) ≥

(−Δ)−γ/2 (Vn/2 − 1)q (−Δ)−γ/2 Vn/2 q 1 1 ≥  nd( p − q )−γ , Vn/2 − 1p Vn/2 p + (2π)d/p

If γ ≥ d(1/p − 1/q) and q < ∞, we have η(n) ≥

D(ψ)T ∗ q

1, D(ϕ)T ∗ p

T ∗ (x) = cos x1 ,

and the desired result follows. Assume now that q = ∞. Set Tn,λ (x) =

 ei(k,x) , |k|λ

λ > 0.

|k|∞ ≤n

Note that Lemma 5.6 implies that T ∞ T ∞ (5.31) η(n) = sup  sup . γ/2 T  T ∈Tn D(ϕ/ψ)(T )p T ∈Tn (−Δ) p We divide the rest of the proof into three cases. 1) Let first γ = d/p. Since  1 Tn,d ∞   ln(n + 1), |k|d |k|∞ ≤n

then, by (5.31) and Lemma 5.7, we have η(n) 

Tn,d ∞ (−Δ)γ/2 Tn,d p



ln(n + 1)   i(k,x)     |k|∞ ≤n |k|ed(1−1/p) 



ln(n + 1)

p

(−Δ)−d(1−1/p)/2 Vn p 1

 ln p (n + 1).

n ≥ 2.

46

5. HARDY–LITTLEWOOD–NIKOL’SKII INEQUALITIES

2) Let now γ > d/p. Considering Tn,d+γ and using again (5.31) and Lemma 5.7, we get η(n) 

Tn,d+γ ∞ Tn,d+γ ∞   1. (−Δ)γ/2 Tn,d+γ p (−Δ)−d/2 Vn p

3) Finally, if 0 ≤ γ < d/p, considering Tn,λ with λ = γ + (d − d/p)/2 and using the same procedure, we arrive at η(n)  

Tn,λ ∞ (−Δ)γ/2 Tn,λ p Tn,λ ∞ (−Δ)−(λ−γ)/2 Vn p nd−λ n

1 d(1− p )−(λ−γ)

= n p −γ . d

Here, we take into account that 0 < λ − γ < d(1 − 1/p), 0 < λ < d, and  1  nd−λ . Tn,λ ∞  |k|λ |k|∞ ≤n

 5.3. Hardy–Littlewood–Nikol’skii (Lp , Lq ) inequalities for directional derivatives For applications, in particular, to obtain the sharp Ulyanov inequality for moduli of smoothness, it is important to use the Hardy-Littlewood-Nikol’skii inequalities for directional derivatives. Here the interesting case is p ≤ 1, otherwise see equivalence (3.8) in Corollary 3.3. Lemma 5.12. Let 0 < p ≤ 1, 1 < q < ∞, α > 0, γ > 0, and α + γ = 2k + 1, k ∈ Z+ . Then    ∂ α   sup  ∂ξ Tn   |ξ|=1, ξ∈Rd q    σ(n), (5.32) sup α+γ    Tn ∈Tn ∂  sup  T n  ∂ξ d |ξ|=1, ξ∈R

p

where σ(·) is given as follows: (1) if 0 < p ≤ 1 and 1 < q < ∞, then ⎧

1 ⎪ td( p −1) , γ > d 1 − 1q ; ⎪ ⎪ ⎨

1 1 σ(t) := td( p −1) ln q (t + 1), 0 < γ = d 1 − 1q ; ⎪

⎪ ⎪ ⎩ td( p1 − q1 )−γ , 0 ≤ γ < d 1 − 1q , (2) if 1 < p ≤ q < ∞, then & 1, γ ≥ d( p1 − 1q ), q < ∞; 1 1 σ(t) := td( p − q )−γ , 0 ≤ γ < d( p1 − 1q ).

5.3. HARDY–LITTLEWOOD–NIKOL’SKII (Lp , Lq ) INEQUALITIES

47

Proof. First, we note that   α+γ  α+γ  d    ∂  ∂   α+γ D Tn p  Tn   sup  Tn     , ∂xj ∂ξ p |ξ|=1, ξ∈Rd p j=1 where Dα+γ Tn (x) :=

α+γ d    ∂ i(k,x) ( Tn (x) = ϕ(k)(T , n )k e ∂x j j=1 |k|∞ ≤n

α+γ ( (T + · · · + (iyd )α+γ . n )k is the k-th Fourier coefficient of Tn , and ϕ(y) := (iy1 ) Thus, using Corollary 3.3, we obtain    ∂ α   sup  T n  ∂ξ (−Δ)α/2 Tn q |ξ|=1, ξ∈Rd q    sup . (5.33) sup α+γ T   ∂ α+γ  n p Tn ∈Tn T ∈Tn D  sup  T n  ∂ξ d |ξ|=1, ξ∈R

p

Note that  (α + γ)π  α+γ (5.34) ϕ(y) = cos |y1 | + · · · + |yd |α+γ 2   (α + γ)π sign y1 (α + γ)π sign yd |y1 |α+γ + · · · + sin |yd |α+γ . + i sin 2 2 Hence, it is easy to see that Re ϕ(y) = 0 for y = 0 and α + γ = 2k + 1, k ∈ Z+ . Next, by using Lemma 5.1, we obtain for 0 < p ≤ 1 and 1 < q < ∞ (5.35) where

sup

T ∈Tn

(−Δ)α/2 Tn q  nd(1/p−1) D(ψ/ϕ)Vn q , Dα+γ Tn p ψ(y) |y|α = . α+γ ϕ(y) (iy1 ) + · · · + (iyd )α+γ

Let us show that (5.36)

D(ψ/ϕ)Vn q  (−Δ)−γ/2 Vn q .

For this, it is sufficient to verify that the function h(y) =

(iy1

)α+γ

|y|α+γ + · · · + (iyd )α+γ

is a Fourier multiplier in Lq (Td ). This easily follows from (5.34) and Lemma 2.4. Thus, combining (5.33), (5.35), and (5.36) and using the bounds for (−Δ)−γ/2 Vn q from Lemma 5.7, we complete the proof.   Recall that the homogeneous Sobolev norm is given by f W˙ pr = |ν|1 =r Dν f p . Lemma 5.13. Let d ≥ 1. (i) We have, for 1/q ∗ = (d − 1)/d, (5.37)

T q∗

 d  ∂T  1    , ≤ d j=1  ∂xj 1

T ∈ T .

48

5. HARDY–LITTLEWOOD–NIKOL’SKII INEQUALITIES

(ii) We have T ∞  T W˙ d ,

(5.38)

1

T ∈ T .

Proof. (i) To prove (5.37) see [94, pp. 129–130]. (ii) First, we use the limiting case of the Sobolev embedding theorem (see, e.g., [7, Ch. 10], [77, Theorem 33]) given by T ∞  T W˙ d + T 1 . 1

Then by (i), we estimate T 1  T q∗ 

 d    ∂T     ∂xj  . j=1

1

Thus, (5.38) follows from (5.39)

 ∂T   ∂2T   ∂dT            2   ···   d  , ∂xj 1 ∂ xj 1 ∂xj 1

T ∈ T ,

j = 1, . . . , d.

It remains to show (5.39). We write  xj 2 ∂ ∂T (x) = 2 T (x1 , . . . , xj−1 , tj , xj+1 , . . . , xd )dtj ∂xj zj ∂xj +

∂T (x) (x1 , . . . , xj−1 , zj , xj+1 , . . . , xd ). ∂xj

(x) Then integrating over (0, 2π) by zj and using the fact that ∂T ∂xj as a polynomial of xj belongs to T  , we obtain  2π  xj 2 ∂ T ∂T (x) 2π = 2 T (x1 , . . . , xj−1 , tj , xj+1 , . . . , xd ) dtj dzj , ∂xj zj ∂xj 0

which gives

 ∂T   ∂2T         2 . ∂xj 1 ∂xj 1

Repeating this procedure d − 1 times, we complete the proof of (5.39).



Now, we proceed with the case α + γ ∈ N and γ ≥ 1. It turns out that, in this case, an interesting effect in the Hardy-Littlewood inequality occurs. More precisely, in the limiting case γ = d(1 − 1/q), we can obtain an improved version of Lemma 5.12. Lemma 5.14. Let 0 < p ≤ 1, 1 < q ≤ ∞, d ≥ 2, α > 0, γ ≥ 1, and α + γ ∈ N. Suppose also that α ∈ N if q = ∞. Then    ∂ α   sup  ∂ξ Tn   |ξ|=1, ξ∈Rd d( 1 −1)+(d(1− q1 )−γ ) q +.   (5.40) sup n p α+γ   Tn ∈Tn ∂  sup  T n  ∂ξ d |ξ|=1, ξ∈R

p

5.3. HARDY–LITTLEWOOD–NIKOL’SKII (Lp , Lq ) INEQUALITIES

49

Remark 5.8. It is worth mentioning that inequality (5.40) in the case of γ = d(1 − 1/q) gives shaper bound than both inequality (5.32) and the following relation 1 1 D(ψ)Tn q

nd( p −1) ln q (n + 1) if 0 < p ≤ 1 < q < ∞, (5.41) sup  D(ϕ)T  Tn ∈Tn n q which follows from Theorem 5.1. We assume in (5.41) that ψ ∈ Hα , ϕ ∈ Hα+γ , ϕ ∞ d and ψ ϕ , ψ ∈ C (R \ {0}). Note that the left-hand sides of (5.40) and (5.41) are not equivalent. Proof. By Corollary 3.3 and Nikol’skii’s inequality, we have   α+γ    α+γ   ∂   ∂  1 d( p −1)   Tn W˙ α+γ  sup  T  n sup Tn  n    1 ∂ξ ∂ξ |ξ|=1, ξ∈Rd |ξ|=1, ξ∈Rd 1 p and

   ∂ α   sup  ∂ξ Tn   1 |ξ|=1, ξ∈Rd q d( p −1)   I(Tn ),  ∂ α+γ   n  sup  T n  ∂ξ  d

|ξ|=1, ξ∈R

where

p

⎧ (−Δ)α/2 Tn q ⎪ ⎪ , 1 < q < ∞; ⎪ ⎨ Tn W˙ α+γ 1 I(Tn ) := ⎪ Tn W˙ qα ⎪ ⎪ , q = ∞. ⎩ Tn W˙ α+γ 1

Let us first consider the case 1 < q < ∞. The proof of (5.40) is in five steps. 1) Suppose that γ = d (1 − 1/q) ≥ 1. Choose q ∗ ∈ (1, q] such that   1 1 γ−1=d − . q∗ q Then, using the Hardy-Littlewood inequality for fractional integrals and Lemma 5.13 (i), we have (5.42)

(−Δ)α/2 Tn q  (−Δ)(α+γ−1)/2 Tn q∗  Tn W˙ α+γ−1 ∗ q

 Tn W˙ α+γ , 1

that is, I(Tn )  1. 2) If 1 ≤ γ < d(1 − 1/q), then we can choose q% ∈ (1, q) such that   1 γ =d 1− . q% Then, using Nikol’skii’s inequality and (5.42), we get (−Δ)α/2 Tn q  nd( q − q ) (−Δ)α/2 Tn q  nd(1− q )−γ Tn W˙ α+γ , 1

1

1

1

i.e., I(Tn )  n . 3) If d (1 − 1/q) < γ < d, then we can choose q∗ > q such that     1 1 γ =d 1− >d 1− . q∗ q d(1−1/q)−γ

50

5. HARDY–LITTLEWOOD–NIKOL’SKII INEQUALITIES

Then, using H¨ older’s inequality and (5.42), we have (−Δ)α/2 Tn q  (−Δ)α/2 Tn q∗  Tn W˙ α+γ , 1

i.e., I(Tn )  1. 4) If γ > d, then we have (−Δ)α/2 Tn (x) = (−Δ)(α+γ−d)/2 Tn ∗ fγ−d (x), where fγ−d (x) =

 ei(k,x) . |k|γ−d

k =0

Note that fγ−d ∈ Lq (Td ) (see (5.11)). Thus, from the above and inequality (5.38) we obtain (−Δ)α/2 Tn q  (−Δ)(α+γ−d)/2 Tn q fγ−d 1  Tn W˙ qα+γ−d  Tn W˙ ∞ α+γ−d  Tn  ˙ α+γ , W 1

which gives I(Tn )  1. 5) Assume that γ = d. Noting that α ∈ N, Lemma 5.13 (ii) gives (−Δ)α/2 Tn q  Tn W˙ α  Tn W˙ α+d , q

1

i.e., I(Tn )  1. Finally, let us consider the case q = ∞. If γ ≥ d, then applying Lemma 5.13 (ii), and using (5.39) in the case γ > d, we obtain Tn W˙ ∞ α  Tn W ˙ α+d  Tn W ˙ α+γ . 1

1

If 1 ≤ γ < d, then we choose q% > 1 such that   1 γ =d 1− . q% Now, applying the Nikol’skii inequality and (5.42), we get d

d−γ q  Tn W˙ α  nd−γ (−Δ)α/2 Tn q  nd−γ Tn W˙ α+γ . Tn W˙ ∞ α  n Tn W ˙ α =n q 

q 

Thus, we have proved (5.40) for all cases.

1



CHAPTER 6

General form of the Ulyanov inequality for moduli of smoothness, K-functionals, and their realizations Let ψ ∈ Hα , α > 0, and let Wp (ψ) be the space of ψ-smooth functions in Lp , that is,   Wp (ψ) = g ∈ Lp (Td ) : D(ψ)g ∈ Lp (Td ) . We define the generalized K-functional by Kψ (f, δ)p =

inf

g∈Wp (ψ)

{f − gp + δ α D(ψ)gp } .

It is known that Kψ (f, δ)p = 0 if 0 < p < 1. This was shown for d = 1, ψ(ξ) = (iξ)r , r ∈ N, in [23] and in the general case in [82]. The suitable substitution of the K-functional is given by the realization concept Rψ (f, δ)p =

inf

T ∈T[1/δ]

{f − T p + δ α D(ψ)T p } .

We list below several important properties of the K-functionals and their realizations. Lemma 6.1. ([82, Theorem 4.21]) Let ψ ∈ Hα , α > 0, and 1 ≤ p ≤ ∞. We have Rψ (f, δ)p Kψ (f, δ)p . The next lemma is a simple corollary of Lemma 6.1. Lemma 6.2. ([82, Lemma 4.10]) Let ψ ∈ Hα , α > 0, and 1 < p < ∞. We have  f − gp + δ α (−Δ)α/2 gp }. inf Rψ (f, δ)p

(−Δ)α/2 g∈Lp

The following result was proved in [82, Theorem 4.24] (with the remark that if ψ is a polynomial one can consider any p > 0). Lemma 6.3. Let ψ ∈ Hα , α > 0, and either ψ be a polynomial and 0 < p ≤ ∞ or d/(d + α) < p ≤ ∞. Then Rψ (f, δ)p f − T p + δ α D(ψ)T p , where T ∈ Tn , n = [1/δ], is such that f − T p  En (f )p . 51

52

6. GENERAL FORM OF THE ULYANOV INEQUALITY

The proof is standard using the Bernstein inequality (6.1)

D(ψ)T p  nα T p

(see [83]). The next lemma easily follows from the definition of the generalized K-functional. Lemma 6.4. Let ψ ∈ Hα , α > 0, and 1 ≤ p ≤ ∞. We have Kψ (f, δ)p  δ α D(ψ)f p . Lemma 6.5. ([82, Theorem 4.22]) Let ψ ∈ Hα , α > 0, and 0 < p ≤ ∞. We have   1 α+d p −1 + Rψ (f, δ)p , n ∈ N. Rψ (f, δ)p ≤ Rψ (f, nδ)p  n From Lemma 6.3 and inequality (6.1), it is easy to obtain the following result. Lemma 6.6. Let α1 ≥ α2 > 0, ψ1 ∈ Hα1 , ψ2 ∈ Hα2 , and 0 < p ≤ ∞. If ψ1 /ψ2 ∈ C ∞ (Rd \ {0}) and either ψ1 /ψ2 is a polynomial or d/(d + α1 − α2 ) < p ≤ ∞. Then Rψ1 (f, δ)p  Rψ2 (f, δ)p . The next result provides the general form of the sharp Ulyanov inequality for realizations. It will play the key role in our further study. Theorem 6.1. Let f ∈ Lp (Td ), 0 < p < q ≤ ∞, α > 0, γ ≥ 0, ψ ∈ Hα , and ϕ ∈ Hα+γ . (A) For any δ ∈ (0, 1), we have  q1    δ  q 1 Rϕ (f, t)p 1 dt 1 Rϕ (f, δ)p (6.2) Rψ (f, δ)q  η , + 1 1 δγ δ t 0 td( p − q ) where η(t) := η(t; ψ, ϕ, p, q, d) := sup

 T ∈T[t]

D(ψ)T q . D(ϕ)T p

(B) Let either ψ be a polynomial or d/(d + α) < q ≤ ∞. Let also α + γ > d(1/p − 1/q), and nd(1/p−1/q)−γ  η(n) as n → ∞. Then there exists a sequence of nontrivial polynomials Tn ∈ Tn , n ∈ N, such that  q1    δ  q 1 Rϕ (Tn , t)p 1 dt 1 Rϕ (Tn , δ)p (6.3) Rψ (Tn , δ)q

η , + 1 1 δγ δ t 0 td( p − q ) where n = [1/δ]. Remark 6.1. (i) The condition α + γ > d(1/p − 1/q) is natural since by Lemma 6.5 it is clear that tα+γ Rϕ (f, 1)p  Rϕ (f, t)p , t ∈ (0, 1), 1 ≤ p ≤ ∞. Therefore, if q  1 Rϕ (f, t)p 1 dt < ∞, 1 1 t 0 td( p − q ) then α + γ > d(1/p − 1/q). (ii) The condition nd(1/p−1/q)−γ  η(n) as n → ∞ is also natural since it always ∞ d holds if ψ ϕ ∈ C (R \ {0}) (see Theorem 7.1).

6. GENERAL FORM OF THE ULYANOV INEQUALITY

53

Proof. (A) Let Tn ∈ Tn , n ∈ N, be such that (6.4)

f − Tn p + n−(α+γ) D(ϕ)Tn p ≤ 2Rϕ (f, n−1 )p .

From the Nikol’skii inequality (1.19), it follows that (see [25, Lemma 4.2]) ∞  q1 1  νq d( 1 − 1 ) 2 1 p q f − T2ν qp1 f − T2n q  (6.5)

 

ν=n ∞ 

 q1 1 νq1 d( p − 1q )

2

Rϕ (f, 2−ν )qp1

1

.

ν=n

By the definition of η(t) and (6.4), we get (6.6)

2−αn D(ψ)T2n q ≤ η(2n )2−αn D(ϕ)T2n p ≤ 2η(2n )2γn Rϕ (f, 2−n )p .

Thus, taking into account (6.4), (6.5), and (6.6), we obtain Rψ (f, 2−n )q ≤ f − T2n q + 2−αn D(ψ)T2n q ∞ 1

q1 q1  1 νd( p − q1 ) γn −n n −ν 2  2 Rϕ (f, 2 )p η(2 ) + Rϕ (f, 2 )p . ν=n

From the last inequality, (6.2) follows immediately. (B) Let us prove the second part of the theorem. We choose a sequence of polynomials Tn , n ∈ N, such that D(ψ)Tn q (6.7) η(n)  . D(ϕ)Tn p Further, by Lemma 6.3, we have (6.8)

Rψ (Tn , 1/n)q  n−α D(ψ)Tn q .

Using the definition of the realization of K-functional, we get  q1  q 1/n  Rϕ (Tn , t)p 1 dt 1 γ Bn : = Rϕ (Tn , 1/n)p n η(n) + 1 1 t 0 td( p − q )   1/n  α+γ q1  q11 (6.9) t dt −α  D(ϕ)Tn p n η(n) + D(ϕ)Tn p 1 d( p − q1 ) t 0 t  D(ϕ)Tn p n−α η(n). Thus, from (6.7), (6.8), and (6.9) we derive Rψ (Tn , 1/n)q  n−α D(ψ)Tn q  n−α η(n)D(ϕ)Tn p  Bn . This yields (6.3) by the monotonicity property given in Lemma 6.5.



Taking into account Lemma 6.1, we obtain the following result. Corollary 6.1. Under all conditions of Theorem 6.1 if p ≥ 1, we have  q1    δ  q 1 Kϕ (f, t)p 1 dt 1 Kϕ (f, δ)p Kψ (f, δ)q  η . + 1 1 δγ δ t 0 td( p − q ) The next corollary easily follows from (6.2) and Nikol’skii’s inequality (1.19).

54

6. GENERAL FORM OF THE ULYANOV INEQUALITY

Corollary 6.2. Under all conditions of Theorem 6.1, if ψ(x) = ϕ(x), we have  q1   q δ Rϕ (f, t)p 1 dt 1 (6.10) Rϕ (f, δ)q  . 1 1 t 0 td( p − q ) Following the proof of Theorem 6.1 and taking into account Lemmas 6.3 and 6.6, it is easy to prove a slightly more general form of Theorem 6.1. Theorem 6.1 . Suppose that under conditions of Theorem 6.1, the function ϕ is either a polynomial or d/(d + α + γ) < p ≤ ∞. Let also m > 0 and the function φ ∈ Hα+m be either a polynomial or d/(d + α + m) < p ≤ ∞. Then  q1    δ  q 1 Rφ (f, t)p 1 dt 1 Rϕ (f, δ)p η . Rψ (f, δ)q  + 1 1 δγ δ t 0 td( p − q ) An analogue of Theorem 6.1 for the moduli of smoothness reads as follows. Theorem 6.2. Let f ∈ Lp (Td ), d ≥ 1, 0 < p < q ≤ ∞, α > 0, γ, m ≥ 0, α ∈ N ∪ ((1/q − 1)+ , ∞), and α + γ, α + m ∈ N ∪ ((1/p − 1)+ , ∞). Then, for any δ ∈ (0, 1), we have  q1    δ  q 1 ωα+m (f, t)p 1 dt 1 ωα (f, δ)p  ωα+γ (f, δ)p η , + 1 1 δ t 0 td( p − q ) where η(t) := sup

 T ∈T[t]

ωα (T, 1/t)q . ωα+γ (T, 1/t)p

The proof goes along the same line as in Theorem 6.1 for the realization of the K-functionals.

CHAPTER 7

Sharp Ulyanov inequalities for K-functionals and realizations The goal of this chapter is to prove the following theorem, which provides an explicit form of the sharp Ulyanov inequality. Here, we assume that ψ/ϕ is a smooth function (cf. Theorem 6.1). Theorem 7.1. Let 0 < p < q ≤ ∞, α > 0, γ ≥ 0, ψ ∈ Hα , ϕ ∈ Hα+γ , and ∈ C ∞ (Rd \{0}). (A) Let f ∈ Lp (Td ). Then, for any δ ∈ (0, 1), we have  q1    δ  q 1 Rϕ (f, t)p 1 dt 1 Rϕ (f, δ)p σ , (7.1) Rψ (f, δ)q  + 1 1 δγ δ t 0 td( p − q ) ψ ϕ

where (A1)

if 0 < p ≤ 1 and p < q < ∞, then

⎧ d( 1 −1) 1 p ⎪ t , γ > d 1 − ; ⎪ ⎪ q + ⎪ ⎪ 1 1 ⎪ ⎪ td( p −1) ln q (t + 1), 0 < γ = d 1 − 1q ; ⎪ ⎪ ⎪

+ ⎪ 1 1 ⎪ ⎪ 0 < γ < d 1 − 1q ; ⎨ td( p − q )−γ , + 1 σ(t) := d( p − q1 ) ⎪ t , γ = 0, 0 < p ≤ 1 < q < ∞; ⎪ ⎪ ⎪ d( p1 − q1 ) ⎪ t , γ = 0, 0 < q ≤ 1, and ψ(ξ) ⎪ ϕ(ξ) ≡ const; ⎪ ⎪ d( 1 −1) ⎪ ψ(ξ) ⎪ p , γ = 0, 0 < q < 1, and ϕ(ξ) ≡ const; ⎪ ⎪ t ⎪ ⎩ d( p1 −1) t ln(t + 1), γ = 0, q = 1, and ψ(ξ) ϕ(ξ) ≡ const,

(A2) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ σ(t) := ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (A3)

if 0 < p ≤ 1 and q = ∞, then td( p −1) , 1 td( p −1) , 1

t

1 d( p −1)

γ > d; γ = d = 1 and

ln(t + 1), γ = d = 1 and

1 d( p −1)

ψ(ξ) ϕ(ξ) ψ(ξ) ϕ(ξ)

= A|ξ|−γ sign ξ for some A ∈ C \ {0}; = A|ξ|−γ sign ξ for any A ∈ C \ {0};

t ln(t + 1), γ = d ≥ 2; 1 td( p −γ) , 0 ≤ γ < d, if 1 < p < q ≤ ∞, then ⎧ 1, ⎪ ⎪ ⎪ ⎨ 1, 1 σ(t) := ln p (t + 1), ⎪ ⎪ ⎪ ⎩ d( p1 − 1q )−γ , t

γ ≥ d( p1 − 1q ), q < ∞; γ > dp , q = ∞; γ = dp ,

q = ∞;

0 ≤ γ < d( p1 − 1q ). 55

56

7. SHARP ULYANOV INEQUALITIES FOR K-FUNCTIONALS AND REALIZATIONS

(B) Inequality (7.1) is sharp in the following sense. Let ψ be either a polynomial ϕ ∈ C ∞ (Rd \{0}) and α + γ > d(1 − 1/q)+ . Let also or d/(d + α) < q ≤ ∞. Let ψ ψ(x) = Cϕ(x) for some constant C in the case γ = 0 and 0 < p < q ≤ 1 or γ = 0 and 0 < p ≤ 1, q = ∞. Then there exists a function f0 ∈ Lq (Td ), f0 ≡ const, such that  q1    δ  q 1 Rϕ (f0 , t)p 1 dt 1 Rϕ (f0 , δ)p σ (7.2) Rψ (f0 , δ)q

+ 1 1 δγ δ t 0 td( p − q ) as δ → 0. In light of Lemma 6.1, Theorem 7.1 implies the following result. Corollary 7.1. Under all conditions of Theorem 7.1 if p ≥ 1, we have  q1    δ  q 1 Kϕ (f, t)p 1 dt 1 Kϕ (f, δ)p σ . Kψ (f, δ)q  + 1 1 δγ δ t 0 td( p − q ) Remark 7.1. (i) Note that in the proof of Theorem 7.1, in (A1) with γ = 0, ∞ d 0 < p < q ≤ 1 and in (A2) we may not assume that ψ ϕ ∈ C (R \{0}). (ii) It is important to remark that in part (B) of Theorem 6.1, we show sharpness of the corresponding Ulyanov type inequality by constructing a sequence of functions which depends on δ, while in Theorem 7.1 we construct a function f0 which is independent of δ. Similarly to Theorem 6.1 , one can obtain the following more general analogue of Theorem 7.1. Theorem 7.1 . Suppose that under conditions of Theorem 7.1, the function ϕ is either a polynomial or d/(d + α + γ) < p ≤ ∞. Let also m > 0 and the function φ ∈ Hα+m be either a polynomial or d/(d + α + m) < p ≤ ∞. Then  q1    δ  q 1 Rφ (f, t)p 1 dt 1 Rϕ (f, δ)p (7.3) Rψ (f, δ)q  σ . + 1 1 δγ δ t 0 td( p − q ) Proof of Theorem 7.1. (A) Taking into account Theorem 6.1, it is enough to estimate D(ψ)T q η(t) = sup .  T ∈T[t] D(ϕ)T p Since

ψ ϕ

∈ C ∞ (Rd \{0}), the inequality η(t)  σ(t)

follows from Theorem 5.1 and Remark 5.3 in all cases except the case 0 < p < q ≤ 1, γ = 0, d ≥ 2, and ψ(ξ) ϕ(ξ) ≡ const. In the latter case, the proof of this inequality follows from Corollary 10.1 stated below. To prove (B), we will construct a nontrivial function f0 ∈ Lq (Td ) such that equivalence (7.2) holds as δ → 0. Part (B1): 0 < p ≤ 1 and p < q ≤ ∞. We set (7.4)

f0 (x) = Fϕ (x) −

1 Fϕ (N x), N α+γ

7. SHARP ULYANOV INEQUALITIES FOR K-FUNCTIONALS AND REALIZATIONS

57

where Fϕ (x) ∼

(7.5)

 ei(k,x) k =0

ϕ(k)

and N is a fixed sufficiently large integer that will be chosen later. Let us show that Fϕ ∈ Lq∗ (Td ), where q ∗ = max(q, 1). For this, we consider the function  ei(k,x) Fα+γ (x) ∼ . |k|α+γ k =0

It is known that Fα+γ ∈ Lq∗ (Td ) for α + γ > d(1 − 1/q ∗ ) (see (5.11)). At the same time, the function u(y) =

|y|α+γ−ε (1 − v(2|y|)) , ϕ(y)

0 < ε < α + γ,

is a Fourier multiplier in Lr (Td ) for any 1 ≤ r ≤ ∞. This follows from Lemma 2.4 (see also Lemma 2.2) and the fact that ϕ(y) = 0 if and only if y = 0. Take ε such that α + γ − ε > d(1 − 1/q ∗ ). We have that  ei(k,x) ∈ Lq∗ (Td ). |k|α+γ−ε

k =0

Thus, taking into account that Fϕ (x) ∼

 k =0

u(k) ei(k,x) , |k|α+γ−ε

we obtain that Fϕ ∈ Lq∗ (Td ) and, therefore, f0 ∈ Lq∗ (Td ). Next, (7.6)

Rϕ (f0 , n−1 )p  f0 − f0 ∗ Vn p + n−α−γ D(ϕ)(f0 ∗ Vn )p .

Recall that Vn is given by (5.2). Let us consider the second summand in (7.6). In light of   d  N |kj | ei(N k,x) v (Fϕ (N ·) ∗ Vn )(x) = n ϕ(k) j=1 k =0

and since ϕ is homogeneous of order α + γ, we obtain     d d  |kj | i(k,x)   N |kj | i(N k,x) v − v D(ϕ)(f0 ∗ Vn )(x) = e e n n j=1 j=1 k =0

k =0

= Vn (x) − Vn/N (N x). Using the last equalities and Corollary 2.1, we derive that   1 (7.7) D(ϕ) f0 ∗ Vn p  Vn p + Vn/N p  nd(1− p ) . Now, let us consider the first summand in (7.6). We estimate (7.8) f0 − f0 ∗ Vn pp ≤ Fϕ − Fϕ ∗ Vn pp + N −p(α+γ) Fϕ (N ·) − Fϕ (N ·) ∗ Vn pp .

58

7. SHARP ULYANOV INEQUALITIES FOR K-FUNCTIONALS AND REALIZATIONS

It is enough to estimate the first term in (7.8). We have

Fϕ − Fϕ ∗ Vn p

   i(k,x)  d    |kj | e   =  v 1−  n ϕ(k) p d j=1 k∈Z      k 1   = ξ ei(k,x)    , α+γ n n p d k∈Z

where ξ(y) =

1−

$d j=1

v(|yj |)

ϕ(y)

.

d( − ) 2 By Lemma 2.5, using the fact that ξ ∈ B˙ 2,pp 2 ∩B˙ 2,1 (Rd ) (see, e.g., Lemma 2.2), and the homogeneity property of the Besov (quasi-)norm (see Lemma 2.3), we get 1

(7.9)

1

d

 y    Fϕ − Fϕ ∗ Vn p  n−α−γ ξ  d( 1 − 1 ) n B˙ 2,pp 2 (Rd ) 1

 nd(1− p )−α−γ ξ

1

d( 1 − 1 ) 2

B˙ 2,pp

 nd(1− p )−α−γ .

(Rd )

Hence, from (7.8) we have 1

f0 − f0 ∗ Vn p  nd(1− p )−α−γ .

(7.10)

Thus, combining (7.6), (7.7), and (7.10), we arrive at Rϕ (f0 , n−1 )p  nd(1− p )−α−γ . 1

(7.11) Let us assume that (7.12)

Rψ (f0 , 1/n)q  n−α D(ψ/ϕ)Vn q ,

n ∈ N,

and (7.13)

td(1/p−1/q)−γ = O(η(t))

as

where η is given by η(t) = sup

 T ∈T[t]

We will prove (7.12) and (7.13) later.

D(ψ)T q . D(ϕ)T p

t → ∞,

7. SHARP ULYANOV INEQUALITIES FOR K-FUNCTIONALS AND REALIZATIONS

59

Now, comparing estimates (7.11) and (7.12), we derive the following inequalities for δ → 0 Rψ (f0 , δ)q  δ α D(ψ/ϕ)V1/δ q 1 1 = δ α+d( p −1) d( 1 −1) D(ψ/ϕ)V1/δ q p δ 1  δ α+d( p −1) η(1/δ) 1 α+d( p −1)

(by

(7.12))

(by Lemma 5.1)

1 α+d( p −1)

 δ σ(1/δ) + δ η(1/δ) 1 Rϕ (f0 , δ)p σ(1/δ) + δ α+d( p −1) η(1/δ)  δγ 1 Rϕ (f0 , δ)p σ(1/δ) + δ α+γ+d( q −1)  γ δ  q1   q δ Rϕ (f0 , t)p 1 dt 1 Rϕ (f0 , δ)p (7.14)  σ(1/δ) + 1 1 δγ t 0 td( p − q )

(by Theorem (by

(7.11))

(by

(7.13))

(by

5.1)

(7.11)),

where in the last inequality we used the fact that α + γ > d(1/p − 1/q) ≥ d(1 − 1/q). Therefore, (7.2) follows. To complete the proof of (B1), we need to verify (7.12) and (7.13). Proof of estimate (7.12). We divide the proof of this fact into three steps. Step 1. Assume that 0 < q ≤ 1 and γ > 0. Denoting Hϕ,n (x) := Fϕ ∗ Vn (x) − we estimate (7.15)

1 Fϕ (N ·) ∗ VN n (x), N α+γ

Rψ (f0 , 1/n)q ≥ C(q) Rψ (Hϕ,n , 1/n)q − Rψ (f0 − Hϕ,n , 1/n)q .

Since Hϕ,n is a trigonometric polynomial of degree 2N n, using Lemma 6.3, we get Rψ (Hϕ,n , 1/n)q  n−α D(ψ)Hϕ,n q . Using the representation     d d   |kj | ψ(k) i(k,x) |kj | ψ(k) i(N k,x) −γ D(ψ)Hϕ,n (x) = e e v −N v , n ϕ(k) n ϕ(k) j=1 j=1 k =0

we obtain

k =0

 Rψ (Hϕ,n , 1/n)q 

(7.16)

1 1 − γq N

1/q

n−α D(ψ/ϕ)Vn q .

Further, applying Holder’s inequality and Lemma 6.4 yields Rψ (f0 − Hϕ,n , 1/n)q  Rψ (f0 − Hϕ,n , 1/n)1  Kψ (f0 − Hϕ,n , 1/n)1 n−α D(ψ)(f0 − Hϕ,n )1    n−α Fϕ/ψ − Fϕ/ψ ∗ Vn 1 + Fϕ/ψ (N ·) − Fϕ/ψ ∗ VN n (N ·)1 ,  i(k,x) where Fϕ/ψ (x) = k =0 ψ(k) (see (7.5)). ϕ(k) e 

60

7. SHARP ULYANOV INEQUALITIES FOR K-FUNCTIONALS AND REALIZATIONS

Since Fϕ/ψ 1  Fϕ/ψ 1+ε  Fγ 1+ε  1 (see (5.11)), we have that Fϕ/ψ ∈ L1 (Td ) and, therefore, Rψ (f0 − Hϕ,n , n−1 )q = o(n−α ) as

(7.17)

n → ∞.

Thus, combining (7.15)–(7.17), we immediately obtain Rψ (f0 , 1/n)q  n−α D(ψ/ϕ)Vn q − o(n−α ). Note that by Lemma 5.2, there exists a constant C independent of n such that D(ψ/ϕ)Vn q ≥ C. Hence,

Rψ (f0 , 1/n)q  n−α D(ψ/ϕ)Vn q , which implies (7.12) for 0 < q ≤ 1 and γ > 0. Step 2. Let 0 < q ≤ 1 and γ = 0. In this case, we assume that ϕ(y) = Cψ(y) with some constant C. By Lemma 6.5, one has that Rψ (f0 , δ)q  Rψ (f0 , 1)q δ α+d(1/q−1) .

On the other hand, (5.9) in Remark 5.5 implies that D (ψ/ϕ) V1/δ q δ d(1−1/q) and hence Rψ (f0 , δ)q  δ α D (ψ/ϕ) V1/δ q , which is (7.12). Step 3. Assume that 1 < q ≤ ∞. Taking into account that f0 ∗ Vn is the near best approximant of f0 in Lq , that is, f0 ∗ Vn − f0 q ≤ CEn (f0 )q , we have that Lemma 6.3 implies Rψ (f0 , 1/n)q  n−α D(ψ)(f0 ∗ Vn )q (7.18)   1  n−α D(ψ/ϕ)Vn q − γ D(ψ/ϕ)Vn/N q . N If 1 < q < ∞ and γ ≥ 0, Theorem 5.1 gives the exact growth order of D(ψ/ϕ)Vn q as n → ∞, which yields 1 (7.19) D(ψ/ϕ)Vn q − γ D(ψ/ϕ)Vn/N q  D(ψ/ϕ)Vn q N for sufficiently large N . Therefore, (7.12) follows. If q = ∞ and γ > 0, we take take into account that   d  ψ(k)  N |kj | i(k,x) D(ψ/ϕ)Vn/N (x) = v e ϕ(k) j=1 n k =0

    d  ψ(k)  N |kj | |kj | i(k,x) = v v e ϕ(k) j=1 n n k =0

$ and that the function B(y) = dj=1 v (yj ) is a Fourier multiplier in L∞ (Td ) (see Lemmas 2.4 and 2.2). Hence, we have D(ψ/ϕ)Vn/N ∞  B(N y)L∞ →L∞ D(ψ/ϕ)Vn ∞ = BL∞ →L∞ D(ψ/ϕ)Vn ∞  D(ψ/ϕ)Vn ∞ , which yields (7.19) for sufficiently large N .

7. SHARP ULYANOV INEQUALITIES FOR K-FUNCTIONALS AND REALIZATIONS

61

Finally, suppose that q = ∞ and γ = 0. In this case, we assume that ϕ(y) = Cψ(y) with some constant C. Therefore, D(ψ/ϕ)Vn ∞ Vn ∞ nd . Then, using (7.18) and (7.19) for sufficiently large N , we arrive at (7.12). Thus, the proof of (7.12) is complete. Proof of estimate (7.13). First, assume that γ > 0. By Lemma 5.1 (iii), η(n) nd(1/p−1) D(ψ/ϕ)Vn q . If 1 < q < ∞, then Lemmas 5.6 and 5.7 describe the sharp growth order of D(ψ/ϕ)Vn q and, in particular, 1

D(ψ/ϕ)Vn q  nd(1− q )−γ . This gives (7.13). If p < q ≤ 1, Lemma 5.2 yields η(n)  nd( p −1)  nd( p − q )−γ . 1

1

1

Finally, let q = ∞. In this case, by Lemma 5.8, we have D(ψ/ϕ)Vn ∞  nd−γ , which implies the desired result. Second, assume that γ = 0 and ϕ(ξ) = Cψ(ξ). Then (7.13) is equivalent to nd( p − q )  sup 1

1

Tn ∈Tn

T q . T p

It is enough to consider extremizers for the classical Nikol’skii inequality, for example, the Jackson-type kernel (see [104, § 4.9]). Hence, the proof of inequality (7.13) and the part (B1) are complete. Part (B2): 1 < p < q < ∞. In this case, if γ ≥ d(1/p − 1/q), then we can take as f0 any non-trivial function from C ∞ (Td ). Indeed, by Lemma 6.2 and (1.15), for any 1 < r < ∞, we have (7.20)

Rψ (f0 , δ)r inf (f0 − gr + δ α (−Δ)α/2 gr ) ωα (f0 , δ)r δ α . g

Then, Rψ (f0 , δ)q  δ α  (7.21)

δ α+γ δγ

Rϕ (f0 , δ)p  + δγ



δ



Rϕ (f0 , t)p

0

t

1 d( p − 1q )

q1

dt t

 q1

If 0 ≤ γ < d(1/p − 1/q), then we take f0 (x) = Fϕ (x) ∼

 ei(k,x) k =0

ϕ(k)

.

Let us first estimate Rϕ (f0 , δ)p from above. We have Rϕ (f0 , 2−n )p  f0 − f0 ∗ V2n p + 2−n(α+γ) D(ϕ)(f0 ∗ V2n )p , where V2n is given by (5.2).

1

.

62

7. SHARP ULYANOV INEQUALITIES FOR K-FUNCTIONALS AND REALIZATIONS

Note that by (7.9), f0 − f0 ∗ V2n 1  2−(α+γ)n . Then, using Lemma 4.2 from [25] and taking into account that α + γ > d(1 − 1/q) > d(1 − 1/p), we obtain 1/p ∞  p 1  νpd(1− p ) f0 − f0 ∗ V2ν 1 2 f0 − f0 ∗ V2n p  ν=n 1 −(α+γ+d( p −1))n

 2

(7.22)

.

Further, using the multidimensional Hardy-Littlewood theorem for series with monotone coefficients (see Lemma 2.6), we derive that D(ϕ)(f0 ∗ V2n )p



V2n − 1p     i(k,x)    e  

p

|k|∞ ≤2n



(7.23)

 2n

d/p

ν p−2

1

2dn(1− p ) .

ν=1

Therefore, Rϕ (f0 , δ)p  δ α+γ+d( p −1) . 1

(7.24)

Let us estimate Rψ (f0 , 1/n)q from below. Taking into account that f0 ∗ Vn is the near best approximant of f0 in Lq and using Lemmas 5.7 and 6.3 with 0 ≤ γ < d(1/p − 1/q) < d(1 − 1/q), we get Rψ (f0 , 1/n)q  n−α D(ψ)(f0 ∗ Vn )q

(7.25)

 n−α (−Δ)−γ/2 Vn q  nd(1− q )−(α+γ) . 1

Therefore, combining (7.24) and (7.25), we finally obtain   q  1q δ Rϕ (f0 , t)p dt Rϕ (f0 , δ)p σ(1/δ) + 1 1 δγ t 0 td( p − q )  1q  δ 1 α+d( p −1) q(α+γ+d( 1q −1)) dt  δ σ(1/δ) + t (by t 0  δ α+d( p −1) σ(1/δ) + δ α+γ+d( q −1) 1

1

(7.24))

(since α + γ > d(1 − 1/q))

α+γ+d( 1q −1)

 δ  Rψ (f0 , δ)q

(by the definition of (by (7.25)),

σ)

which proves (7.2). Part (B3): 1 < p < ∞ and q = ∞. First, let γ > d/p. We take a non-trivial function f0 ∈ C ∞ (Td ). Then, by Lemma 6.2, for any 1 < r < ∞, we have Rψ (f0 , δ)∞  Rψ (f0 , δ)r  δ α (−Δ)−α/2 f0 r  δ α . By (7.20), Rϕ (f0 , δ)p δ α+γ and (7.2) follows since in this case σ(t) = 1 (see (7.21)). Now, we assume that 0 ≤ γ < d/p and σ(t) = td/p−γ . We consider the function  ei(k,x) . f0 (x) = |k|γ ψ(k) d k∈Z

7. SHARP ULYANOV INEQUALITIES FOR K-FUNCTIONALS AND REALIZATIONS

63

Then, by Lemma 6.3, we have Rψ (f0 , 2−n )∞

 2−αn D(ψ)(f0 ∗ V2n )∞ = 2−αn (−Δ)−γ/2 V2n ∞  2n(d−α−γ) .

(7.26)

On the other hand, as above, by (7.22) and (7.23), we obtain that f0 − f0 ∗ V2n p  2−(α+γ+d( p −1))n 1

and 1

D(ϕ)(f0 ∗ V2n )p  2dn(1− p ) . Hence, Rϕ (f0 , 2−n )p  2−(α+γ+d( p −1))n . 1

(7.27)

Thus, (7.26) and (7.27) give (7.2).  Finally, let γ = d/p and σ(t) = ln1/p (t + 1). In this case the proof is more technical. Consider  1 f0 (x) = ei(k,x) . d ψ(k)|k| d k∈Z

First, by Lemma 6.3, (7.28)

Rψ (f0 , 1/n)∞  n−α D(ψ)(f0 ∗ Vn )∞  n−α

 |k|∞ ≤n

1  n−α ln n. |k|d

Note that Rϕ (f0 , δ)p Rϕ (f0∗ , δ)p , where f0∗ (x) =



1

|k|α+d d

ei(k,x) .

k∈Z

Denoting by Dn the cubic Dirichlet kernel, i.e.,  Dn (x) = ei(k,x) , |k|∞ ≤n

we have Rϕ (f0 , 1/n)p  Rϕ (f0∗ , 1/n)p  f0∗ − f0∗ ∗ Dn p + n−α−γ D(ϕ)(f0∗ ∗ Dn )p   ei(k,x)    ei(k,x)     −α−γ   + n .   α+d p d−γ p |k| |k| |k| ≥n |k| ≤n

(7.29)

∞

By Lemma 5.7, we get      ei(k,x)  −α−γ    n−α−γ ln p1 (n + 1), n  d−γ  |k| p |k| ≤n ∞

Let us prove that (7.30)

     ei(k,x)     n−α−γ .  α+d  |k| p |k| ≥n ∞

∞

γ=

d . p

64

7. SHARP ULYANOV INEQUALITIES FOR K-FUNCTIONALS AND REALIZATIONS

For simplicity, we consider only the case d = 2. Taking into account the Hardy– Littlewood theorem for multiple series (Lemma 2.6), we get ∞  ∞   cos k1 x1 cos k2 x2      |k|α+d p k1 =n k2 =n  ∞ 1/p ∞ ∞ ∞   p−2    (2)  p p−2   Δ aν1 ,ν2  k1 k2 =: D, k1 =1

k2 =1

where

 aν1 ,ν2 =

ν1 =k1 ν2 =k2

|ν|−(α+d) , |ν|∞ ≥ n; 0, |ν|∞ < n.

We divide this series into four parts (D  D1 + D2 + D3 + D4 ) and estimate each of them separately. We have 1/p  n−2 n−2 ∞ ∞   p−2 p−2    (2)  p Δ aν ,ν  k1 k2 D1 := 1 2 ν1 =n−1 ν2 =n−1

k1 =1 k2 =1 ∞ 

 n2(p−1)/p

∞   (2)  Δ aν1 ,ν2 

ν1 =n−1 ν2 =n−1 1

 n2(p−1)/p an−1,n−1 

n2(1− p ) , nα+d

since p > 1 and Δ(2) aν1 ,ν2 ≥ 0 (see (5.17)). Further,  n−2 ∞ ∞   p−2 p−2  k1 k2 D2 :=



k1 =1



n

1 1− p

 n

1 1− p

k2p−2 apn−1,k2

k2 =n−1 ∞ 

k2 =n−1

1/p

k2p−2 (n2 + k22 )

∞ 

(α+d)p 2

1/p

1

.

(α+d)p−p+2

k2 =n−1

1/p

1/p

∞ 

k1p−2

1 ,ν2

 p 

ν1 =n−1 ν2 =k2

k1 =1 k2 =n−1

 n−2 

∞   (2)  Δ aν

k2

In light of α + γ = α + d/p > d, we have (α + d)p − p + 1 > 0 and, therefore, 1

D2  Similarly, we derive that  ∞ n−2 ∞   p−2 p−2  D3 := k1 k2 k1 =n−1 k2 =1

n2(1− p ) . nα+d ∞   (2)  p Δ aν1 ,ν2 

ν1 =k1 ν2 =n−1

1/p

1

n2(1− p )  . nα+d

7. SHARP ULYANOV INEQUALITIES FOR K-FUNCTIONALS AND REALIZATIONS

65

Finally, 

∞ 

D4 := 



∞  k1 =n−1 ∞ 



=:



(k12 + k22 )

k1 

k1 =n−1 ∗ D4 + D4∗∗ .

(k12 + k22 )

∞ 

k1p−2

1/p

1/p

(α+d)p 2

1/p

k2p−2

k2 =n−1

∞ 

 (2)  p Δ aν1 ,ν2 

∞ 

k2p−2

k2 =n−1

k1p−2

∞ 

ν1 =k1 −1 ν2 =k2 −1

∞ 

k1p−2

k1 =n−1

+

k1p−2 k2p−2

k1 =n−1 k2 =n−1

 

∞ 

(α+d)p 2

1/p

k2p−2 (k12 + k22 )

k2 =k1

(α+d)p 2

We estimate each term separately. First,  D4∗

∞ 



1/p

k1 

k1p−2

k2p−2

(α+d)p

k1 =n−1



k1

∞ 



k2 =n−1

1/p

k12p−3

.

(α+d)p

k1 =n−1

k1

Using again the assumption α + d/p > d, it is easy to see that (α + d)p − 2p + 2 > 0, which implies that 1

D4∗ 

n2(1− p ) . nα+d

Further,  D4∗∗



∞ 

k1p−2

k1 =n−1

 

∞ 

∞ 

k2p−2

1/p

(α+d)p

k2 =k1

k12p−3

k2

1/p

(α+d)p

k1 =n−1

k1

1

n2(1− p )  . nα+d Combining these estimates, we arrive at inequality (7.30). Thus, by (7.29) and (7.30), we obtain that 1

(7.31)

Rϕ (f0 , 1/n)p  n

−α−γ

d 1 n2(1− p ) ln (n + 1) + α+d  n−α− p ln p (n + 1). n 1 p

66

7. SHARP ULYANOV INEQUALITIES FOR K-FUNCTIONALS AND REALIZATIONS

Hence, Rϕ (f0 , δ)p σ(1/δ) + δγ



Rϕ (f0 , t)p dt d t 0 tp  δ Rϕ (f0 , t)p dt Rϕ (f0 , δ)p 1/p ln (1/δ) +

d t δ d/p 0 tp    δ 1 1 dt tα ln p  δ α ln(1/δ) + t t 0 α  δ ln(1/δ)  Rψ (f0 , δ)∞

and (7.2) follows.

δ

(by (by

(7.31)) (7.28)) 

CHAPTER 8

Sharp Ulyanov inequalities for moduli of smoothness 8.1. One-dimensional results For d = 1, by using Theorem 7.1 with ψ(ξ) = (iξ)α and ϕ(ξ) = (iξ)α+γ , which correspond to the fractional Weyl derivatives, we obtain the following sharp inequality. Theorem 8.1. Let 0 < p < q ≤ ∞, α ∈ N ∪ ((1/q − 1)+ , ∞) and γ, m ≥ 0 be such that α + γ, α + m ∈ N ∪ ((1/p − 1)+ , ∞). (A) Let f ∈ Lp (T). Then, for any δ ∈ (0, 1), we have (8.1)

ωα+γ (f, δ)p ωα (f, δ)q  σ δγ

 q1    δ  q 1 ωα+m (f, t)p 1 dt 1 , + 1 1 δ t 0 tp−q

where (A1) if 0 < p ≤ 1 and p < q ≤ ∞, then ⎧ 1 ⎪ t p −1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ t p1 −1 ln q1 (t + 1), σ(t) := 1 1 ⎪ ⎪ t p − q −γ , ⎪ ⎪ ⎪ ⎪ ⎩ p1 − 1q t , (A2) if 1 < p ≤ q ≤ ∞, then ⎧ 1, ⎪ ⎪ ⎪ ⎨ 1, 1 σ(t) := ln p (t + 1), ⎪ ⎪ ⎪ ⎩ ( p1 − q1 )−γ , t

γ > 1 − 1q ; + 0 < γ = 1 − 1q ;

+ 0 < γ < 1 − 1q ; +

γ = 0,

γ ≥ p1 − 1q , q < ∞; γ > p1 , q = ∞; γ = p1 ,

q = ∞;

0≤γ
(1 − 1/q)+ and m − γ ∈ Z+ ∪ ((1/p − 1)+ , ∞). There exists a function f0 ∈ Lq (T), f0 ≡ const, such that the following equivalence holds  q1    δ  q 1 ωα+m (f0 , t)p 1 dt 1 ωα+γ (f0 , δ)p σ ωα (f0 , δ)q

+ 1 1 δγ δ t 0 tp−q as δ → 0. 67

68

8. SHARP ULYANOV INEQUALITIES FOR MODULI OF SMOOTHNESS

Corollary 8.1. Under the conditions of Theorem 8.1, inequality ( 8.1) with m = γ implies the following inequality     q1  q11 δ 1 ωα+γ (f, t)p dt σ . (8.2) ωα (f, δ)q  γ t t t 0 Moreover, if α + γ > (1 − 1/q)+ , then there exists a nontrivial function f0 ∈ Lq (T) such that     q1  q11 δ 1 ωα+γ (f0 , t)p dt σ . ωα (f0 , δ)q

γ t t t 0 Proof of Theorem 8.1. Part (A) follows from the realization result (1.14) and Theorem 7.1 with ψ(ξ) = (iξ)α and ϕ(ξ) = (iξ)α+γ . The appearance of m in the integral in the right-hand side of (8.1) follows from the fact that inequality (6.5) can be written as  f − T2n q 

∞ 

 q1 1 νq1 ( p − q1 )

2

1

f −

T2ν qp1

ν=n

 

(8.3)

∞

q1  1 1 2νq1 ( p − q ) ωα+m (f0 , 2−ν )p

ν=n



2−n





ωα+m (f, t)p t

0

q1

1 1 p−q

dt t

 q1

1

 q1

1

,

and, therefore, (8.1) holds. Part (B) follows from Theorem 7.1 (B) noting that in our case (7.14) is given by  q1   q δ ωα+γ (f0 , t)p 1 dt 1 ωα+γ (f0 , δ)p ωα (f0 , δ)q  σ(1/δ) + . 1 1 δγ t 0 tp−q Using the inequality ωα+γ (f0 , t)p  ωα+m (f0 , δ)p for m ≥ γ, we arrive at the statement of part (B). 

8.2. Comparison between the sharp and classical Ulyanov inequalities Let us compare the obtained inequality (8.1) and the classical Ulyanov inequality given by  (8.4)

ωα (f, δ)q 

δ



ωα (f, t)p tp−q 1

0

1

q1

dt t

 q1

1

,

0 < p < q ≤ ∞.

First, if 1 < p < q < ∞, the sharp version of this inequality, i.e.,   q  1q δ ωα+θ (f, t)p dt 1 1 (8.5) ωα (f, δ)q  , θ= − , 1 1 − t p q 0 tp q

8.2. COMPARISON BETWEEN THE SHARP AND CLASSICAL ULYANOV INEQUALITIES 69

which coincides with (8.1), clearly gives a better estimate than (8.4). Moreover, (8.5) is sharp over the class     Lip ω(·), α + θ, p = f ∈ Lp (T) : ωα+θ (f, δ)p = O(ω(δ)) (see [90]). In more detail, ω ∈ Ωα+θ , there exists a function  for any function  f0 (x) = f0 (x, p, ω) ∈ Lip ω(·), α + θ, p such that for any q ∈ (p, ∞) and for any δ>0 1  δ

q dt q −θ ωα (f0 , δ)q ≥ C t ω(t) , t 0 where a constant C is independent of δ and ω. Let us give several examples showing essential differences between (8.1) and (8.4). Example 8.1. Let 1 < p < q < ∞ and α > θ. Define ∞  1 f0 (x) ∼ am cos mx, am = α+ε+1− 1 , q m m=1

ε > 0.

The Hardy–Littlewood theorem (see (2.3)) implies that f0 ∈ Lq (T). Moreover, realization (1.14) and Theorem 6.1 from [35] give  n 1/ν  ∞ 1/ν   −ξ ν ξν+ν−2 ν ν−2 ak k + ak k , ξ > 0, p ≤ ν ≤ q. ωξ (f0 , 1/n)ν n k=1

k=n+1

Then it is easy to see that ωα (f0 , δ)q δ α ,  0

and

δ



ωα+θ (f0 , t)p t

1 1 p−q

 0

δ



q

dt t

 q1

ωα (f0 , t)p t

1 1 p−q



q

dt t

1

δ α ln p 1δ , ε = 0; ε > 0, δα ,

 q1

δ α−θ ,

that is, (8.5) is essentially sharper than (8.4). Second, if 0 < p < 1, then an Lp -function may have certain pathological behavior in the sense of its smoothness properties. This phenomenon was observed earlier (see, e.g., [23], [57], and [75]). Let us now consider functions, which are smooth in Lp , 0 < p < 1 (in the sense of behaviour of their moduli of smoothness), and show that (8.1), unlike (8.4), provides the best possible estimate for such functions. Example 8.2. Let 0 < p < 1, p < q ≤ ∞, α ∈ N ∪ ((1/q − 1)+ , ∞), and let γ > 0 be such that α + γ ∈ N ∪ ((1/p − 1)+ , ∞) and α + γ > (1 − 1/q)+ . Let, for example, f = f0 , where f0 is given by (7.4) with ϕ(ξ) = (iξ)α+γ . Then, by (1.14), (7.11), and Lemma 6.5, we have ωα+γ (f, δ)p δ α+γ+ p −1 . 1

At the same time, (1.14) and (7.12) with ψ(x) = (ix)α imply   1 (−γ) α α ωα (f, δ)q  δ V1/δ q δ σ , δ

70

8. SHARP ULYANOV INEQUALITIES FOR MODULI OF SMOOTHNESS

where V1/δ is given by (5.2) and σ is defined in Theorem 8.1. By using the above formulas for the moduli of smoothness, it is easy to calculate that    δ  q  1q 1 ωα+γ (f, t)p dt ωα+γ (f, δ)p ωα (f, δ)q

σ + 1 1 δγ δ t 0 tp−q   1

δα σ , δ i.e., (8.1) is sharp for the function f0 . On the other hand, inequality (8.4) for f0 implies only the following (non-sharp) estimate:     q  1q δ 1 ωα (f, t)p dt α δ σ

ωα (f, δ)q  1 1 δ t 0 tp−q

δ α−( p − q ) . 1

1

We conclude this section by the following example dealing with the case p = 1. Example 8.3. Let 1 = p < q < ∞. We define f0 such that  a|ν| f0 (x) = eiνx , α+1−1/q (iν) ν∈Z where {aν } is a convex sequence of positive numbers, which can tend to zero very (α+1−1/q) slowly. Then, by [117, Ch. V, (1.5), p. 183], one gets f0 ∈ L1 (T). By the realization result, we then have that ωα+1−1/q (f0 , t)1  tα+1−1/q . Hence, using the Marchaud inequality, we obtain that ωα (f0 , t)1  tα . Then the classical inequality (8.4) yields ωα (f0 , t)q  tα−(1−1/q) . On the other hand, sharp Ulyanov inequality (8.5) gives a much better estimate: 1 ωα (f0 , t)q  tα ln1/q . t Moreover, this estimate is sharp in the sense that ωα (f0 , t)q  tα ln1/q 1t a[1/t] . 8.3. Multidimensional results Since for 1 < p < ∞ one has (8.6)

ωα (f, δ)p Rψ (f, δ)p ,

ψ(ξ) = |ξ|α

(see Lemma 6.2), the sharp Ulyanov inequalities for moduli of smoothness in the case 1 < p < q < ∞ immediately follows from Theorem 7.1. Note that inequality (8.6) is not true anymore if p = 1 or p = ∞. Therefore, since Theorems 6.1 and 7.1 cannot be applied, we will provide a direct proof of the sharp Ulyanov inequalities for moduli of smoothness. Our main result of this section reads as follows. Theorem 8.2. Let f ∈ Lp (Td ), d ≥ 2, 0 < p < q ≤ ∞, α ∈ N∪((1−1/q)+ , ∞), and γ, m ≥ 0 be such that α + γ, α + m ∈ N ∪ ((1/p − 1)+ , ∞). Then, for any δ ∈ (0, 1), we have  q1    δ  q 1 ωα+m (f, t)p 1 dt 1 ωα+γ (f, δ)p (8.7) ωα (f, δ)q  σ , + 1 1 δγ δ t 0 td( p − q )

8.3. MULTIDIMENSIONAL RESULTS

where (1) if 0 < p ≤ 1 and p < q ≤ ∞, ⎧ 1 ⎪ td( p −1) , ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ td( p −1) , ⎪ ⎪ ⎪ ⎪ d( p1 −1) q1 ⎨ t ln 1 (t + 1), σ(t) := 1 1 ⎪ ⎪ td( p −1) ln q (t + 1), ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ td( p − q )−γ , ⎪ ⎪ ⎪ ⎩ d( p1 − q1 ) t ,

71

then

γ > d 1 − 1q ;

+ γ = d 1 − 1q ≥ 1 and α + γ ∈ N;

+ 1 γ =d 1− q ≥ 1 and α + γ ∈ N; + 0 < γ = d 1 − 1q < 1;

+ 0 < γ < d 1 − 1q ; +

γ = 0;

(2) if 1 < p ≤ q ≤ ∞, then ⎧ 1, ⎪ ⎪ ⎪ ⎨ 1, 1 σ(t) := ln p (t + 1), ⎪ ⎪ ⎪ ⎩ d( p1 − 1q )−γ , t

γ ≥ d( p1 − 1q ), q < ∞; γ > dp , q = ∞; γ = dp ,

q = ∞;

0 ≤ γ < d( p1 − 1q ).

Remark 8.1. Equivalence (8.6) and Theorem 7.1 (B) give the sharpness of inequality (8.7) in the case 1 < p < ∞. Corollary 8.2. Under all conditions of Theorem 8.2, we have    q1  q11 δ 1 ωα+γ (f, t)p dt ωα (f, δ)q  σ . γ t t t 0 Proof of Theorem 8.2. By Theorem 6.2 and Corollary 3.1, we have that    ∂ α   sup|ξ|=1, ξ∈Rd  ∂ξ Tn   q   . η(n) nγ sup α+γ    Tn ∈Tn ∂  sup|ξ|=1, ξ∈Rd  T n  ∂ξ  p

Hence, it is enough to prove that

(8.8)

   ∂ α   sup|ξ|=1, ξ∈Rd  ∂ξ Tn   q   sup  σ(n). α+γ    Tn ∈Tn ∂   sup|ξ|=1, ξ∈Rd  ∂ξ Tn  p

Note that for γ = 0 and 0 < p < q ≤ ∞, this follows immediately from the classical Nikol’skii inequality (1.19). The rest of the proof for γ > 0 is divided into three cases. Case 1. Let 1 < p < q < ∞. In this case, Theorem 5.1 yields (8.8) since, by Corollary 3.1, we have that   α   ∂  α/2 sup  Tn    (−Δ) Tn q ∂ξ d |ξ|=1, ξ∈R

q

72

8. SHARP ULYANOV INEQUALITIES FOR MODULI OF SMOOTHNESS

and sup |ξ|=1, ξ∈R

  α+γ   ∂  (α+γ)/2  Tn  Tn p .   (−Δ) ∂ξ d p

Case 2. Let 1 < p < q = ∞. To show (8.8), we note that by Theorem 5.1 and Remark 5.3 with ϕ(y) ≡ 1 and ψ(y) = |y|−γ , we have, for any Tn ∈ Tn and ξ ∈ Rd , |ξ| = 1,    α   α      ∂ ∂ γ/2    T  σ(n) (−Δ) Tn  n  .   ∂ξ ∂ξ ∞ p Applying twice Corollary 3.3, we continue estimating as follows   α+γ   ∂  Tn   σ(n)(−Δ)(α+γ)/2 Tn p  σ(n) sup    , ∂ξ ξ∈Rd , |ξ|=1 p which is the desired estimate. Case 3. Let 0 < p ≤ 1, p < q ≤ ∞, and γ > 0. In this case, we use the Marchaud-type inequality (see (10.16) below):  q1  1  τ  τ1  δ  q ωα+γ (f, t)p ωα+m (f, t)p 1 dt 1 dt α (8.9) ωα (f, δ)q  δ + , 1 1 1 1 t t δ 0 tα+d( p − q ) td( p − q ) where

 τ=

min(q, 2), q < ∞; 1, q = ∞.

By (4.9), it is easy to see that (8.10)

⎧

1 ⎪ 1, γ > d 1 − ⎪ q ; ⎪  1  τ   ωα+γ (f, t)p dt ωα+γ (f, δ)p ⎨ 1/τ  1 1 α ln  γ+d( 1 −1) δ 1 1 δ +1 , γ =d 1− q ; ⎪ t

p δ tα+d( p − q ) δ ⎪  1 d(1− 1q )−γ ⎪ 1 ⎩ , 0 < γ < d 1− δ q . 1 τ

Hence, (8.9) and (8.10) imply that (8.7) holds in the following four cases: (i) γ > 0 and 0 < q ≤ 1; (ii) γ = d(1 − 1/q) and 1 < q ≤ 2 with α + γ ∈ N; (iii) γ = d(1 − 1/q) and q = ∞ with α + γ ∈ N; (iv) 0 < γ < d (1 − 1/q)+ . Thus, it remains to consider: (v) γ = d (1 − 1/q) ≥ 1, α + γ ∈ N, and 2 < q < ∞ (note that for such q and d ≥ 2 we always have d(1 − 1/q) ≥ 1); (vi) γ = d (1 − 1/q) ≥ 1, 1 < q ≤ ∞, and α + γ ∈ N. We have that inequality (8.8) follows from Lemma 5.12 in the case (v) and from Lemma 5.14 in the case (vi).  We finish this section with an important corollary.

8.3. MULTIDIMENSIONAL RESULTS

73

Corollary 8.3. Let 0 < p ≤ 1 < q ≤ ∞ and d ≥ 2. We have   q 1 δ ωα+d(1− q1 ) (f, t)p 1 dt q1 (8.11) ωα (f, δ)q  1 1 t 0 td( p − q ) provided that α + d(1 − 1/q) ∈ N. In particular, for any d, α ∈ N, we have  δ ωα+d (f, t)1 dt (8.12) ωα (f, δ)∞  . td t 0 Note that (8.11) does not hold in the one-dimensional case. Comparing inequality (8.11) for p = 1 with (1.7), we observe new effects in the multivariate case. For d = 1, inequality (8.12) is known (see [102]).

CHAPTER 9

Sharp Ulyanov inequalities for realizations of K-functionals related to Riesz derivatives and corresponding moduli of smoothness Let d ≥ 1, ψ(ξ) = |ξ|α , and ϕ(ξ) = |ξ|α+γ . dimensional case, we define Rα (f, δ)p =

inf

T ∈T[1/δ]

where (−Δ)α/2 T (x) =

By analogy with the one-

{f − T p + δ α (−Δ)α/2 T p },



|k|α ck ei(k,x) ,

x ∈ Td .

|k|∞ ≤n

Theorem 7.1 for such ψ and ϕ immediately implies the following sharp Ulyanov inequality for Rα (f, δ)p . Theorem 9.1. Let f ∈ Lp (Td ), d ≥ 1, 0 < p < q ≤ ∞, α > 0, and γ ≥ 0. (A) For any δ ∈ (0, 1), we have  q1    δ   Rα+γ (f, δ)p Rα+γ (f, t)p q1 dt 1 1 (9.1) Rα (f, δ)q  σΔ , + 1 1 δγ δ t 0 td( p − q ) where (1) if 0 < p ≤ 1 and p < q ≤ ∞, then ⎧ 1 ⎪ td( p −1) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ td( p1 −1) ln q11 (t + 1), σΔ (t) := 1 1 ⎪ ⎪ td( p − q )−γ , ⎪ ⎪ ⎪ ⎪ ⎩ d( p1 − q1 ) t , (2) if 1 < p ≤ q ≤ ∞, then ⎧ 1, ⎪ ⎪ ⎪ ⎨ 1, 1 σΔ (t) := ln p (t + 1), ⎪ ⎪ ⎪ ⎩ d( p1 − q1 )−γ , t



γ > d 1 − 1q ; + 0 < γ = d 1 − 1q ;

+ 0 < γ < d 1 − 1q ; +

γ = 0,

γ ≥ d( p1 − 1q ), q < ∞; γ > dp , q = ∞; γ = dp ,

q = ∞;

0 ≤ γ < d( p1 − 1q ).

(B) Inequality (9.1) is sharp in the following sense. Let α ∈ (2N)∪(d(1/q −1)+ , ∞) and α + γ > d(1 − 1/q)+ . Then there exists a function f0 ∈ Lq (Td ), f0 ≡ const, such that  q1    δ   Rα+γ (f0 , δ)p Rα+γ (f0 , t)p q1 dt 1 1 σΔ Rα (f0 , δ)q

+ 1 1 δγ δ t 0 td( p − q ) 75

76

9. SHARP ULYANOV INEQUALITIES

as δ → 0. Remark 9.1. We would like to emphasize that the main difference in the definitions of σ in Theorems 8.1, 8.2, and 9.1 is when γ is the critical parameter, i.e., γ = d (1 − 1/q)+ . Under certain additional restrictions, similarly to Theorem 7.1 , one can prove the following more accurate inequality. Remark 9.2. Let f ∈ Lp (Td ), d ≥ 1, 0 < p < q < ∞, α > 0, and γ, m ≥ 0 be such that α + γ, α + m ∈ (2N) ∪ (d(1/p − 1)+ , ∞). Then, for any δ ∈ (0, 1), we have  q1    δ   Rα+γ (f, δ)p Rα+m (f, t)p q dt 1 Rα (f, δ)q  σΔ , + 1 1 δγ δ t 0 td( p − q ) where σΔ (·) is defined in Theorem 9.1. Note that in some cases the realization Rα can be replaced by special moduli of smoothness (see, for example, [59] for the case q ≥ 1 and [24] for the case 0 < q < 1; see also [85–87]). Example 9.1. Let α > 0, r ∈ N, and 1 ≤ q ≤ ∞. Denote   ˙ 2r f (·)   Δ δu  , wα (f, δ)q :=  du r = [(d − 1 + α)/2] + 1,   |u|d+α |u|≥1

q

where ˙ rh = Δ ˙ h (Δ ˙ r−1 ), Δ ˙ rh f (x) = f (x + h) − f (x − h), h ∈ Rd . Δ h It follows from [59], [84], and Lemma 6.1 that, for 1 ≤ q ≤ ∞, one has

f − T q + δ α (−Δ)α/2 T q wα (f, δ)q Rα (f, δ)q = inf T ∈T[1/δ]

f − gq + δ α (−Δ)α/2 gq . inf

Kα (f, δ)q = (−Δ)α/2 g∈Lq (Td )

Example 9.2. Now, we consider the following moduli of smoothness, which were studied in [86] for any 0 < q ≤ ∞ (see also [24] for the case m = 1):     m d   ξm    (−1)ν 2m (m)  , w %2 (f, δ)q := sup  ) − f (·)) (f (· + νhe j  d  2 m − |ν| ν |h|≤δ q j=1 ν=−m where m, d ∈ N and

 ξm =

Set

  m  2m (−1)ν 2 m−ν ν2 ν=1

−1 .

⎧ ⎨ 0,

d = 1; d = 2, m = 2k, k ∈ N; ⎩ otherwise. It is known [86] (see also [24, Theorem 3.2]) that qm,d =

(m)

w %2

d d+2(m+1) , d d+2m ,

(f, δ)q R2 (f, δ)q ,

qm,d < q ≤ ∞.

Thus, for all 1 ≤ q ≤ ∞ and m ∈ N, we have (m)

%2 w2 (f, δ)q w

(f, δ)q R2 (f, δ)q .

9. SHARP ULYANOV INEQUALITIES

77

Using the above equivalences and Remark 9.2, we obtain the following result. Corollary 9.1. Let f ∈ Lp (Td ), d ≥ 1. For any δ ∈ (0, 1), one has the following three inequalities: (i) if 1 ≤ p < q ≤ ∞ and α, γ > 0, then  q1    δ  q 1 wα+γ (f, t)p 1 dt 1 wα+γ (f, δ)p σΔ ; wα (f, δ)q  + 1 1 δγ δ t 0 td( p − q ) (ii) if pm,d < p < q, 1 ≤ q ≤ ∞, and 0 < γ < 2, then  q1    δ  (m) q (m) 1 w %2 (f, t)p 1 dt 1 w %2 (f, δ)p w2−γ (f, δ)q  σΔ ; + 1 1 δγ δ t 0 td( p − q ) (iii) if pm,d < p < q ≤ ∞, then  (m)

w %2

δ



(f, δ)q  0

(m)

w %2 t

(f, t)p

q1

1 d( p − q1 )

dt t

 q1

1

,

where σΔ (·) is defined in Theorem 9.1. Example 9.3. A more complete picture can be obtained when d = 1. We will deal with the following special modulus of smoothness, which has been recently introduced by Runovski and Schmeisser [85] (see also [87]). For α > 0 and f ∈ Lp (T), we define        β|ν| (α)  ωα (f, δ)p := sup  − f (· + νh) − f (·)  , β0 (α) 0 0 and 1/(α + 1) < p ≤ ∞, one has (9.2)

ωα (f, δ)p Rα (f, δ)p ,

δ > 0.

Note also that, for all α ∈ 2N, the modulus ωα (f, δ)p coincides with the classical modulus of smoothness ωα (f, δ)p . From equivalence (9.2) and Theorem 9.1, we have Corollary 9.2. Let f ∈ Lp (T), 0 < p < q ≤ ∞, γ ≥ 0, α ∈ (2N) ∪ ((1/q − 1)+ , ∞), and α + γ ∈ (2N) ∪ ((1/p − 1)+ , ∞). Then, for any δ ∈ (0, 1), we have  q1    δ   ωα+γ (f, δ)p ωα+γ (f, t)p q1 dt 1 1 + ωα (f, δ)q  σΔ , 1 1 δγ δ t 0 tp−q where σ(·) is defined in Theorem 9.1 in the case d = 1.

CHAPTER 10

Sharp Ulyanov inequality via Marchaud inequality This chapter is devoted to the study of new types of Ulyanov inequalities with the use of the Marchaud inequalities. Such inequalities are of great importance when investigating the embedding theorems in Chapter 13. Throughout this Chapter, we assume that  min(q, 2), q < ∞; τ = τ (q) = 1, q = ∞. 10.1. Inequalities for realizations of K-functionals First, we obtain an analogue of the Marchaud inequality (cf. Theorem 4.4) for the realizations of the K-functionals. Lemma 10.1. Let f ∈ Lq (Td ), 0 < q ≤ ∞, α > 0, γ ≥ 0, ψ ∈ Hα , and ϕ ∈ Hα+γ . Let also either ψ be a polynomial or d/(d + α) < q ≤ ∞. Then, for any δ ∈ (0, 1/2), one has  1  τ  τ1 Rϕ (f, t)q dt α . (10.1) Rψ (f, δ)q  δ α t t δ Proof. We start with the inverse estimate given by  1   1 E[1/t] (f )q τ dt τ α . (10.2) Rψ (f, δ)q  δ tα t δ In the case 0 < q ≤ 1 and q = ∞, the proof of (10.2) can be found in [82, Theorem 4.26, p. 203]. In the case 1 < q < ∞, it follows from Theorem 4.4 (see also [107, Theorem 2.1]) and (8.6). To finish the proof of (10.1), we apply Jackson’s inequality, which follows from Lemma 6.3.  One of the main results in this chapter is the following sharp Ulyanov inequality via Marchaud inequality. Theorem 10.1. Let 0 < p < q ≤ ∞, α > 0, γ ≥ 0, ψ ∈ Hα , and ϕ ∈ Hα+γ . Let also either ψ be a polynomial or d/(d + α) < q ≤ ∞. (A) Let f ∈ Lp (Td ). Then, for any δ ∈ (0, 1/2), we have  1  τ  τ1  δ  q1  q11 R R (f, t) dt (f, t) dt ϕ p ϕ p (10.3) Rψ (f, δ)q  δ α + . 1 1 α+d( p − 1q ) d( p − q1 ) t t δ 0 t t (B) Inequality (10.3) is sharp in the following sense. Let α + γ > d(1 − 1/q)+ , γ > 0, and (1) γ = d(1 − 1/q) > 0 if 2 < q < ∞ and 0 < p ≤ 1; 79

ϕ ψ

and

ψ ϕ

∈ C ∞ (Rd \{0}),

80

10. SHARP ULYANOV INEQUALITY VIA MARCHAUD INEQUALITY

(2) γ = 1 if d = 1, q = ∞, 0 < p ≤ 1, and A ∈ C \ {0}; (3) γ = d(1/p − 1/q) if 1 < p < q ≤ ∞.

ψ(ξ) ϕ(ξ)

= A|ξ|−γ sign ξ

for any

Then there exists a function f0 ∈ Lq (Td ), f0 ≡ const, such that  (10.4) Rψ (f0 , δ)q δ α δ

1



Rϕ (f0 , t)p t

1 α+d( p − 1q )

τ

dt t



 τ1

δ

+ 0



Rϕ (f0 , t)p t

1 d( p − q1 )

q1

dt t

 q1

1

as δ → 0. Similarly to Theorem 6.1 , one can obtain the following analogue of Theorem 10.1. Theorem 10.1 Under all conditions of Theorem 10.1, suppose that the function ϕ is either a polynomial or d/(d + α + γ) < p ≤ ∞. Let also m > 0 and the function φ ∈ Hα+m be either a polynomial or d/(d + α + m) < p ≤ ∞. Then, for any δ ∈ (0, 1/2), we have  q1  1  τ  τ1  δ  q Rϕ (f, t)p Rφ (f, t)p 1 dt 1 dt α (10.5) Rψ (f, δ)q  δ + . 1 1 1 1 t t δ 0 tα+d( p − q ) td( p − q ) It is worth mentioning the following important corollary for the case γ = 0. Corollary 10.1. Let f ∈ Lp (Td ), 0 < p < q ≤ 1, ψ, ϕ ∈ Hα , α > 0. Let either ψ be a polynomial or d/(d + α) < q ≤ 1 and let either ϕ be a polynomial or d/(d + α) < p < 1. Let also m > 0 and the function φ ∈ Hγ+m be either a polynomial or d/(d + α + m) < p < 1. Then, for any δ ∈ (0, 1), we have (10.6)   q  q1  δ Rφ (f, t)p dt Rϕ (f, δ)p 1,   0 < q < 1; Rψ (f, δ)q  + d( 1 −1) 1 1 d( p − q1 ) ln + 1 , q = 1. t p 0 t δ δ Proof. The proof of (10.6) follows from (10.5) and Lemma 6.5.



Remark 10.1. (i) We note that all three cases (1)–(3) in part (B) of Theorem 10.1, where the optimality of inequality (10.3) cannot be proved, are those cases where the Ulyanov inequality, given by Theorem 7.1, provides a sharper estimate than inequality (10.3). Moreover, one can construct examples showing that equivalence (7.2) holds in these cases. Indeed, in cases (1) and (2), taking f = f0 such that Rϕ (f0 , t)p td(1/p−1)+α+γ , we see that the equivalence in (7.2) is true, while in (10.4) the equivalence for f0 is not valid. In the case (3), we obtain the same conclusion by taking any f0 ∈ C ∞ (Td ). (ii) At the same time, taking f0 ∈ C ∞ (Rd ) and letting 0 < p < 1 and γ = d(1 − 1/q)+ , we have that the sharp Ulyanov inequality given by (10.5) becomes the equivalence  q1  1  τ  τ1  δ  q Rϕ (f0 , t)p Rφ (f0 , t)p 1 dt 1 dt α α Rψ (f0 , δ)q δ δ + . 1 1 1 1 t t δ 0 tα+d( p − q ) td( p − q )

10.1. INEQUALITIES FOR REALIZATIONS OF K-FUNCTIONALS

81

On the other hand, the sharp Ulyanov inequality given by (7.3) implies only Rϕ (f0 , δ)p 1

Rψ (f0 , δ)q δ  δ ln σ δ δγ α

1 q

α

 q1    δ  q 1 Rφ (f0 , t)p 1 dt 1 . + 1 1 δ t 0 td( p − q )

Proof. (A) Using the Ulyanov type inequality given by (6.10) and then the Marchaud inequality given by (10.1), we get 

1

Rψ (f, δ)q  δ α



u−ατ

(10.7)



u−ατ



δ

+ δα

Rϕ (f, t)p



u−ατ

u



1

Rϕ (f, t)p

q1

δ

1

 qτ

dt t

1

dt t dt t

td( p − q ) 1

δ

 q1

td( p − q )

0

1

1

1

δ



td( p − q )

0 1

q1

Rϕ (f, t)p 1

δ

⎛  α⎝ δ



u

 qτ

1

 qτ

1

du u

 τ1

⎞ τ1 du ⎠ u du u

 τ1 := I1 + I2 .

We start with the estimate of the first integral:  1  τ1  δ  q1  q11 R du (f, t) dt ϕ q I1 = δ α 1 1 1+ατ d( − ) t δ u 0 t p q (10.8) 1    q q δ Rϕ (f, t)q 1 dt 1  . 1 1 t 0 td( p − q ) To estimate I2 , we consider two cases: τ > q1 and τ ≤ q1 . In the first case, Hardy’s inequality (see Lemma 2.7) yields  I2  δ

(10.9)

1



Rϕ (f, t)p

α

τ

tα+d( p − q ) 1

δ

1

dt t

 τ1 .

In the second case, we use the discretization of integrals. Let n, N ∈ Z+ be such that 2−N < δ ≤ u ≤ 2−n . Since by Lemma 6.5 we have Rϕ (f, t)p Rϕ (f, 2t)p , then 

u



Rϕ (f, t)p td( p − q ) 1

δ

1

q1

dt t

 qτ

1

   

2−n



Rϕ (f, t)p

q1

td( p − q ) 1

2−N

1

dt t

 qτ

1

N

q1  1 1 2kd( p − q ) Rϕ (f, 2−k )p

 qτ

1

k=n N

τ  1 1  2kd( p − q ) Rϕ (f, 2−k )p k=n

 

2−n 2−N



Rϕ (f, t)p t

1 d( p − q1 )

τ

dt  t

 δ

u



Rϕ (f, t)p t

1 d( p − q1 )

τ

dt . t

82

10. SHARP ULYANOV INEQUALITY VIA MARCHAUD INEQUALITY

Thus, 

(10.10)



τ

1 dt du τ I2  δ u 1 1 t u δ δ td( p − q )  1  τ  1 1 Rϕ (f, t)p du dt τ α =δ 1 1 1+ατ t δ t u td( p − q )  1  τ  τ1 Rϕ (f, t)p dt α δ . 1 α+d( p − 1q ) t δ t 1

α

−ατ



u

Rϕ (f, t)p

Finally, combining (10.7)–(10.10), we get (10.3). (B) The proof of this part goes along the same lines as the proof of part (B) of Theorem 7.1. In what follows, we denote for the sake of simplicity  U M (f, δ)p := δ

1

α δ



Rϕ (f, t)p t

1 α+d( p − q1 )

τ

dt t



 τ1

δ



+

Rϕ (f, t)p t

0

q1

1 d( p − 1q )

dt t

 q1

1

.

Part (B1): 0 < p ≤ 1 and p < q ≤ ∞. Let f0 be given by (7.4), i.e., f0 (x) = Fϕ (x) −

1

N

F (N x), α+γ ϕ

Fϕ (x) =

 ei(k,x) k =0

ϕ(k)

,

and N is a fixed sufficiently large integer which will be chosen later. Let δ [1/n]. By (7.11) and (7.12), we have, respectively, (10.11)

Rϕ (f0 , δ)p  δ d( p −1)+α+γ 1

and (10.12)

Rψ (f0 , 1/n)q  n−α D(ψ/ϕ)Vn q .

Note that we have already obtained the following two-sided bounds of D(ψ/ϕ)Vn q : (a) if 0 < q ≤ 1, Lemma 5.2 gives D(ψ/ϕ)Vn q 1, (b) if 1 < q < ∞, Lemmas 5.6 and 5.7 imply ⎧

1 ⎪ 1, γ > d 1 − ⎪ q ; ⎪ ⎨ 1 1 ln q (n + 1), γ = d 1 − q ; D(ψ/ϕ)Vn q

⎪

⎪ ⎪ ⎩ nd(1− q1 )−γ , 0 < γ < d 1 − 1 , q

10.1. INEQUALITIES FOR REALIZATIONS OF K-FUNCTIONALS

(c) if q = ∞, by Lemma 5.8, ⎧ 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ D(ψ/ϕ)Vn q

ln(n + 1), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ln(n + 1), ⎪ ⎩ d−γ n ,

83

γ > d; ψ(ξ) = A|ξ|−γ sign ξ ϕ(ξ) for some A ∈ C \ {0}; ψ(ξ) = A|ξ|−γ sign ξ γ = d = 1 and ϕ(ξ) for any A ∈ C \ {0}; γ = d ≥ 2; 0 < γ < d.

γ = d = 1 and

Now, using (10.11) and taking into account that α +γ > d(1−1/q)+ , we obtain  1

τ dt  τ1 α+γ−d(1− 1q ) γ−d(1− q1 ) α U M (f0 , δ)p  δ +δ t t δ ⎧

⎪ 1, γ > d 1 − 1q and 0 < q ≤ ∞; ⎪ (10.13) ⎪ ⎨ 1

 ln τ 1δ + 1 , γ = d 1 − 1q and 1 < q ≤ ∞;  δα ⎪

⎪ ⎪ ⎩ δ γ−d(1− q1 ) , 0 < γ < d 1 − 1q and 1 < q ≤ ∞. Thus, combining (10.12) and (10.13), we derive that for γ > d(1−1/q)+ the required inequality U M (f0 , δ)p  Rψ (f0 , δ)q holds in all cases except (i) 2 < q < ∞ and γ = d(1 − 1/q), −γ (ii) γ = d = 1, q = ∞, and ψ(ξ) sign ξ for some A ∈ C \ {0}. ϕ(ξ) = A|ξ| Part (B2): 1 < p < q < ∞ and γ = d(1/p − 1/q). In this case, if γ > d(1/p − 1/q), then we can take as f0 any function from C ∞ (Td ). Then (7.20) implies (10.14)

Rψ (f0 , δ)q δ α

Rϕ (f0 , δ)q δ α+γ .

and

Hence, (10.15)

U M (f0 , δ)p Rψ (f0 , δ)q δ α .

If 0 ≤ γ < d(1/p − 1/q), then we take f0 (x) = Fϕ (x) =

 ei(k,x) k =0

ϕ(k)

.

By (7.24), we have Rϕ (f0 , δ)p  δ α+γ+d( p −1) 1

and by (7.25) Rψ (f0 , δ)q  δ α+γ+d( q −1) . 1

Therefore, since γ < d(1/p − 1/q) < d(1 − 1/q), we have U M (f0 , δ)p  δ α+γ+d( q −1)  Rψ (f0 , δ)q . 1

84

10. SHARP ULYANOV INEQUALITY VIA MARCHAUD INEQUALITY

Part (B3): 1 < p < ∞, q = ∞, and γ = d(1/p − 1/q). Let first γ > d/p. Then we consider f0 ∈ C ∞ (Td ). In this case, we have the same equivalences as in (10.14) and (10.15). Finally, assume that 0 < γ < d/p. Here we deal with the function f0 (x) =

 ei(k,x) . |k|γ ψ(k)

k =0

Then, (7.26) yields Rψ (f0 , δ)∞  δ α+γ−d . On the other hand, by (7.27), Rϕ (f0 , δ)p  δ α+γ+d( p −1) . 1

Thus, applying the last two estimates, we have U M (f0 , δ)p  δ α+γ−d  Rψ (f0 , δ)q .  10.2. Inequalities for moduli of smoothness Using Theorem 4.4, the Ulyanov inequality (Theorems 8.1 and 8.2 for γ = 0), and repeating arguments that were used in the proof of Theorem 10.1 (A), we obtain the following result. Theorem 10.2. Let f ∈ Lp (Td ), d ≥ 1, 0 < p < q ≤ ∞, α ∈ N ∪ ((1/q − 1)+ , ∞), and γ, m > 0 be such that α + γ, α + m ∈ N ∪ ((1/p − 1)+ , ∞). We have (10.16)  q1  1  τ  τ1  δ  q ωα+γ (f, t)p ωα+m (f, t)p 1 dt 1 dt α ωα (f, δ)q  δ + . 1 1 1 1 t t δ 0 tα+d( p − q ) td( p − q ) Remark 10.2. (i) Using Theorem 10.1 (B) and realization result (1.14), we obtain that inequality (10.16) is sharp in the following sense. Let d = 1 and α + γ > (1 − 1/q)+ . Let also (1) γ = 1 − 1/q > 0 if 2 < q ≤ ∞ and 0 < p ≤ 1, (2) γ =  1/p − 1/q if 1 < p < q ≤ ∞. Then there exists a function f0 ∈ Lq (T), f0 ≡ const, such that we have  ωα (f0 , δ)q δ

α δ

1



ωα+γ (f0 , t)p t

1 α+ p − q1

τ

dt t



 τ1 +

0

δ



ωα+γ (f0 , t)p t

1 1 p−q

q1

dt t

 q1

1

as δ → 0. (ii) Note that two cases (1) and (2) where the sharpness of inequality (10.3) cannot be proved are those cases where the Ulyanov inequality, given by Theorem 8.2, provides a sharper bound than the one given by inequality (10.16). Let us formulate an important corollary which will be used to obtain embedding theorems in Chapter 13.

10.2. INEQUALITIES FOR MODULI OF SMOOTHNESS

Corollary 10.2. Under the conditions of Theorem 10.2,    q1 q δ ωα+m (f, t)p 1 dt 1 ωα (f, δ)q  1 1 t 0 td( p − q ) ⎧ ⎪ 1, (10.17) ⎪ ⎪ ⎨   ωα+γ (f, δ)p ln1/τ 1δ + 1 , + γ+d( 1 −1) ⎪ p δ ⎪ 1 ⎪ ⎩  1 d(1− q )−γ , δ

85

we have



γ > d 1 − 1q ;

+ γ = d 1 − 1q ;

+ γ < d 1 − 1q . +

The proof of the corollary follows from inequality (10.16) and property (e) of moduli of smoothness from Chapter 4. We conclude this chapter with the proof of Remark 5.2. Proof of Remark 5.2. Let φ(ξ) = ϕ(ξ)|ξ|m , m ∈ 2N, and m > d(1/p − 1). First, by Corollary 10.1, we obtain the following analogue of (10.17): (10.18)   q  1q  δ Rφ (f, t)p dt Rϕ (f, δ)p 1, 0 < q < 1; Rψ (f, δ)q  + 1 1 1 d( p − 1q ) d( p −1) + 1), q = 1. ln( t 0 t δ δ Next, let f = Tn ∈ Tn , n = [1/δ]. Then by the definition of the realization, we have (10.19)

Rψ (Tn , 1/n)q n−α D(ψ)Tn q ,

(10.20)

Rϕ (Tn , 1/n)p n−α D(ϕ)Tn p ,

and Rφ (Tn , t)q  tα+m D(φ)Tn p ,

(10.21)

0 < t < δ.

Combining (10.18)–(10.21), we derive (10.22) n−α D(ψ)Tn q  n−α−m+d( p − q ) D(φ)Tn p 1

1

+ n−α+d( p −1) D(ϕ)Tn p 1



1, 0 < q < 1; ln n, q = 1.

Finally, taking into account that by the classical Bernstein inequality D(φ)Tn p  nm D(ϕ)Tn p , we get from (10.22) that D(ψ)Tn q  D(ϕ)Tn p which gives (5.3).

&

0 < q < 1; nd( p −1) , 1 d( p −1) n ln n, q = 1, 1



CHAPTER 11

Sharp Ulyanov and Kolyada inequalities in Hardy spaces In this chapter, we obtain sharp Ulyanov and Kolyada type inequalities in Hardy spaces. We start with the real Hardy spaces on Rd and continue by Subsection 11.2 dealing with analytic Hardy spaces. 11.1. Sharp Ulyanov and Kolyada inequalities in Hp (Rd ) Let us recall that the real Hardy spaces Hp (Rd ), 0 < p < ∞, is the class of tempered distributions f ∈ S  (Rd ) such that   f Hp = f Hp (Rd ) =  sup |ϕt ∗ f (x)|L (Rd ) < ∞, t>0

p

where ϕ ∈ S (Rd ), ϕ(0)

= 0, and ϕt (x) = t−d ϕ(x/t) (see [95, Ch. III]). The K-functionals and moduli of smoothness in Hp (Rd ) are defined, respectively, as follows (see [115]):

Kα (f, δ)Hp := inf f − gHp + δ α (−Δ)α/2 gHp g

and ωα (f, δ)Hp := sup Δα h f Hp , |h| 0 we have (11.1)

ωα (f, δ)Hp Kα (f, δ)Hp .

In the case α ∈ N, this equivalence was proved in [62, p. 175]. The case α > (1/p − 1)+ can be obtained similarly by using the Fourier multiplier technique (see [114]). In this chapter, we restrict ourselves to consideration of K-functionals defined by the Laplacian operators since the general case Kψ (f, δ)Hp with ψ ∈ Hα can be reduced to K(−Δ)α/2 (f, δ)Hp . This follows from an analogue of Lemma 5.6 in Hp (Rd ), 0 < p < ∞ (see, e.g., [31, Ch. III, § 7]). Using boundedness properties of the fractional integrals in the Hardy spaces (see [56]) and following the proof of Theorem 6.1, we obtain the sharp Ulyanov inequality in Hp (Rd ). 87

88

11. SHARP ULYANOV AND KOLYADA INEQUALITIES IN HARDY SPACES

Theorem 11.1. Let f ∈ Hp (Rd ), 0 < p < q < ∞, α > 0, and θ = d(1/p−1/q). Then, for any δ ∈ (0, 1), we have 

δ



Kα (f, δ)Hq  0

q

Kα+θ (f, t)Hp tθ

dt t

 1q .

In addition, if α ∈ N ∪ ((1/q − 1)+ , ∞) and α + θ ∈ N ∪ ((1/p − 1)+ , ∞), then 

δ

ωα (f, δ)Hq  0



ωα+θ (f, t)Hp tθ

q

dt t

 q1 .

Hence, unlike the case of Lebesgue spaces (cf. [102]), the sharp Ulyanov inequality in Hardy spaces has the same form both in quasi-Banach spaces (0 < p < 1) and Banach spaces (p ≥ 1). Now, we concern with the Kolyada-type inequality. We recall that in the Lebesgue spaces Lp (Td ), 1 < p < q < ∞, inequality (1.2) was improved by Kolyada [50] as follows: (11.2)  q1    1  p1  δ    1 1 dt dt p q 1−θ θ−1 −θ − t ω1 (f, t)q t ω1 (f, t)p  , θ=d δ . t t p q δ 0 This estimate is sharp over the classes   Lip (ω(·), 1, p) = f ∈ Lp (Td ) : ω1 (f, δ)p = O (ω(δ)) ,   that is, for any ω ∈ ω(0) = 0, ω ↑, ω(δ1 + δ2 ) ≤ ω(δ1 ) + ω(δ2 ) , there exists a function f0 ∈ Lip (ω(·), 1, p) such that, for any δ > 0,  δ

1−θ

1

t

δ

θ−1

p dt ω1 (f0 , t)q t



 p1

δ



t

0

−θ

q dt ω(t) t

 q1 .

It is worth mentioning that (11.2) is not valid for p = 1, d = 1 but is true for p = 1, d ≥ 2. Note that recently, Trebels [107] (see also [34]) obtained an analogue of inequality (11.2) for moduli of smoothness of fractional order in 1 ≤ p < ∞. Kolyada [50] also proved (11.2) for functions belonging to the analytic Hardy spaces on the disc, that is for f ∈ Hp (D) and 0 < p < q < ∞. Below, we extended this results to the real Hardy spaces Hp (Rd ). To study Kolyada’s inequalities in Hp (Rd ), we need the following straightforward modification of the Holmstedt formulas (see [5] and [39])  (11.3)

K(f, t ; X, (X, Y )θ,q ) t θ

∞

θ

[s

−θ

t

ds K(f, s)] s

 1q

and % K(f, t1−θ ; (X, Y )θ,q , Y ) :=

inf {|f − g|θ,q + t1−θ |g|Y }

g∈Y



(11.4)

t

[s 0

−θ

ds K(f, s)] s

 q1 ,

11.1. SHARP ULYANOV AND KOLYADA INEQUALITIES IN Hp (Rd )

89

where (X,  · X ) is a quasi-Banach space, Y ⊂ X is a complete subspace with seminorm | · |Y and  · Y =  · X + | · |Y , K(f, t) ≡ K(f, t; X, Y ) := inf (f − gX + t|g|Y ) g∈Y

is Peetre’s K-functional, and & (X, Y )θ,q :=



f ∈ X : |f |θ,q =

∞

[t

−θ

0

dt K(f, t)] t

 q1

' < ∞,

0 θ. Then    ∞  p  p1 q  q1 δ Kα (f, t)Hq Kα (f, t)Hp dt dt α−θ (11.7) δ  . α−θ θ t t t t δ 0 In addition, if α ∈ N ∪ ((1/p − 1)+ , ∞), then    ∞  p  p1 q  1q δ ω ω (f, t) (f, t) dt dt α H α H q p (11.8) δ α−θ  . α−θ θ t t t t δ 0 Proof. By using (11.6) and (11.3), we obtain  ∞ 1  θ−α p dt p θ 1− α I:=δ t Kα (f, t)Hq t δ 1/α  ∞

p ds  p1 θ θ 1− α −1 α α s δ K(f, s; Hq , Fq,2 ) s δ

θ α

K f, δ 1− α ; Hq , (Hq , Fq,2 )1− θ ,p . α

Using now f Hq (Rd )  f B˙ p,q θ (Rd ) and f B˙ q,p α−θ (Rd )  f F˙ α

p,2 (R

d)

(−Δ)α/2 f Hp (Rd ) ,

(see Lemma 2.1), we get θ

α−θ I ≤ f − gHq + δ 1− α |g|Bq,p θ

 f − gB˙ p,q + δ 1− α gF˙ α θ

p,2

 |f − g|Bp,q θ +δ

θ 1− α

α |g|Fp,2

90

11. SHARP ULYANOV AND KOLYADA INEQUALITIES IN HARDY SPACES

α for all g ∈ Fp,2 . α Next, taking infimum over all g ∈ Fp,2 and applying (11.4) and (11.5) as well as (11.6), we have

α α % f, δ 1− αθ ; (Hp , Fp,2 IK ) θ ,q , Fp,2 α 1  δ

q ds q θ −α α s K(f, s; Hp , Fp,2 )

s 0  q1  1/α δ  −θ q dt t Kα (f, t)Hp

. t 0

The last inequality implies (11.7). Finally, (11.1) yields (11.8).



Remark 11.1. Note that inequality (11.8) implies the sharp Ulyanov inequality for f ∈ Hp (Rd ), 0 < p < q ≤ 2, given by    q1  δ ωα (f, t)Hp q dt ωα−θ (f, δ)Hq  , α > θ. tθ t 0 This follows from the Marchaud inequality in Hq (Rd ):  ∞  q  1q ωα (f, t)Hq dt α−θ ωα−θ (f, δ)Hq  δ , α−θ t t δ

0 < q ≤ 2.

The latter can be proved by using the standard technique with the help of Bernstein’s and Jackson’s inequalities (see, e.g., [115] and [108, Ch. 1]). The above results in this chapter remain true in the periodic real Hardy classes Hp (Td ). Recall that the real periodic Hardy spaces Hp (Td ), 0 < p < ∞, are defined as a class of the tempered distributions f ∈ S  (Td ) such that       i(k,x)     f Hp (Td ) =  sup  ϕ(tk)a

< ∞, k (f )e  d t>0 k∈Zd

Lp (T )

where ak (f ) are Fourier coefficients of the distribution f (see, e.g., [108, Ch. 9] and [29, p. 156]). In particular, we have the following analogue of Theorem 11.2. Theorem 11.3. Let f ∈ Hp (Td ), 0 < p < q < ∞, θ = d (1/p − 1/q), and α > θ. Then, for any δ ∈ (0, 1), we have     p  p1 q  1q 1 δ Kα (f, t)Hq (Td ) Kα (f, t)Hp (Td ) dt dt α−θ  . (11.9) δ α−θ θ t t t t δ 0 In addition, if α ∈ N ∪ ((1/p − 1)+ , ∞), then, for any δ ∈ (0, 1), we have     1 1   1 δ ωα (f, t)Hq (Td ) p dt p ωα (f, t)Hp (Td ) q dt q α−θ  . (11.10) δ tα−θ t tθ t δ 0 Since Hp (Td ) = Lp (Td ) for 1 < p < ∞, both (11.9) and (11.10) remain true for the Lebesgue spaces. In fact, inequality (11.10) also holds for Lp (Td ), p ≥ 1, when d ≥ 2. This was proved by Kolyada [50] in the case α = 1. We extend this

11.2. SHARP ULYANOV AND KOLYADA INEQUALITIES IN ANALYTIC HARDY SPACES 91

result to moduli of smoothness of arbitrary integer order. For the case p > 1 see also [33, 69, 107]. Theorem 11.4. Let f ∈ Lp (Td ), d ≥ 2, 1 ≤ p < q < ∞, θ = d (1/p − 1/q), and r ∈ N, r > θ. Then, for any δ ∈ (0, 1), we have 1 1       1 δ ωr (f, t)Lq (Td ) p dt p ωr (f, t)Lp (Td ) q dt q r−θ δ  . tr−θ t tθ t δ 0 Proof. As it was mentioned above, the case p > 1 follows from Theorem 11.3. Let us consider the case p = 1. It follows from [52, Theorem 4] that for any f ∈ W1r (Td ), r ∈ N, and 1 < q < ∞ the following embedding holds f B˙ r−θ  f W˙ r .

(11.11)

1

q,1

Note also that f q  f B˙ θ

(11.12)

1,q

for f ≡ const (see (2.1)). At the same time, we have (11.13)

r−θ (Lq , Wqr )1− θ ,1 = Bq,1 r

and

θ (L1 , W1r ) θ ,q = B1,q r

(see, e.g., [108, Sec. 2.5]). Thus, using (11.11)–(11.13) and repeating the proof of Theorem 11.2, we obtain (11.9). 

11.2. Sharp Ulyanov and Kolyada inequalities in analytic Hardy spaces Let us recall that an analytic function f on the unit disc D = {z ∈ C : |z| < 1} belongs to the space Hp = Hp (D), if  p1  2π it p |f (ρe )| dt < ∞. f Hp = sup 0 0. Both inequalities clearly hold also for 1 < p < ∞. In the case 0 < p ≤ 1, it is shown in [26] that for α ∈ N one has 1  δ p dt p (r) −r (12.2) t ωr+α (f, t)p ωα (f , δ)p  , t 0 95

96

12. (Lp , Lq ) INEQUALITIES OF ULYANOV-TYPE INVOLVING DERIVATIVES

where f (r) is the derivative in the sense of Lp , which is defined as follows. A function f ∈ Lp (T), 0 < p < ∞, has the derivative f (r) of order r ∈ N in the sense Lp if  r   Δh f  (r)    hr − f  → 0 as h → 0 . p

Below we study sharp Ulyanov (Lp , Lq ) inequalities for moduli of smoothness involving derivatives. Our goal is to improve the known estimate ([26]) (12.3)

 ωα (f

(r)

δ

, δ)q  0



q1 dt 1 1 t−r−( p − q ) ωr+α (f, t)p t

 q1

1

, 0 < p < q ≤ ∞,

where 1/p − 1/q < α, d = 1, and α, r ∈ N. Theorem 12.2. Let f ∈ Lp (T), 0 < p < q ≤ ∞, r ∈ N, α ∈ N∪((1/q−1)+ , ∞), and γ ≥ 0 be such that α + γ ∈ N ∪ ((1/p − 1)+ , ∞). Then, for any δ ∈ (0, 1), we have  q1   q δ ωr+α+γ (f, t)p 1 1 dt 1 (r) (12.4) σ , ωα (f , δ)q  tγ+r t t 0 where (1) if 0 < p ≤ 1 and p < q ≤ ∞, then ⎧ 1 ⎪ t p −1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ t p1 −1 ln q1 (t + 1), σ(t) := 1 1 ⎪ ⎪ t p − q −γ , ⎪ ⎪ ⎪ ⎪ ⎩ p1 − 1q t , (2) if 1 < p ≤ q ≤ ∞, then ⎧ 1, ⎪ ⎪ ⎪ ⎨ 1, 1 σ(t) := ln p (t + 1), ⎪ ⎪ ⎪ 1 ⎩ ( p − q1 )−γ , t



γ > 1 − 1q ; + 0 < γ = 1 − 1q ;

+ 0 < γ < 1 − 1q ; +

γ = 0,

γ ≥ p1 − 1q , q < ∞; γ > p1 , q = ∞; γ = p1 ,

q = ∞;

0≤γ
0, 0 < λ < ∞, and φ be a non-negative monotone function on R+ . Then the following inequality holds λ  ∞  ∞  t  −ξ λ dt dt ds −ξ t φ(t)  . φ(s) t s t t 0 0 0 Noting that this inequality also holds for weakly decreasing functions φ, that is, satisfying φ(x) ≤ Cφ(y) for x ≥ y ≥ 2x (see, e.g., [103]), we arrive at (12.6).  Remark 12.3. (i) Note that in Theorem 12.3, q ≥ 1 and we deal with the usual partial derivatives Dβ f . (ii) Assuming in Theorem 12.3 that p ≥ 1, inequality (12.6) holds if we replace  by q1 , which gives a stronger result. This can be shown as in the proof of Theorem 12.2. Indeed, first, we use Theorem 8.2 for the derivative Dβ f and then we apply inequality (12.5). (iii) If 1 < q < ∞, then (12.6) holds for α > 0 (see Remark 12.2). We finish this section chapter with the following result related to the discussion in Subsection 1.3. We show that while dealing with absolutely continuously

12. (Lp , Lq ) INEQUALITIES OF ULYANOV-TYPE INVOLVING DERIVATIVES

99

functions the pathological behavior of smoothness properties in Lp , 0 < p < 1, disappears. Proposition 12.1. Let 0 < p < 1, α ∈ N, f (α−1) ∈ AC(T), and ωα (f, δ)p = o(δ α ),

δ → 0,

then f ≡ const. Proof. Let 0 < p < 1, f ∈ AC(T) and ω1 (f, δ)p = o(δ),

δ → 0,

then f ≡ const (see [99]). Using this and the estimate   1/p p δ ωα (f, p)p (α−1) p−1 , δ)p  t dt = o(δ), ω1 (f tα 0

δ → 0,

(see (12.2)), we arrive at f (α−1) ≡ const and hence the desired result follows.



CHAPTER 13

Embedding theorems for function spaces As mentioned earlier, both sharp Ulyanov’s and Kolyda’s inequalities are closely related to embedding theorems for smooth function spaces (Lipschitz, Nikol’skii– Besov, etc.). In what follows, we restrict ourselves to the classical Lipschitz spaces   Lip(α, k, p) = f ∈ Lp (Td ) : ωk (f, δ)p = O(δ α ) and their generalizations given by   Lipβ (α, k, p) = f ∈ Lp (Td ) : ωk (f, δ)p = O(δ α lnβ 1/δ) . The latter spaces are the well-known logarithmic Lipschitz spaces, widely used in functional analysis (see, e.g., [11, 81]) and differential equations (see, e.g., [15, 116]). The theory of embedding theorems for function spaces has been studied for a long time, beginning with the work of Hardy and Littlewood (see [36, 37]). They proved that Lip(α, 1, p) → Lip(α − θ, 1, q), where 1 ≤ p < q < ∞,

θ = 1/p − 1/q,

θ < α ≤ 1,

d = 1.

Later, this result was extended by many authors mostly in the case 1 ≤ p < q ≤ ∞ (see [7, 34, 38, 50, 51, 69, 89, 90, 96, 102, 107, 112]). In particular, Kolyada’s inequality (see (11.2)) implies that α−θ , Lip(α, α, p) → Bq,p

1 < p < q < ∞.

Recall that for an important limit case α = k, we have for 1 < p ≤ ∞ and ⎧ α−1 f ∈ BV (T), ⎨ D f ∈ BV (T), f ∈ Lip(α, α, 1) iff ⎩ 1−α f ∈ BV (T), I

Lip(α, α, p) ≡ Wpα (T) α > 1; α = 1; α < 1,

where BV is the space of all functions which are of bounded variation on every finite interval and I 1−α is the fractional integral (see [117, XII, § 8, (8.3), p. 134]). unwald-Letnikov derivative of order α > 0 of a By Dα f we denote the Liouville-Gr¨ function f in the Lp -norm: if for f ∈ Lp (T) there exists g ∈ Lp (T) such that    lim h−α α h f (·) − g(·) p = 0, h→0+

then g = D f (see [13], [88, § 20, (20.7)]). Note that for any f ∈ Lp (T), p ≥ 1, we have f (α) (x) = D α f (x) a.e., where f (α) is the Weyl derivative of f . α

101

102

13. EMBEDDING THEOREMS FOR FUNCTION SPACES

In the case 0 < p < 1, the space Lip(α, k, p) with limiting smoothness (that is α = 1/p + k − 1) consists only of ”trivial” functions. Let us state it more precisely. We have f ∈ Lip(1/p + k − 1, k, p) if and only if after correction of f on a set of measure zero its (k − 1)-st derivative is of the form  dl , f (k−1) (x) = d0 + xl α + d(1/p − 1) consists only of a.e. constant functions. This follows from Theorem 4.3. Note also that the embedding Lip(η, α, p) → Lq holds for 0 < p < q ≤ ∞ if and only if η > θ = d(1/p − 1/q). Thus, it is natural to assume below that η > θ and η ≤ α + d(1/p − 1) if 0 < p < 1. Theorem 13.1. Let d ≥ 1, 0 < p < q ≤ ∞, α ∈ N ∪ ((1/p − 1)+ , ∞), θ = d(1/p − 1/q). (A) If 0 < p ≤ 1 and p < q ≤ ∞, then, for any θ < η ≤ α + d(1/p − 1), one has Lip(η, α, p) → ⎧ d ⎪ Lip(η − θ, α − d(1 − 1q ), q), ⎪ d−1 ≤ q ≤ ∞ and α ∈ N; ⎪ ⎪ d ⎪ Lip (η − θ, α, q) , ⎪ d−1 ≤ q ≤ 2 and α ∈ N; ⎪ ⎪ d ⎪ and d ≥ 2; Lip (η − θ, α, q) , q < d−1 ⎪ ⎪ ⎪ ⎪ ⎨ Lip (η − θ, α, q) , q ≤ 2 and d = 1;

1 1 q Lip(η − θ, α, q) ∩ Lip η − θ, α − d(1 − q ), q , 2 < q < ∞ and α ∈ N; ⎪ ⎪

⎪ 1 ⎪ 1 ⎪ q ⎪ η − θ, α − d(1 − Lip(η − θ, α, q) ∩ Lip ), q , 2 < q < ∞ and d = 1; ⎪ q ⎪ ⎪ ⎪ ⎪ Lip (η − θ, α, q) , q = ∞, d ≥ 2, and α ∈ N; ⎪ ⎪ ⎩ Lip(η − θ, α − d(1 − 1 ), q), q = ∞, d = 1, and α ∈ N, q (B) If 1 < p < q ≤ ∞, then, for θ < η = α, one has (13.1)

⎧ α−θ 1 < p ≤ min(2, q); ⎨ Bq p , , min(2, q) < p < q < ∞; Lip(η − θ, α − θ, q) ∩ Bqα−θ Lip(η, α, p) → p  ⎩ Lip(η − θ, α, q) ∩ Lip1/p (η − θ, α − θ, q), 1 < p < q = ∞, while, for any θ < η < α, one has (13.2)

Lip(η, α, p) → Lip (η − θ, α, q) .

13.1. NEW EMBEDDING THEOREMS FOR LIPSCHITZ SPACES

103

Remark 13.1. In the special case 1 = p < q < ∞, d ≥ 2, and η = α ∈ N, some embeddings from Theorem 13.1 (A) can be improved as follows (see Theorem 11.4) 1 θ = d(1 − ). q

α−θ Lip(η, α, p) → Bq,1 ,

Note that by the Marchaud inequality (4.13), we have   1 α−θ Bq,1 → Lip η − θ, α − d(1 − ), q , q see also the proof of part (B) below. Remark 13.2. In the scale of Lipschitz spaces, embeddings in Theorem 13.1 are sharp. This follows from Theorem 13.2. Let us illustrate the embeddings, which follow from Theorem 13.1, for the most important case η = α and 0 < p < q ≤ ∞. In Fig. 1 and Fig. 2, we use the following notation: X1 = Lip (α − θ, α − d (1 − 1/q) , q) , X2 = Lip(α − θ, α, q), α−θ , X3 = Bq,p

X4 = Lip(α − θ, α − θ, q) ∩ X3 , 1

X5 = Lip(α − θ, α − θ, q) ∩ Lip p (α − θ, α − θ, q). We use the solid line —— (or the dashed line - - - -) to emphasize that the corresponding boundary is included (or excluded).

1 q

X2 1 1 − 1/d 1/2 X3

X1

X4 1/2

1

1 p

X5 Fig. 1: The sets of pairs (1/p, 1/q) for which the embeddings Lip(α, α, p) → Xi , i = 1, 5, from Theorem 13.1 are fulfilled in the case d ≥ 2 and η = α ∈ N.

104

13. EMBEDDING THEOREMS FOR FUNCTION SPACES 1 q

X2 1

1/2 X3

X5

X4 1 p

1

1/2 X5

X2

Fig. 2: The sets of pairs (1/p, 1/q) for which the embeddings Lip(α, α, p) → Xi , i = 1, 5, from Theorem 13.1 are fulfilled in the case d ≥ 2 and η = α ∈ N.

Proof. (A) We need to compare the embeddings, which three main inequalities (the classical Ulyanov inequality, the sharp Ulyanov inequality, and the Ulyanov– Marchaud inequality) provide. First of all, let us compare the sharp Ulyanov inequality and the Ulyanov– Marchaud inequality. In the case 0 < p ≤ 1, we have from the sharp Ulyanov inequality given by Theorems 8.1 and 8.2 that

(13.3)



δ

ωα−γ (f, δ)q 



ωα (f, t)p

q1

dt t

 q1

1

td( p − q ) ⎧ ⎪ 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ln1/q1 1δ + 1 , ⎪ ⎨ ωα (f, δ)p + γ+d( 1 −1) 1, ⎪ p  δ ⎪ 1/q  1 ⎪ ⎪ ⎪ δ +1 , ⎪ ln ⎪ ⎪  1 d(1− q1 )−γ ⎪ ⎪ ⎪ , ⎪ δ ⎪ ⎪ ⎩  1 d(1− q1 ) , δ 1

0

1



γ > d 1 − 1q ;

+ γ = d 1 − 1q ≥ 1, d ≥ 2, and α ∈ N;

+ γ = d 1 − 1q ≥ 1, d ≥ 2, and α ∈ N; +

γ = d = 1 and q =

∞; 1 0 0, and ε > 0. Then there exists a function f1 ∈ Lipε (α, α, q) Lip(α, α + β, r). Proof. Let us define the lacunary series f1 (x) =

∞ 

λn+1 /λn ≥ λ > 1.

cn cos λn x,

n=1

Zygmund’s theorem (see [117, Ch. 8]) implies that f1 q f1 2 for 1 ≤ q < ∞ and f1 ∞ {cn }∞ n=1 1 (see [92]). Taking cn = 2nε λ−α n

n

λn = 22 ,

and choosing s for any natural N such that λs ≤ N < λs+1 , we get by the realization results (1.14) that    (α)  ωα (f1 , 1/N )q  N −α SN (f1 ) + f1 − SN (f1 )q ⎛  N −α ⎝  =N

−α

q



⎠ c2ν λ2α ν

λν ≤N s 

⎞1/2

2

+⎝ 

1/2

2νε

⎛

+

ν=0 ε



⎞1/2 c2ν ⎠

λν ≥N ∞  22νε λ2α ν=s+1 ν

1/2

 N −α ln (N + 1), i.e., f1 ∈ Lipε (α, α, q) for q < ∞. If q = ∞, similar calculation implies that    (α)  ωα (f1 , 1/N )q  N −α SN (f1 ) + f1 − SN (f1 )q q   −α α N c ν λν + cν  N −α lnε (N + 1). λν ≤N

λν ≥N

13.2. EMBEDDING PROPERTIES OF BESOV AND LIPSCHITZ-TYPE SPACES

109

On the other hand, Jackson’s inequality (4.1) for N = λs gives ∞ 1/2  22νε 2sε ≥ . ωα+β (f1 , 1/N )r  EN −1 (f1 )r  f1 − SN −1 (f1 )r  λ2α λα s ν=s ν Assuming that f1 ∈ Lip(α, α + β, r), we arrive at the contradiction 2sε 1  ω (f , 1/N )  . α+β 1 r λα λα s s / Lip(α, α + β, r) for r < ∞. If r = ∞ Thus, f1 ∈ ωα+β (f1 , 1/λs )r  Eλs −1 (f1 )r  cλs =

2sε . λα s 

Proposition 13.2. Let 1 ≤ q, r ≤ ∞, α, β > 0, and ε ∈ (0, 1/2) for r < ∞ and ε ∈ (0, 1) for r = ∞. Then there exists a function f2 ∈ Lip(α, α + β, q) Lipε (α, α, r). Proof. We consider f2 (x) =

∞ 

cn cos 2n x,

cn =

n=1

1 . 2αn

It is clear that f2 ∈ Lq (T), 1 ≤ q ≤ ∞. Moreover, since f2 q f2 2 for 1 ≤ q < ∞, we get by (1.14) that for integer N ∈ [2n , 2n+1 )    (α+β)  ωα+β (f2 , 1/N )q  N −α−β SN (f2 ) + f − SN (f2 )q  −n(α+β)

2

q

n 



1/2

c2k 22k(α+β)

+

k=1

∞ 

1/2 c2k

 2−αn  N −α ,

k=n+1

i.e., f2 ∈ Lip(α, α + β, q) if q < ∞. For q = ∞ the same holds using f2 ∞

{cn }∞ n=1 1 . Moreover, for 1 ≤ r < ∞,  N 1/2     −α  (α) −nα 1  N −α ln1/2 (N + 1) ωα (f2 , 1/N )r  N SN (f2 )  2 r

and

k=1

    (α) ωα (f2 , 1/N )∞  N −α SN (f2 )

∞

 N −α ln(N + 1).

/ Lipε (α, α, r) for any ε ∈ (0, 1/2) when r < ∞ and for any ε ∈ (0, 1) Then f2 ∈ when r = ∞.  Proposition 13.3. Let 1 < q ≤ 2, α, β > 0, and ε ∈ (0, 1/τ ). Then there exists a function f3 ∈ Lip(α, α + β, q) Lipε (α, α, q). Proof. Define f3 (x) =

∞  n=1

an cos nx,

aqn =

1 nαq+q−1

.

110

13. EMBEDDING THEOREMS FOR FUNCTION SPACES

Since

∞ 

aqn nq−2 =

n=1

∞ 

1 < ∞, αq+1 n n=1

by the Hardy–Littlewood theorem (2.3) we get that f3 ∈ Lq (T). Moreover, realization (1.14) and Theorem 6.1 from [35] gives  n 1/q  ∞ 1/q  p  q −α−β (α+β)q+q−2 q−2 ωα+β (f3 , 1/n)q n ak k + ak k  n−α , k=1

k=n+1

i.e., f3 ∈ Lip(α, α + β, q). On the other hand,     ωα (f3 , 1/n)q  n−α Sn (f3 )(α)   n

−α

q

n 

1/q

aqk kαq+q−2

 n

−α

k=1

n  1 k

1/q 

k=1

which yields that f3 ∈ / Lipε (α, α, q) for any ε ∈ (0, 1/τ ).

ln1/q (n + 1) , nα 

The next theorem shows the relationship for the spaces Lip(α − θ, α − θ, q) and α−θ Bq,p . Theorem 13.3. Let 1 < p < q < ∞, θ = d(1/p − 1/q), and α > θ. We have α−θ → Lip(α − θ, α − θ, q); 1) if p ≤ min(2, q), then Bq,p α−θ are not 2) if min(2, q) < p, then the spaces Lip(α − θ, α − θ, q) and Bq,p comparable. The first part is a simple consequence of the Marchaud inequality (see Theorem 4.4). To prove the second part, we consider the following functions: ∞  1 f1 (x) = aν cos νx, aν = 1− 1 +α−θ 1 ν q (ln(ν + 1)) q ν=1 and f2 (x) =

∞  ν=1

a2ν cos 2ν x,

a2ν =

1 1 2ν(α−θ) ν 2

,

for which we have, α−θ if p ≤ min(2, q), f1 ∈ Lip(α − θ, α − θ, q) \ Bq,p α−θ α−θ and f2 ∈ Bq,p \Lip(α−θ, α−θ, q) if min(2, q) < p f1 ∈ Lip(α−θ, α−θ, q)\Bq,p (see [90]).

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List of symbols Sets N, set of positive integers, vii Rd , d-dimensional Euclidean space, vii Zd , integer lattice in Rd , vii Td , d-dimensional torus, vii D, unit disc, 91 T , set of all trigonometric polynomials, vii Tn , set of all trigonometric polynomials of degree at most n, vii Tn = Tn \ T0 , vii Numbers, relations A  B, the estimate A ≤ C B, where C is a positive constant, vii Y → X, continuous embedding, vii [a], the largest integer not greater than a, viii a α+ = max{a, 0}, viii ν , binomial coefficients, 3 ρ = ρ(p) = max(p, 2), 26 τ = τ (p) = min(p, 2) if p < ∞ and 1 if p = ∞, 26   ν = ν =0 , 31 q1 = q if q < ∞ and 1 if q = ∞, viii Spaces Lp (X), Lebesgue space on X, vii S (Rd ), Schwartz space, 11 of all continuous functionals on S (Rd ), 11  S  (Rd ), space  d ˙ S (R ) = ϕ ∈ S (Rd ) : (Dν ϕ)(0)

= 0 for all ν ∈ Nd ∪ {0} , 11 S˙  (Rd ), space of all continuous functionals on S˙ (Rd ), 11 s Bp,q (Rd ), Besov space, 11 s ˙ Bp,q (Rd ), homogeneous Besov space, 11 s (Rd ), Triebel-Lizorkin space, 11 Fp,q s F˙ p,q (Rd ), homogeneous Triebel-Lizorkin space, 12 Hp (D), Hardy space on the unit disk D, 91 Hp (Rd ), real Hardy space on Rd , 87 Hp (Td ), real Hardy space on Td , 90 ˙ pα (Td ), homogeneous Sobolev space, 8 W Wp (ψ), space of ψ-smooth functions in Lp , 51 AC(T), space of absolutely continuously functions on T, 7 117

118

LIST OF SYMBOLS

Hα , class of homogeneous functions of degree α, 31 Lip(α, k, p), Lipschitz space, 101 Lipβ (α, k, p), logarithmic Lipschitz space, 101 Functionals and functions En (f )p , error of the best trigonometric approximation, 25 ωk (f, δ)p , modulus of smoothness of integer order k, 2 ωα (f, δ)p , modulus of smoothness of fractional order α, 2 ωα (f, δ)Hp , modulus of smoothness in the Hardy spaces, 87 Kα (f, δ)Hp , K-functional in the Hardy spaces, 87 Kψ (f, δ)p , K-functional generated by a homogeneous function ψ, 51 ∂ α Rα (f, δ)p , realization generated by the directional derivative ( ∂ξ ) , 27 α/2 Rα (f, δ)p , realization generated by the Laplacian (−Δ) , 75 Rψ (f, δ)p , realization generated by a homogeneous function ψ, 51 f k , Fourier coefficients, 13 Vn (x), de la Vall´ee Poussin type kernel, 32 Operators Ff = f , Fourier transform of f , vii Dν , partial derivative, 8 α/2 (−Δ) , Laplace operator of order α, 8 ∂ ∂ξ (α)

α

, directional derivative, 19

f , fractional derivative in the sense of Weyl, 6 D(ψ), Weyl-type differentiation operator, 31 Δkh , finite difference of order k ∈ N, 2 Δα h , fractional difference of order α > 0, 3

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Selected Published Titles in This Series 1316 Ulrich Bunke and David Gepner, Differential Function Spectra, the Differential Becker-Gottlieb Transfer, and Applications to Differential Algebraic K-Theory, 2021 ´ Matheron, and Q. Menet, Linear Dynamical Systems on Hilbert 1315 S. Grivaux, E. Spaces: Typical Properties and Explicit Examples, 2021 1314 Pierre Albin, Fr´ ed´ eric Rochon, and David Sher, Resolvent, Heat Kernel, and Torsion under Degeneration to Fibered Cusps, 2021 1313 Paul Godin, The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners, 2021 1312 Patrick Delorme, Pascale Harinck, and Yiannis Sakellaridis, Paley-Wiener Theorems for a p-Adic Spherical Variety, 2021 1311 Lyudmila Korobenko, Cristian Rios, Eric Sawyer, and Ruipeng Shen, Local Boundedness, Maximum Principles, and Continuity of Solutions to Infinitely Degenerate Elliptic Equations with Rough Coefficients, 2021 1310 Hiroshi Iritani, Todor Milanov, Yongbin Ruan, and Yefeng Shen, Gromov-Witten Theory of Quotients of Fermat Calabi-Yau Varieties, 2021 1309 J´ er´ emie Chalopin, Victor Chepoi, Hiroshi Hirai, and Damian Osajda, Weakly Modular Graphs and Nonpositive Curvature, 2020 1308 Christopher L. Douglas, Christopher Schommer-Pries, and Noah Snyder, Dualizable Tensor Categories, 2020 1307 Adam R. Thomas, The Irreducible Subgroups of Exceptional Algebraic Groups, 2020 1306 Kazuyuki Hatada, Hecke Operators and Systems of Eigenvalues on Siegel Cusp Forms, 2020 1305 Bogdan Ion and Siddhartha Sahi, Double Affine Hecke Algebras and Congruence Groups, 2020 1304 Matthias Fischmann, Andreas Juhl, and Petr Somberg, Conformal Symmetry Breaking Differential Operators on Differential Forms, 2020 1303 Zhi Qi, Theory of Fundamental Bessel Functions of High Rank, 2020 1302 Paul M. N. Feehan and Manousos Maridakis, L  ojasiewicz-Simon Gradient Inequalities for Coupled Yang-Mills Energy Functionals, 2020 1301 Joachim Krieger, On Stability of Type II Blow Up for the Critical Nonlinear Wave Equation in R3+1 , 2020 1300 Camille Male, Traffic Distributions and Independence: Permutation Invariant Random Matrices and the Three Notions of Independence, 2020 1299 David M. J. Calderbank, Michael G. Eastwood, Vladimir S. Matveev, and Katharina Neusser, C-Projective Geometry, 2020 1298 Sy David Friedman and David Schrittesser, Projective Measure Without Projective Baire, 2020 1297 Jonathan Gantner, Operator Theory on One-Sided Quaternion Linear Spaces: Intrinsic S-Functional Calculus and Spectral Operators, 2020 1296 Christophe Cornut, Filtrations and Buildings, 2020 1295 Lisa Berger, Chris Hall, Ren´ e Pannekoek, Jennifer Park, Rachel Pries, Shahed Sharif, Alice Silverberg, and Douglas Ulmer, Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields, 2020 1294 Jacob Bedrossian, Pierre Germain, and Nader Masmoudi, Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case, 2020 1293 Benjamin Jaye, Fedor Nazarov, Maria Carmen Reguera, and Xavier Tolsa, The Riesz Transform of Codimension Smaller Than One and the Wolff Energy, 2020

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/memoseries/.

Memoirs of the American Mathematical Society

9 781470 447588

MEMO/271/1325

Number 1325 • May 2021

ISBN 978-1-4704-4758-8