JFSAJournal of Function Spaces and Applications0972-68022090-8997Hindawi Publishing Corporation64313510.1155/2012/643135643135Research ArticleSmoothness and Function Spaces Generated by Homogeneous MultipliersRunovskiKonstantin1SchmeisserHans-Jürgen2StepanovV.1Faculty of Mathematics and InformaticsTaurida National V. I. Vernadsky University95007 SimferopolUkraine2Mathematical InstituteFriedrich-Schiller University Jena07737 JenaGermanyuni-jena.de20122912012201223032010300420102012Copyright © 2012 Konstantin Runovski and Hans-Jürgen Schmeisser.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Differential operators generated by homogeneous functions ψ of an arbitrary real order s>0 (ψ-derivatives) and related spaces of s-smooth periodic functions of d variables are introduced and systematically studied. The obtained scale is compared with the scales of Besov and Triebel-Lizorkin spaces. Explicit representation formulas for ψ-derivatives are obtained in terms of the Fourier transform of their generators. Some applications to approximation theory are discussed.

1. Introduction

Smoothness is one of the basic concepts of analysis, having a long history. A fundamental observation is its strong connection with the decay of the Fourier coefficients of a given function. This paved the way to apply methods of Fourier analysis to the further development of smoothness concepts. Let us consider two directions of the Fourier analytic approach to describe the differentiability properties of functions. The first one is related to the scales of Besov spaces   Bp,qs   and Triebel-Lizorkin spaces Fp,qs which are constructed by means of decomposition of the Fourier series into dyadic blocks with the help of an appropriate resolution of unity (see, e.g.,  for periodic and nonperiodic setting). The second direction is based on the interpretation of a derivative as an operator of multiplier type. In this case one also deals with the Fourier coefficients, but not with decomposition into blocks. Following this way the concept of classical derivative was essentially extended to fractional derivatives such as Riesz and Weyl derivatives (see, e.g., ) and later on to the concept of generalized derivatives and the corresponding scale of the Stepanets classes in the one-dimensional case (see [7, 8]).

Both the theory of function spaces as it has been developed by the Russian (S. M. Nikol'skij) or German (H. Triebel) school and the theory of generalized smoothness elaborated by Stepanets and his coworkers have found many applications in various fields of modern mathematics. More precisely, the first direction is mainly applied to the theory of (partial) differential equations, computational mathematics, stochastic processes, and fractal and nonlinear analysis. The second one turned out to be important for many problems of approximation theory, in particular, constructing optimal linear approximation methods on various classes of smooth functions and obtaining approximation relations with (asymptotically) sharp constants.

It is well known (see , Ch. 3, or , Ch. 2, in the nonperiodic case), for 1<p<+ and s>0, that the space Fp,2s coincides with the (fractional) Sobolev Hps which is related to the operator (I-Δ)s/2, where Δ is the Laplace operator and I is the identity operator. Here “relation to operator” means that Hps consists of periodic functions f in Lp such that (I-Δ)s/2f belongs to Lp as well. In other words, we have Hps=(I-Δ)-s/2(Lp) which means that Hps is characterized as the image of Lp by the operator (I-Δ)-s/2. With this exception both Besov spaces Bp,qs and Triebel-Lizorkin spaces Fp,qs are not related to any operator acting on Lp in general. For this reason the scales of Besov and Triebel-Lizorkin are not well adapted to the study of problems which are directly connected with concrete operators. In contrast to these function spaces, the classes of Stepanets are related to certain operators of multiplier type which are connected to the so-called (ψ,β)-derivatives; see, for example . In Stepanets' theory the one-dimensional case is considered only. The smoothness generator ψ is an arbitrary function satisfying some natural conditions. On the one hand, it gives quite a lot of freedom and allows the description of many interesting properties of functions which are smooth in this sense. On the other hand, such a “poor” information on ψ does not enable us to obtain explicit representation formulas for related derivatives in terms of the functions under consideration itself in place of their Fourier coefficients. This fact prevents the application of this approach to many important problems of numerical approximation.

In the present paper, we will introduce and study operators 𝒟(ψ) of multiplier type generated by homogeneous functions and related spaces Xp(ψ) of periodic functions of d variables. On the one hand, homogeneity seems to be a rather general assumption. Practically all known differential operators as, for instance, the classical derivatives, Weyl and Riesz derivatives, mixed derivatives, and the Laplace operator and its (fractional) powers are generated by homogeneous multipliers. On the other hand, taking into account that the Fourier transform (in the sense of distributions) of a homogeneous function of order s is also a homogeneous function of order -(d+s) (see, e.g., [9, Theorem  7.1.6]), one can derive quite substantial statements concerning the corresponding operators and related function spaces. In particular, we will prove that there is unique space Xp(ψ) which coincides with the fractional Sobolev space Hps if ψ is homogeneous of degree s>0 and 1<p<+ (Theorems 3.1 and 4.1). However, we get infinitely many new spaces in the cases p=1 and p=+ (Theorem 3.4). Moreover, we find an explicit representation formula for 𝒟(ψ)f for functions f belonging to the periodic Besov space Bp,1s if ψ is homogeneous of degree s,  0<s<1 (Theorem 5.1).

Let us also mention that it is aimed to extend the approach to generalized smoothness based on the introduction of operators 𝒟(ψ) and spaces Xp(ψ) in further works by

finding representation formulae for operators generated by homogeneous functions of degree s1 using higher-order differences,

studying generalized K-functionals and their realizations,

constructing new moduli of smoothness related to 𝒟(ψ),

studying the same concept in nonperiodic case,

investigating generalized differential equations.

The paper is organized as follows. Section 2 deals with notations and preliminaries. The basic properties of operators 𝒟(ψ) and spaces Xp(ψ) are described in Section 3. Some relations between spaces Xp(ψ) and Besov and Triebel-Lizorkin spaces are discussed in Section 4. Finally, Section 5 is devoted to the derivation of an explicit formula for 𝒟(ψ) in terms of the Fourier transform of the generator ψ.

2. Notations and Preliminaries2.1. Numbers and Vectors

By the symbols , 0, , , , 0d, d, and d we denote the sets of natural, nonnegative integer, integer, real and complex numbers, d-dimensional vectors with non-negative integer, integer and real components, respectively. The symbol 𝕋d is reserved for the d-dimensional torus [0,2π)d. We will also use the notations xy=x1y1++xdyd,|x||x|2=(x12++xd2)1/2, for the scalar product and the l2-norm of vectors and Br={xRd:|x|<r},B¯r={xRd:|x|r} for the open and closed balls, respectively.

2.2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M73"><mml:mrow><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>-Spaces

As usual, LpLp(𝕋d), where   1p<+,   𝕋d=[0,2π)d, is the space of measurable real-valued functions   f=f(x), x=(x1,,xd) which are 2π-periodic with respect to each variable satisfying fp=(Td|f(x)|pdx)1/p<+. In the case p=+, we consider the space   CC(𝕋d)  (p=+) of real-valued   2π-periodic continuous functions equipped with the Chebyshev norm f=maxxTd|f(x)|. For   Lp-spaces of non-periodic functions defined on a measurable set   Ωd   we will use the notation Lp(Ω).

2.3. Fourier Coefficients and Fourier Transform

The Fourier coefficients of fL1 are defined by f(k)=(2π)-dTdf(x)e-ikxdx,kZd. Let per and per be the space of infinitely differentiable periodic functions and its dual the space of periodic distributions, respectively. The Fourier coefficients of fper are given by f(k)=(2π)-df,e-ik,kZd, where, as usual, f,g means the value of the functional f at gper.

The Fourier transform and its inverse are given by f̂(ξ)=Rdf(x)e-ixξdx,f(x)=(2π)-dRdf(ξ)eixξdξ,fL1(Rd). For an element f belonging to the space of tempered distributions 𝒮=𝒮(d), which is the dual of the Schwartz space 𝒮=𝒮(d) of rapidly decreasing infinitely differentiable functions, the Fourier transform is defined by setting f̂,g=f,ĝ,gS.

2.4. Trigonometric Polynomials

Let   σ   be a real nonnegative real number. By 𝒯σ we denote the space of all real-valued trigonometric polynomials of (spherical) order   σ. It means Tσ={t(x)=|k|σckeikx:c-k=ck¯,  |k|σ}, where c¯ is the complex conjugate to c. Further, 𝒯 stands for the space of all real-valued trigonometric polynomials of arbitrary order. Let 1p+. As usual, we put Eσ(f)p=inftTσf-tp,σ>0, for the best approximation of f in Lp (fC if p=+) by trigonometric polynomials of order at most σ in the metric of Lp.

2.5. Homogeneous Functions

A complex-valued function ψ defined on d{0} is called homogeneous of order s if ψ(tξ)=tsψ(ξ) for t>0 and ξd{0}. An element ψ of the space 𝒮 is called homogeneous distribution of order s (see, e.g., [9, Def. 3.2.2, page 74]) if for any   t>0ψ,g(t)=t-(s+d)ψ,g,gS. It is well known (see, e.g., [9, Theorem  7.1.16, page 167]) that the Fourier transform of a homogeneous distribution ψ of order s is also a homogeneous distribution of order -(s+d).

Let now s>0. By the symbol s we denote the class of functions ψ satisfying the following conditions:

ψ is continuous on d and complex valued;

ψ is infinitely differentiable on d{0};

ψ is homogeneous of order s;

ψ(-ξ)=ψ(ξ)¯ for each ξd;

ψ(ξ)0 for ξd{0}.

It follows that ψ𝒮 and that the restriction of ψ̂ to d{0} belongs to C(d{0}) (see [9, Theorem  7.1.18, page 168]).

2.6. Periodic Besov and Triebel-Lizorkin Spaces

The Fourier analytical definition is based on dyadic resolutions of unity (see, e.g., [5, Chapter 3], or  for the nonperiodic case). Let   φ0   be a real-valued centrally symmetric (φ0(-ξ)=φ(ξ) for all ξd) infinitely differentiable function satisfying φ0(ξ)={1,|ξ|12,0,|ξ|1. We put θ(ξ)=φ0(ξ)-φ0(2ξ),φj(ξ)=θ(2-jξ),jN. Clearly, these functions are also centrally symmetric and infinitely differentiable with compact support. We have suppθB¯1B1/4;suppφjB¯2jB2j-2,jN. By (2.14) we obtain φj(ξ)=φ0(2-jξ)-φ0(2-j+1ξ),jN. Combining (2.13) and (2.16), one has j=0+φj(ξ)=1,ξRd. In view of (2.15) and (2.17), the system   Φ={φj}j=0+   is called a smooth dyadic resolution of unity.

Let s>0, 1p+, and 1q+. The periodic Besov space Bp,qs   and the periodic Triebel-Lizorkin space Fp,qs are given by (cf., [5, Chapter 3]) Bp,qs={fLp:fBp,qs(j=0+2jsqfjpq)1/q<+} if q<, Bp,s={fLp:fBp,ssupjN02jsfjp<+}, and by Fp,qs={fLp:gFp,qs(j=0+2jsq|fj(x)|q)1/qp<+} if p<+,q<+, Fp,s={fLp:fFp,ssupjN02js|fj(x)|p<+}, if p<+, where the function  fj, j0, are defined by fj(x)=kZdφj(k)f(k)eikx,xRd,  jN0, and φj, j0, are given by (2.13) and (2.14). It is well known that Definitions (2.18)–(2.21) are independent of the choice of the resolution of unity Φ. The associated norms are mutually equivalent. Therefore, we do not indicate Φ in the notation of norms and spaces. For the details, further properties, and natural extensions to parameters s,  0<p,  and  q+, we refer to [5, Chapter 3].

2.7. Fourier Means

Let φ be a real-valued centrally symmetric continuous function with a compact support. It generates the operators σ(φ) which are given by Fσ(φ)(f;x)=(2π)-dTdf(h)Wσ(φ)(x-h)dh,σ0, for fL1. The function σ(φ)(f) is called Fourier mean of f generated by φ. The functions Wσ(φ) in (2.23) are defined as W0(φ)(h)=1;Wσ(φ)(h)=kZdφ(kσ)eikx,σ>0. The Fourier means describe classical methods of trigonometric approximation which are well defined for functions in Lp, where 1p+. They are well studied and investigated in detail in many books and papers on approximation theory; see, for example, [6, 10]. Let us also mention [5, Chapter 3], for a treatment within the framework of periodic Besov and Triebel-Lizorkin spaces. Following , we recall and state the following properties.

The set of the norms Fσ(φ)(f)(p)=supfp1Fσ(φ)p of operators defined on Lp by (2.23) is uniformly bounded, and we have supσ0Fσ(φ)(p)<+, for p=1, p=+ or for all 1p+ if and only if φ̂ belongs to L1(d);

for fL1 it holds that Fσ(φ)(f;x)=kZdφ(kσ)f(k)eikx,σ>0;

if φ̂L1(d) and φ(0)=1, then the Fourier means σ(φ) converge in Lp for all 1p+, and we have limσ+f-Fσ(φ)(f)p=0,fLp;

if φ̂L1(d) and φ(ξ)={1,|ξ|ρ,0,|ξ|>1, for some ρ>0, then for 1p+f-Fσ(φ)(f)pcEρσ(f)p,fLp,  σ0, where the positive constant c does not depend on f and σ.

If θ is given by (2.14), then in view of (2.13) and (2.27) the functions fj which are well defined for fL1 by (2.22) can be represented as f0(x)=f(0),fj(x)=F2j(θ)(f;x),jN. Moreover, using (2.16) and (2.27) we get j=0Jfj(x)=kZd(j=0Jφj(k))f(k)eikx=kZdφ0(2-Jk)f(k)eikx=F2J(φ0)(f;x) for J. Taking into account that φ̂0L1(d), we obtain therefrom by the convergence property (2.28) of the Fourier means the decomposition f=j=0+fj,fLp, into a series of trigonometric polynomials being convergent in the space Lp,  1p+.

2.8. Inequalities of Multiplier Type for Trigonometric Polynomials

Let ψs for some s>0. It generates the family of operators (Aσ(ψ))σ defined by Aσ(ψ)t(x)=kZdψ(kσ)t(k)eikx,σ>0,  tT, on the space 𝒯 of trigonometric polynomials. We have proved in [12, 13] that the inequality Aσ(ψ1)tpcAσ(ψ2)tp, where   Aσ(ψ1)   is generated by   ψ1s1   and   Aσ(ψ2)   is generated by   ψ2s2, is valid for all t𝒯σ and σ1 with a certain constant c independent of t and σ for all 1p+ if s1>s2. If s1=s2, then (2.35) is valid for 1<p<+ and for arbitrary generators ψ1,  ψ2. If s1=s2 and p=1 or p=+, then the validity of (2.35) implies that the functions ψ1 and ψ2 are proportional.

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M247"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow></mml:math></inline-formula>-Smoothness and Basic Properties

Let ψs for some s>0. It generates an operator 𝒟(ψ) by setting D(ψ)t(x)=kZdψ(k)t(k)eikx,tT, which is initially defined on the space of trigonometric polynomials. The domain of definition can be extended within the spaces Lp, 1p+. To this end we introduce the space Xp(ψ) which consists of functions f in Lp having the property that the set {ψ(k)f(k),kZd} is the system of the Fourier coefficients of a certain function in  Lp. This function will be called ψ-derivative of f in the following. By this definition we have (D(ψ)f)(k)=ψ(k)f(k),kZd, for the Fourier coefficients of the  ψ-derivative of f. Its uniqueness follows from the well-known fact that each L1-function is uniquely determined by the set of its Fourier coefficients. If fXp(ψ), then the series in (3.1) with f in place of t converges in Lp (see the proof of Theorem 3.1). In this sense we can reformulate Xp(ψ)={fLp:D(ψ)f=kZdψ(k)f(k)eikxLp}.

We give some examples. Let   d=1, s, 1<p<+, and ψ(ξ)=(iξ)s. Then, 𝒟(ψ) is the operator of the usual derivative of order  s. In this case Xp(ψ) is the Sobolev space Wps of (s-1)-times differentiable functions f such that f(s-1) is absolutely continuous, f(s) exists almost everywhere and belongs to Lp. If   d=1, s, and ψ(ξ)=(iξ)s=|ξ|sesgnξ·(sπi)/2, then 𝒟(ξ)   is the Weyl derivative (·)(s) of fractional order s and Xp(ψ) is the corresponding Weyl class. For   d=1   and ψ(ξ)=|ξ|, the operator 𝒟(ψ) is the Riesz derivative (·). Taking into account that it is the composition of the usual derivative of the first order and the operator of conjugation, we see that in this case Xp(ψ) coincides with the space W̃p1 of those functions where both the function itself and its conjugate belong to Wp1. It is well known that this space coincides with Wp1 if and only if 1<p<+. For more details concerning the derivatives and spaces mentioned above, we refer to .

In the multivariate case (d>1) the classical Laplace operator Δ and its (fractional) power   (-Δ)s/2, s>0, are operators of type 𝒟(ψ) associated with ψ(ξ)=-|ξ|2 and ψ(ξ)=|ξ|s, respectively. In this case Xp(ψ)Xp((-Δ)s/2) coincides with the periodic version of the Bessel-potential space (the (fractional) Sobolev space if 1<p<+) considered in [14, Chapter 5] (see also the next subsection).

Now we study the properties of the spaces Xp(ψ). Recall that we have to replace Lp(𝕋d) by C(𝕋d) if p=+.

Theorem 3.1.

Let ψs, s>0, and 1p+. Then,

Xp(ψ) is a Banach space with respect to the norm fXp(ψ)=fp+D(ψ)fp;

𝒯 is dense in Xp(ψ);

all spaces Xp(ψ) with ψs  coincide and their norms are equivalent if 1<p<+.

Proof.

In order to prove (i) we only have to check completeness because all other properties of Banach space are obviously fulfilled. Let {fn} be a Cauchy sequence in Xp(ψ). In view of (3.5) fn-fmp,D(ψ)fn-D(ψ)fmp0(n,m+). By completeness of Lp there exist functions  f,  FLp, such that fnLpf,D(ψ)fnLpF(n+). Using (3.3) and Hölder's inequality, we get |F(k)-ψ(k)f(k)||(F-D(ψ)fn)(k)|+|ψ(k)||(f-fn)(k)|(2π)-d/p(F-D(ψ)fnp+|ψ(k)|f-fnp) for any   kd   and n. In view of (3.7) the right-hand side of this inequality tends to   0   if   n+  . This implies F(k)=ψ(k)f(k),kZd. Hence, f is ψ-differentiable, 𝒟(ψ)f=F, and completeness is proved.

In order to show part (ii) we consider the Fourier means σ(φ0) of type (2.23)-(2.24), where φ0 satisfies the conditions described in Section 2 (see, in particular, (2.13)). Let fXp(ψ). Taking into account that D(ψ)(Fσ(φ0)(f))=Fσ(φ0)(D(ψ)f),σ0, by (2.27) and (3.3) we get f-Fσ(φ0)(f)Xp(ψ)=f-Fσ(φ0)(f)p+D(ψ)f-D(ψ)(Fσ(φ0)(f))p=f-Fσ(φ0)(f)p+D(ψ)f-Fσ(φ0)(D(ψ)f)p for σ0. The terms on the right-hand side tend to 0 if σ+ by (2.28). This yields the desired density of 𝒯 in Xp(ψ).

Now we prove part (iii). Let 1<p<+, ψ1,  ψ2s, and let f be an arbitrary function in Xp(ψ2). First we observe that in view of (2.34) and (3.1), inequality (2.35) can be rewritten as D(ψ1)tpcD(ψ2)tp,tT. It is valid for 1<p<+ (see the comment at the end of Section 2). Therefore, we obtain D(ψ1)(Fn(φ0)(f)-Fm(φ0)(f))pcD(ψ2)(Fn(φ0)(f)-Fm(φ0)(f))p=cFn(φ0)(D(ψ2)f)-Fm(φ0)(D(ψ2)f)p for   n,  m. Hence, {𝒟(ψ1)(n(φ0)(f))} is a Cauchy sequence in Lp by (2.28), and there exists   FLp such that D(ψ1)(Fn(φ0)(f))LpF(n+). By the help of (3.3), (2.27), and (3.14) we get F(k)=limn+[D(ψ1)(Fn(φ0)(f))](k)=ψ1(k)f(k). For each kd. Hence, f belongs to Xp(ψ1) and 𝒟(ψ1)f=F. In order to prove that the embedding Xp(ψ2)Xp(ψ1) is continuous, it is enough to notice that D(ψ1)fp=limn+D(ψ1)(Fn(φ0)(f))pclimn+D(ψ2)(Fn(φ0)(f))p=D(ψ2)fp as a consequence of (2.35). This completes the proof.

In other words part (iii) of the theorem means that in the case   1<p<+   there is only one space Xp(s)  of s-smooth functions. It can be characterized as Xp(s)={fLp:D(||s)f=kZd|k|sf(k)eikxLp} and may be equipped with the norm fXp(s)=fp+D(||s)fp.

Theorem 3.2.

Suppose 1p+, ψ1s1, and ψ2s2. If s1>s2>0, then Xp(ψ1)Xp(ψ2) and this embedding is continuous.

Proof.

Let fXp(ψ1). In view of (2.15) the function φ=((θψ2)/ψ1), where θ is given by (2.14), is infinitely differentiable on d and has compact support. Hence, its Fourier transform belongs to L1(d) and the operators σ(φ) are uniformly bounded in Lp as stated in Section 2. Using this fact as well as (3.3), (2.31), and the homogeneity property of ψ1 and ψ2, we get j=mnD(ψ2)fjpj=mnD(ψ2)fjp=j=mnkZdθ(2-jk)ψ2(k)f(k)eikxp=j=mn2-(s1-s2)jk0φ(2-jk)ψ1(k)f(k)eikxp=j=mn2-(s1-s2)jF2j(φ)(D(ψ1)f)pcD(ψ1)fpj=mn2-(s1-s2)j for   n>m1. Here, fj,  j, has the meaning of (2.22). Because of s1>s2 the sequence of the partial sums of the series j=1+𝒟(ψ2)fj is fundamental in Lp and there exists FLp such that F=Lpj=1+D(ψ2)fj. By (2.13) we have φ0(k)0 only for   k=0. Thus, we obtain j=1JD(ψ2)fj(x)=kZdφ0(k2J)ψ2(k)f(k)eikx,xRd for any J in analogy to (2.32). Considering the limit process J+ we find F(ν)=limJ+(j=1JD(ψ2)fj)(ν)=ψ2(ν)f(ν) for any   νd with the help of (2.13) and (3.22). Hence, f belongs to Xp(ψ2) and 𝒟(ψ2)f=F. In order to prove that embedding (3.20) is continuous, it is enough to put m=1  in estimate (3.21) and to consider n to +. This completes the proof.

Theorem 3.3.

Let   1p+   and   ψs, s>0. Then, D(ψ):Xp(ψ)Lp0{fLp:f(0)=0} is a surjective mapping.

Proof.

Let fLp0. We introduce functions gj, j, by setting gj(x)=k0θ(2-jk)(ψ(k))-1f(k)eikx,jN, where θ has the meaning of (2.14), and we put φ=θ/ψ. By the same arguments as in the proof of Theorem 3.2 (see, in particular, (3.21)) we get j=mngjpj=mngjp=j=mn2-jsF2j(φ)(f)pcfpj=mn2-js for n>m1. Hence, there exists gLp such that g=Lpj=1+gj.

Taking into account that φ0(k)0 only for k=0 in view of (2.13), we obtain analogously to (2.32) j=1Jgj=k0φ0(2-Jk)(ψ(k))-1f(k)eikx by (3.26). As a consequence of (3.28) and (3.29) we get g(ν)=limJ+(j=1Jgj)(ν)=(ψ(ν))-1f(ν) for ν0. Finally, because of f(0)=0 we obtain f(ν)=ψ(ν)g(ν),νZd. From (3.30). Hence, gXp(ψ), 𝒟(ψ)g=f, and the proof is complete.

Theorem 3.3 shows that for any ψs an operation which is inverse to 𝒟(ψ) is well defined on Lp0. The operator I(ψ)=(D(ψ))-1:Lp0(Xp(ψ))Lp0 is called operator of ψ-integration. The Fourier series of the function I(ψ)f, fLp0, is given by k0(ψ(k))-1f(k)eikx. As it follows from the proof of Theorem 3.3 the series (3.33) converges in the sense of (3.26) and (3.28). Taking in (3.27) m=1 and considering n to +, we obtain the boundedness of the operator I(ψ) in Lp0.

With the help of Theorem 3.3 we will see that in contrast to the case 1<p<+, where in view of part (iii) of Theorem 3.1 only one space of s-smooth functions exists, in the cases   p=1,+   there are infinitely many ways to define smoothness spaces of order s associated with homogeneous generators by means of operators of multiplier type.

Theorem 3.4.

Suppose that ψ1 and ψ2 are linear independent homogeneous functions belonging to s, s>0. Then, X1(ψ1)X1(ψ2),X(ψ1)X(ψ2).

Proof.

We consider the case p=1. For p=+ the proof is similar. Let us assume (to the contrary) that X1(ψ1)X1(ψ2). Then, by Theorem 3.3 the operators Ln=Fn(φ0)D(ψ2)I(ψ1),nN, where φ0 has the meaning of Section 2 (see also (2.13)), are well defined on the space L10. The operator n acts in accordance with the following chain of mappings and inclusions: L10I(ψ1)X1(ψ1)X1(ψ2)D(ψ2)L10Fn(φ0)Tn. By (2.23), (2.27), (3.3), and (3.33) we get Fn(φ0)D(ψ2)I(ψ1)(f;x)=k0φ0(kn)ψ2(k)(ψ1(k))-1f(k)eikx=(2π)-dTdf(x-h)Φn(h)dh for   n. Here, Φn(h)=k0φ0(kn)ψ2(k)(ψ1(k))-1eikx,nN. In view of (3.38) each operator n is bounded in L10. Moreover, the boundedness of n(φ0) (see Section 2) yields Ln(f)1cD(ψ2)I(ψ1)f1 for any fL10. Applying now the Banach-Steinhaus principle we conclude that the operators n are uniformly bounded in L10. This leads to the estimate k0φ0(kn)ψ2(k)(ψ1(k))-1f(k)eikx1cf1, where the constant c does not depend on fL10 and n. Let now t(x)=|k|mckeikx be an arbitrary trigonometric polynomial. We choose n>2m. Then, it holds that φ0(k/n)ψ2(k)=ψ2(k) for |k|m in view of (2.13). Applying (3.41) to f=𝒟(ψ1)t, we obtain D(ψ2)t1cD(ψ1)t1. This contradicts the statement on the nonvalidity of inequality (2.35) for p=1 pointed out at the end of Section 2. Changing the roles of ψ1 and ψ2 completes the proof.

4. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M481"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow></mml:math></inline-formula>-Smoothness and Besov and Triebel-Lizorkin Spaces

The aim of this section is to compare the spaces Xp(ψ),ψs, with periodic Besov and Triebel-Lizorkin spaces. As we have seen already in part (iii) of Theorem 3.1 there is a unique space Xp(s) if 1<p<+ which has been characterized in (3.18) and (3.19). The following theorem shows that it coincides with the classical fractional Sobolev space defined by (s>0,1<p<+) Hps={fLp:kZd(1+|k|2)s/2f(k)eikxLp} and equipped with the norm fHps=kZd(1+|k|2)s/2f(k)eikxp. Note that Hps=Wps={fLp:DαfLp  for  |α|s} if s (all derivatives in the sense of periodic distributions, see [5, Subsection 3.5.4]).

Theorem 4.1.

Let 1<p<+, and let s>0. Then, one has Xp(s)=Hps=Fp,2s with equivalence of norms.

Proof.

Both statements are known. The second identity can be found in [5, Theorem  3.5.4, page 169]. To prove the first identity and to show the equivalence of norms one can use the Fourier multipliers and the theorem of Mikhlin-Hörmander (see, e.g., [15, Theorem  5.2.7, page 367], for the non-periodic version) which can be transferred to the periodic case using [16, Chapter 7, Theorem  3.1]. The arguments are similar to [17, Subsections  6.2.2, 6.2.3], or [18, Theorem  6.3.2], where equivalent characterizations and connections between nonperiodic homogeneous and inhomogeneous Sobolev spaces are treated. We omit the details.

Next we consider the cases p=1 and p=+.

Theorem 4.2.

Let 1p+, and let ψs for some s>0. Then, Bp,1sXp(ψ)Bp,s and these embeddings are continuous.

Proof.

Let η be an infinitely differentiable positive function defined on d such that η(ξ)=1if|ξ|18,η(ξ)=0if|ξ|14. We put ψ*=ψ(1-η). By (2.15) we have ψ(2-jk)θ(2-jk)=ψ*(2-jk)θ(2-jk),kZd,  jN, where θ is given by (2.14).

Let f be an arbitrary function in Bp,1s, and let fj, j0, be given by (2.22). Using (4.7), the homogeneity property of ψ, and (2.27) and applying a Fourier multiplier theorem which can be found in [5, Theorem  3.3.4, page 150], we obtain j=mnD(ψ)fjpj=mnD(ψ)fjp=j=mnkZdψ(k)θ(2-jk)f(k)eikxp=j=mn2jskZdψ*(2-jk)θ(2-jk)f(k)eikxpcj=mn2jsF2j(θ)(f)p=j=mn2jsfjp for n>m1. In view of (2.18) and (4.8) we can conclude that the series j=0+𝒟(ψ)fj converges in Lp. Now, by standard arguments we see as in the proof of Theorem 3.1 that f belongs to Xp(ψ) and that the first embedding in (4.5) is continuous.

In order to prove the second embedding we introduce an infinitely differentiable function ψ*(ξ) which coincides with ψ(ξ) for |ξ|1/4 and which is not equal to 0 on d. By (2.15) we have ψ(2-jk)θ(2-jk)=ψ*(2-jk)θ(2-jk),kZd,  jN, where θ has the meaning of (2.14).

Let f be an arbitrary function in Xp(ψ), and let fj, j0, be given by (2.22). Applying (4.9), the homogeneity property of ψ, and (2.27), we obtain fj(x)=kZdθ(2-jk)f(k)eikx=kZd(ψ*(2-jk))-1ψ(2-jk)θ(2-jk)f(k)eikx=2-jskZd(ψ*(2-jk))-1ψ(k)θ(2-jk)f(k)eikx=2-jskZd(ψ*(2-jk))-1(F2j(θ)(D(ψ)f))(k)eikx for any j. The right-hand side can be estimated again by means of the Fourier multiplier theorem which can be found in [5, Theorem  3.3.4, page 150]. Using in addition the uniform boundedness of the Fourier means 2j(θ) in Lp, we obtain the inequalities fjpc2-jsF2j(θ)(D(ψ)f)pc2-jsD(ψ)fp for j from (4.10). Obviously, f0pcfp. Hence, by (4.11) and (3.5) fBp,scfXp(ψ) for some constant c>0 and all fXp(ψ). This completes the proof.

Having in mind Theorem 4.1 the embeddings (4.5) are well known in the case 1<p<+. Even a better result holds. This follows also from Theorem 4.1 and the elementary embeddings Bp,1sBp,min(p,2)sFp,2sBp,max(p,2)sBp,s (see [5, Remark  3.5.1/4, page 164]).

5. A Representation Formula for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M539"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow></mml:math></inline-formula>-Derivatives

The following observations give some motivation and pave the way to find explicit representations of the operator 𝒟(ψ) in terms of the Fourier transform of its generator ψs. For the sake of simplicity we restrict ourselves to the case 0<s<1.

Let ψs. Recall that gper if and only if |g(k)|C(m)|k|-m,k0, for each m and note that |ψ(ξ)|c|ξ|s,ξRd, by homogeneity. Hence, it makes sense to consider the periodic distribution Ψ(x)=kZdψ(k)e-ikx (convergence in per). Let gper. Expansion into the Fourier series leads to the representation Ψ(x)=νZdψ̂(x+2πν) (convergence in per). By definition of the Fourier coefficients, the definition of the ψ-derivative and, (5.3) we get D(ψ)(g)(x)=(2π)-dΨ(),g(x+)-g(x),gEper. Using (5.4) and (5.5), we see, at least formally, that D(ψ)(g)(x)=(2π)-dνZdψ̂(+2πν),g(x+)-g(x)=(2π)-dνZdTdψ̂(y+2πν)(g(x+y)-g(x))dy=(2π)-dRd(g(x+h)-g(x))ψ̂(h)dh,gEper. We claim that the integral on the right-hand side of (5.6) exists for all x𝕋d. To this end we first recall that (see Section 2) ψ̂𝒮 is a homogeneous distribution of order -(d+s). Its restriction to d{0} can be identified with a function ψ̂C(d{0}) by [9, Theorem  7.1.18]. Hence, we have the estimate |ψ̂(h)|c|h|-(d+s),hRd{0}. For brevity we use the standard notation Δhg(x)=g(x+h)-g(x). Now, we split RdΔhg(x)ψ̂(h)dh=B1Δhg(x)ψ̂(h)dh+RdB1Δhg(x)ψ̂(h)dh. The second summand is finite because of (5.7) and the boundedness of |Δhg(x)| on d. As for the first one we use the estimate |Δhg(x)|c|h|hB1,xTd, and (5.7) to see that |B1Δhg(x)ψ̂(h)dh|cB1|h||ψ̂(h)|dhc′′B1|h|-d+(1-s)dh<+. The above arguments suggest that formula (5.6) might be true in a stronger sense. Indeed, the following theorem shows that the representation for the derivative 𝒟(ψ)(g) holds pointwise almost everyone under much weaker assumptions with respect to g.

Theorem 5.1.

Suppose 1p+ and ψs, where 0<s<1. Then, D(ψ)(f)(x)=(2π)-dRd(f(x+h)-f(x))ψ̂(h)dh,fBp,1s, for almost all x𝕋d (for all, if p=+).

Proof.

First we recall that Bp,1sXp(ψ) by Theorem 4.2. Hence, the left-hand side of (5.11) makes sense. Let fBp,1s. As is well known (see, e.g., [5, Theorem  3.5.4]), fBp,1s*=fp+Rd|h|-sΔhfpdh|h|d is an equivalent norm in the Besov space Bp,  1s. Using (5.7), (5.12), and the generalized Minkowski inequality we obtain for the integral I(x) at the right-hand side of (5.11) the estimates IpRdΔhf(x)p|ψ̂(h)|dhcRdΔhfp|h|-(d+s)dhcfBp,1α*. Hence, the function I(x) belongs to Lp. To prove (5.11) it is sufficient to show that the Fourier coefficients of the functions on both sides coincide. We have (I())(k)=(2π)-dTdRd(f(x+h)-f(x))ψ̂(h)dhe-ikxdx=f(k)Rd(eikh-1)ψ̂(h)dh by Fubini's theorem. It remains to show that Rd(eikh-1)ψ̂(h)dh=(2π)-dψ(k),kZd. This is obvious if k=0 because of ψ(0)=0. If k0, we have to use appropriate limiting arguments to circumvent the difficulty caused by the fact that ψ̂ is not integrable in a neighbourhood of 0. We do not go into details.

We give some remarks. It is known (see [15, Theorem  2.4.6, page 128]) that for any s,  0<s<2, the restriction of the Fourier transform of ψ(ξ)=|ξ|s to d{0} can be identified with (||s)(x)=c(d,s)|x|-d-s,(c(d,s)=2d+sπd/2Γ(s/2+d/2)Γ(-s/2)), combining (5.7) with (5.16) in the one-dimensional case, we obtain the well-known formula for the Riesz derivative (see, e.g., ) fs(x)=(2π)-1c(1,s)-+f(x+h)-f(x)|h|s+1dh,fBp,1s, which is valid for 0<s<1 and 1p+. In the multivariate case we get under the same conditions with respect to s and p the representation formula (-Δ)s/2f(x)=(2π)-dc(d,s)Rdf(x+h)-f(x)|h|d+sdh,fBp,1s for the fractional power of the Laplace operator.

Let us mention that formulas for ψ-derivatives with s1 can be achieved using differences of higher order. Suppose, for example, that 1s<2 and, in addition to the previous conditions, that ψ is real valued. Analogously to (5.6) we find D(ψ)(g)(x)=(2π)-dRdg(x+h)-2g(x)+g(x-h)2ψ̂(h)dh, and, in particular, (-Δ)s/2g(x)=(2π)-dc(d,s)Rdg(x+h)-2g(x)+g(x-h)2|h|d+sdh for gper. Similarly to the proof of Theorem 5.1 one can show that formulas (5.19) and (5.20) are valid at least for functions belonging to the Besov spaces Bp,1s.

Acknowledgment

This paper was partially supported by the DFG-project SCHM 969/10-1.

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