Differential operators generated by homogeneous functions ๐œ“ of an arbitrary real order ๐‘ >0 (๐œ“-derivatives) and related spaces of ๐‘ -smooth periodic functions of ๐‘‘ variables are introduced and systematically studied. The obtained scale is compared with the scales of Besov and Triebel-Lizorkin spaces. Explicit representation formulas for ๐œ“-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 ๐ต๐‘ ๐‘,๐‘ž and Triebel-Lizorkin spaces ๐น๐‘ ๐‘,๐‘ž 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., [1โ€“5] 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., [6]) 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 [5], Ch. 3, or [4], Ch. 2, in the nonperiodic case), for 1<๐‘<+โˆž and ๐‘ >0, that the space ๐น๐‘ ๐‘,2 coincides with the (fractional) Sobolev ๐ป๐‘ ๐‘ which is related to the operator (๐ผโˆ’ฮ”)๐‘ /2, where ฮ” is the Laplace operator and ๐ผ is the identity operator. Here โ€œrelation to operatorโ€ means that ๐ป๐‘ ๐‘ consists of periodic functions ๐‘“ in ๐ฟ๐‘ such that (๐ผโˆ’ฮ”)๐‘ /2๐‘“ belongs to ๐ฟ๐‘ as well. In other words, we have ๐ป๐‘ ๐‘=(๐ผโˆ’ฮ”)โˆ’๐‘ /2(๐ฟ๐‘) which means that ๐ป๐‘ ๐‘ is characterized as the image of ๐ฟ๐‘ by the operator (๐ผโˆ’ฮ”)โˆ’๐‘ /2. With this exception both Besov spaces ๐ต๐‘ ๐‘,๐‘ž and Triebel-Lizorkin spaces ๐น๐‘ ๐‘,๐‘ž are not related to any operator acting on ๐ฟ๐‘ 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 (๐œ“,๐›ฝ)-derivatives; see, for example [7]. In Stepanets' theory the one-dimensional case is considered only. The smoothness generator ๐œ“ 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 โ€œpoorโ€ information on ๐œ“ 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 ๐’Ÿ(๐œ“) of multiplier type generated by homogeneous functions and related spaces ๐‘‹๐‘(๐œ“) of periodic functions of ๐‘‘ 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 ๐‘  is also a homogeneous function of order โˆ’(๐‘‘+๐‘ ) (see, e.g., [9, Theoremโ€‰โ€‰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 ๐‘‹๐‘(๐œ“) which coincides with the fractional Sobolev space ๐ป๐‘ ๐‘ if ๐œ“ is homogeneous of degree ๐‘ >0 and 1<๐‘<+โˆž (Theorems 3.1 and 4.1). However, we get infinitely many new spaces in the cases ๐‘=1 and ๐‘=+โˆž (Theorem 3.4). Moreover, we find an explicit representation formula for ๐’Ÿ(๐œ“)๐‘“ for functions ๐‘“ belonging to the periodic Besov space ๐ต๐‘ ๐‘,1 if ๐œ“ is homogeneous of degree ๐‘ ,0<๐‘ <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 ๐’Ÿ(๐œ“) and spaces ๐‘‹๐‘(๐œ“) in further works by(i)finding representation formulae for operators generated by homogeneous functions of degree ๐‘ โ‰ฅ1 using higher-order differences, (ii)studying generalized ๐พ-functionals and their realizations, (iii) constructing new moduli of smoothness related to ๐’Ÿ(๐œ“), (iv)studying the same concept in nonperiodic case, (v)investigating generalized differential equations.

The paper is organized as follows. Section 2 deals with notations and preliminaries. The basic properties of operators ๐’Ÿ(๐œ“) and spaces ๐‘‹๐‘(๐œ“) are described in Section 3. Some relations between spaces ๐‘‹๐‘(๐œ“) and Besov and Triebel-Lizorkin spaces are discussed in Section 4. Finally, Section 5 is devoted to the derivation of an explicit formula for ๐’Ÿ(๐œ“) in terms of the Fourier transform of the generator ๐œ“.

2. Notations and Preliminaries

2.1. Numbers and Vectors

By the symbols โ„•, โ„•0, โ„ค, โ„, โ„‚, โ„•๐‘‘0, โ„ค๐‘‘, and โ„๐‘‘ we denote the sets of natural, nonnegative integer, integer, real and complex numbers, ๐‘‘-dimensional vectors with non-negative integer, integer and real components, respectively. The symbol ๐•‹๐‘‘ is reserved for the ๐‘‘-dimensional torus [0,2๐œ‹)๐‘‘. We will also use the notations ๐‘ฅ๐‘ฆ=๐‘ฅ1๐‘ฆ1+โ‹ฏ+๐‘ฅ๐‘‘๐‘ฆd,|๐‘ฅ|โ‰ก|๐‘ฅ|2=๎€ท๐‘ฅ21+โ‹ฏ+๐‘ฅ2๐‘‘๎€ธ1/2,(2.1) for the scalar product and the ๐‘™2-norm of vectors and ๐ต๐‘Ÿ=๎€ฝ๐‘ฅโˆˆโ„๐‘‘๎€พ,โˆถ|๐‘ฅ|<๐‘Ÿ๐ต๐‘Ÿ=๎€ฝ๐‘ฅโˆˆโ„๐‘‘๎€พโˆถ|๐‘ฅ|โ‰ค๐‘Ÿ(2.2) for the open and closed balls, respectively.

2.2. ๐ฟ๐‘-Spaces

As usual, ๐ฟ๐‘โ‰ก๐ฟ๐‘(๐•‹๐‘‘), where 1โ‰ค๐‘<+โˆž, ๐•‹๐‘‘=[0,2๐œ‹)๐‘‘, is the space of measurable real-valued functions ๐‘“=๐‘“(๐‘ฅ), ๐‘ฅ=(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘‘) which are 2๐œ‹-periodic with respect to each variable satisfying โ€–๐‘“โ€–๐‘=๎‚ต๎€œ๐•‹๐‘‘||||๐‘“(๐‘ฅ)๐‘๎‚ถ๐‘‘๐‘ฅ1/๐‘<+โˆž.(2.3) In the case ๐‘=+โˆž, we consider the space ๐ถโ‰ก๐ถ(๐•‹๐‘‘)(๐‘=+โˆž) of real-valued 2๐œ‹-periodic continuous functions equipped with the Chebyshev norm โ€–๐‘“โ€–โˆž=max๐‘ฅโˆˆ๐•‹๐‘‘||||๐‘“(๐‘ฅ).(2.4) For ๐ฟ๐‘-spaces of non-periodic functions defined on a measurable set ฮฉโŠ†โ„๐‘‘ we will use the notation ๐ฟ๐‘(ฮฉ).

2.3. Fourier Coefficients and Fourier Transform

The Fourier coefficients of ๐‘“โˆˆ๐ฟ1 are defined by ๐‘“โˆง(๐‘˜)=(2๐œ‹)โˆ’๐‘‘๎€œ๐•‹๐‘‘๐‘“(๐‘ฅ)๐‘’โˆ’๐‘–๐‘˜๐‘ฅ๐‘‘๐‘ฅ,๐‘˜โˆˆโ„ค๐‘‘.(2.5) Let โ„ฐper and โ„ฐโ€ฒper be the space of infinitely differentiable periodic functions and its dual the space of periodic distributions, respectively. The Fourier coefficients of ๐‘“โˆˆโ„ฐโ€ฒper are given by ๐‘“โˆง(๐‘˜)=(2๐œ‹)โˆ’๐‘‘๎ซ๐‘“,๐‘’โˆ’๐‘–๐‘˜โ‹…๎ฌ,๐‘˜โˆˆโ„ค๐‘‘,(2.6) where, as usual, โŸจ๐‘“,๐‘”โŸฉ means the value of the functional ๐‘“ at ๐‘”โˆˆโ„ฐper.

The Fourier transform and its inverse are given by ๎๎€œ๐‘“(๐œ‰)=โ„๐‘‘๐‘“(๐‘ฅ)๐‘’โˆ’๐‘–๐‘ฅ๐œ‰๐‘‘๐‘ฅ,๐‘“โˆจ(๐‘ฅ)=(2๐œ‹)โˆ’๐‘‘๎€œโ„๐‘‘๐‘“(๐œ‰)๐‘’๐‘–๐‘ฅ๐œ‰๐‘‘๐œ‰,๐‘“โˆˆ๐ฟ1๎€ทโ„๐‘‘๎€ธ.(2.7) For an element ๐‘“ belonging to the space of tempered distributions ๐’ฎ๎…ž=๐’ฎโ€ฒ(โ„๐‘‘), which is the dual of the Schwartz space ๐’ฎ=๐’ฎ(โ„๐‘‘) of rapidly decreasing infinitely differentiable functions, the Fourier transform is defined by setting ๎‚ฌ๎๎‚ญ๐‘“,๐‘”=โŸจ๐‘“,ฬ‚๐‘”โŸฉ,๐‘”โˆˆ๐’ฎ.(2.8)

2.4. Trigonometric Polynomials

Let ๐œŽ be a real nonnegative real number. By ๐’ฏ๐œŽ we denote the space of all real-valued trigonometric polynomials of (spherical) order ๐œŽ. It means ๐’ฏ๐œŽ=โŽงโŽชโŽจโŽชโŽฉ๎“๐‘ก(๐‘ฅ)=||๐‘˜||โ‰ค๐œŽ๐‘๐‘˜๐‘’๐‘–๐‘˜๐‘ฅโˆถ๐‘โˆ’๐‘˜=๐‘๐‘˜,||๐‘˜||โŽซโŽชโŽฌโŽชโŽญโ‰ค๐œŽ,(2.9) where ๐‘ is the complex conjugate to ๐‘โˆˆโ„‚. Further, ๐’ฏ stands for the space of all real-valued trigonometric polynomials of arbitrary order. Let 1โ‰ค๐‘โ‰ค+โˆž. As usual, we put ๐ธ๐œŽ(๐‘“)๐‘=inf๐‘กโˆˆ๐’ฏ๐œŽโ€–๐‘“โˆ’๐‘กโ€–๐‘,๐œŽ>0,(2.10) for the best approximation of ๐‘“ in ๐ฟ๐‘ (๐‘“โˆˆ๐ถ if ๐‘=+โˆž) by trigonometric polynomials of order at most ๐œŽ in the metric of ๐ฟ๐‘.

2.5. Homogeneous Functions

A complex-valued function ๐œ“ defined on โ„๐‘‘โงต{0} is called homogeneous of order ๐‘ โˆˆโ„ if ๐œ“(๐‘ก๐œ‰)=๐‘ก๐‘ ๐œ“(๐œ‰)(2.11) for ๐‘ก>0 and ๐œ‰โˆˆโ„๐‘‘โงต{0}. An element ๐œ“ of the space ๐’ฎโ€ฒ is called homogeneous distribution of order ๐‘  (see, e.g., [9, Def. 3.2.2, page 74]) if for any ๐‘ก>0โŸจ๐œ“,๐‘”(๐‘กโ‹…)โŸฉ=๐‘กโˆ’(๐‘ +๐‘‘)โŸจ๐œ“,๐‘”โŸฉ,๐‘”โˆˆ๐’ฎ.(2.12) It is well known (see, e.g., [9, Theoremโ€‰โ€‰7.1.16, page 167]) that the Fourier transform of a homogeneous distribution ๐œ“ of order ๐‘  is also a homogeneous distribution of order โˆ’(๐‘ +๐‘‘).

Let now ๐‘ >0. By the symbol โ„‹๐‘  we denote the class of functions ๐œ“ satisfying the following conditions: (i)๐œ“ is continuous on โ„๐‘‘ and complex valued; (ii)๐œ“ is infinitely differentiable on โ„๐‘‘โงต{0}; (iii)๐œ“ is homogeneous of order ๐‘ ; (iv)๐œ“(โˆ’๐œ‰)=๐œ“(๐œ‰) for each ๐œ‰โˆˆโ„๐‘‘; (v)๐œ“(๐œ‰)โ‰ 0 for ๐œ‰โˆˆโ„๐‘‘โงต{0}.

It follows that ๐œ“โˆˆ๐’ฎโ€ฒ and that the restriction of ๎๐œ“ to โ„๐‘‘โงต{0} belongs to ๐ถโˆž(โ„๐‘‘โงต{0}) (see [9, Theoremโ€‰โ€‰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 [4] for the nonperiodic case). Let ๐œ‘0 be a real-valued centrally symmetric (๐œ‘0(โˆ’๐œ‰)=๐œ‘(๐œ‰) for all ๐œ‰โˆˆโ„๐‘‘) infinitely differentiable function satisfying ๐œ‘0๎ƒฏ||๐œ‰||โ‰ค1(๐œ‰)=1,2,||๐œ‰||0,โ‰ฅ1.(2.13) We put ๐œƒ(๐œ‰)=๐œ‘0(๐œ‰)โˆ’๐œ‘0(2๐œ‰),๐œ‘๐‘—๎€ท2(๐œ‰)=๐œƒโˆ’๐‘—๐œ‰๎€ธ,๐‘—โˆˆโ„•.(2.14) Clearly, these functions are also centrally symmetric and infinitely differentiable with compact support. We have supp๐œƒโŠ‚๐ต1๐ต1/4;supp๐œ‘๐‘—โŠ‚๐ต2๐‘—๐ต2๐‘—โˆ’2,๐‘—โˆˆโ„•.(2.15) By (2.14) we obtain ๐œ‘๐‘—(๐œ‰)=๐œ‘0๎€ท2โˆ’๐‘—๐œ‰๎€ธโˆ’๐œ‘0๎€ท2โˆ’๐‘—+1๐œ‰๎€ธ,๐‘—โˆˆโ„•.(2.16) Combining (2.13) and (2.16), one has +โˆž๎“๐‘—=0๐œ‘๐‘—(๐œ‰)=1,๐œ‰โˆˆโ„๐‘‘.(2.17) In view of (2.15) and (2.17), the system ฮฆ={๐œ‘๐‘—}+โˆž๐‘—=0 is called a smooth dyadic resolution of unity.

Let ๐‘ >0, 1โ‰ค๐‘โ‰ค+โˆž, and 1โ‰ค๐‘žโ‰ค+โˆž. The periodic Besov space ๐ต๐‘ ๐‘,๐‘ž and the periodic Triebel-Lizorkin space ๐น๐‘ ๐‘,๐‘ž are given by (cf., [5, Chapter 3]) ๐ต๐‘ ๐‘,๐‘ž=โŽงโŽชโŽจโŽชโŽฉ๐‘“โˆˆ๐ฟ๐‘โˆถโ€–๐‘“โ€–๐ต๐‘ ๐‘,๐‘žโ‰ก๎ƒฉ+โˆž๎“๐‘—=02๐‘—๐‘ ๐‘žโ€–โ€–๐‘“๐‘—โ€–โ€–๐‘ž๐‘๎ƒช1/๐‘žโŽซโŽชโŽฌโŽชโŽญ<+โˆž(2.18) if ๐‘ž<โˆž, ๐ต๐‘ ๐‘,โˆž=๎ƒฏ๐‘“โˆˆ๐ฟ๐‘โˆถโ€–๐‘“โ€–๐ต๐‘ ๐‘,โˆžโ‰กsup๐‘—โˆˆโ„•02๐‘—๐‘ โ€–โ€–๐‘“๐‘—โ€–โ€–๐‘๎ƒฐ<+โˆž,(2.19) and by ๐น๐‘ ๐‘,๐‘ž=โŽงโŽชโŽจโŽชโŽฉ๐‘“โˆˆ๐ฟ๐‘โˆถโ€–๐‘”โ€–๐น๐‘ ๐‘,๐‘žโ‰กโ€–โ€–โ€–โ€–๎ƒฉ+โˆž๎“๐‘—=02๐‘—๐‘ ๐‘ž||๐‘“๐‘—||(๐‘ฅ)๐‘ž๎ƒช1/๐‘žโ€–โ€–โ€–โ€–๐‘โŽซโŽชโŽฌโŽชโŽญ<+โˆž(2.20) if ๐‘<+โˆž,๐‘ž<+โˆž, ๐น๐‘ ๐‘,โˆž=๎ƒฏ๐‘“โˆˆ๐ฟ๐‘โˆถโ€–๐‘“โ€–๐น๐‘ ๐‘,โˆžโ‰กโ€–โ€–โ€–โ€–sup๐‘—โˆˆโ„•02๐‘—๐‘ ||๐‘“๐‘—||โ€–โ€–โ€–โ€–(๐‘ฅ)๐‘๎ƒฐ<+โˆž,(2.21) if ๐‘<+โˆž, where the function๐‘“๐‘—, ๐‘—โˆˆโ„•0, are defined by ๐‘“๐‘—(๎“๐‘ฅ)=๐‘˜โˆˆโ„ค๐‘‘๐œ‘๐‘—(๐‘˜)๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ,๐‘ฅโˆˆโ„๐‘‘,๐‘—โˆˆโ„•0,(2.22) and ๐œ‘๐‘—, ๐‘—โˆˆโ„•0, are given by (2.13) and (2.14). It is well known that Definitions (2.18)โ€“(2.21) are independent of the choice of the resolution of unity ฮฆ. The associated norms are mutually equivalent. Therefore, we do not indicate ฮฆ in the notation of norms and spaces. For the details, further properties, and natural extensions to parameters ๐‘ โˆˆโ„,0<๐‘,and๐‘žโ‰ค+โˆž, we refer to [5, Chapter 3].

2.7. Fourier Means

Let ๐œ‘ be a real-valued centrally symmetric continuous function with a compact support. It generates the operators โ„ฑ๐œŽ(๐œ‘) which are given by โ„ฑ๐œŽ(๐œ‘)(๐‘“;๐‘ฅ)=(2๐œ‹)โˆ’๐‘‘๎€œ๐•‹๐‘‘๐‘“(โ„Ž)๐‘Š๐œŽ(๐œ‘)(๐‘ฅโˆ’โ„Ž)๐‘‘โ„Ž,๐œŽโ‰ฅ0,(2.23) for ๐‘“โˆˆ๐ฟ1. The function โ„ฑ๐œŽ(๐œ‘)(๐‘“) is called Fourier mean of ๐‘“ generated by ๐œ‘. The functions ๐‘Š๐œŽ(๐œ‘) in (2.23) are defined as ๐‘Š0(๐œ‘)(โ„Ž)=1;๐‘Š๐œŽ๎“(๐œ‘)(โ„Ž)=๐‘˜โˆˆโ„ค๐‘‘๐œ‘๎‚€๐‘˜๐œŽ๎‚๐‘’๐‘–๐‘˜๐‘ฅ,๐œŽ>0.(2.24) The Fourier means describe classical methods of trigonometric approximation which are well defined for functions in ๐ฟ๐‘, where 1โ‰ค๐‘โ‰ค+โˆž. 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 [11], we recall and state the following properties. (i)The set of the norms โ€–โ€–โ„ฑ๐œŽ(๐œ‘)โ€–โ€–(๐‘“)(๐‘)=supโ€–๐‘“โ€–๐‘โ‰ค1โ€–โ€–โ„ฑ๐œŽ(๐œ‘)โ€–โ€–๐‘(2.25) of operators defined on ๐ฟ๐‘ by (2.23) is uniformly bounded, and we have sup๐œŽโ‰ฅ0โ€–โ€–โ„ฑ๐œŽ(๐œ‘)โ€–โ€–(๐‘)<+โˆž,(2.26) for ๐‘=1, ๐‘=+โˆž or for all 1โ‰ค๐‘โ‰ค+โˆž if and only if ๎๐œ‘ belongs to ๐ฟ1(โ„๐‘‘); (ii)for ๐‘“โˆˆ๐ฟ1 it holds that โ„ฑ๐œŽ(๐œ‘)๎“(๐‘“;๐‘ฅ)=๐‘˜โˆˆโ„ค๐‘‘๐œ‘๎‚€๐‘˜๐œŽ๎‚๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ,๐œŽ>0;(2.27)(iii)if ๎๐œ‘โˆˆ๐ฟ1(โ„๐‘‘) and ๐œ‘(0)=1, then the Fourier means โ„ฑ๐œŽ(๐œ‘) converge in ๐ฟ๐‘ for all 1โ‰ค๐‘โ‰ค+โˆž, and we have lim๐œŽโ†’+โˆžโ€–โ€–๐‘“โˆ’โ„ฑ๐œŽ(๐œ‘)โ€–โ€–(๐‘“)๐‘=0,๐‘“โˆˆ๐ฟ๐‘;(2.28)(iv)if ๎๐œ‘โˆˆ๐ฟ1(โ„๐‘‘) and ๎‚ป||๐œ‰||||๐œ‰||๐œ‘(๐œ‰)=1,โ‰ค๐œŒ,0,>1,(2.29) for some ๐œŒ>0, then for 1โ‰ค๐‘โ‰ค+โˆžโ€–โ€–๐‘“โˆ’โ„ฑ๐œŽ(๐œ‘)โ€–โ€–(๐‘“)๐‘โ‰ค๐‘๐ธ๐œŒ๐œŽ(๐‘“)๐‘,๐‘“โˆˆ๐ฟ๐‘,๐œŽโ‰ฅ0,(2.30) where the positive constant ๐‘ does not depend on ๐‘“ and ๐œŽ.

If ๐œƒ is given by (2.14), then in view of (2.13) and (2.27) the functions ๐‘“๐‘— which are well defined for ๐‘“โˆˆ๐ฟ1 by (2.22) can be represented as ๐‘“0(๐‘ฅ)=๐‘“โˆง(0),๐‘“๐‘—(๐‘ฅ)=โ„ฑ2(๐œƒ)๐‘—(๐‘“;๐‘ฅ),๐‘—โˆˆโ„•.(2.31) Moreover, using (2.16) and (2.27) we get ๐ฝ๎“๐‘—=0๐‘“๐‘—๎“(๐‘ฅ)=๐‘˜โˆˆโ„ค๐‘‘๎ƒฉ๐ฝ๎“๐‘—=0๐œ‘๐‘—๎ƒช๐‘“(๐‘˜)โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ=๎“๐‘˜โˆˆโ„ค๐‘‘๐œ‘0๎€ท2โˆ’๐ฝ๐‘˜๎€ธ๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ=โ„ฑ(๐œ‘0)2๐ฝ(๐‘“;๐‘ฅ)(2.32) for ๐ฝโˆˆโ„•. Taking into account that ๎๐œ‘0โˆˆ๐ฟ1(โ„๐‘‘), we obtain therefrom by the convergence property (2.28) of the Fourier means the decomposition ๐‘“=+โˆž๎“๐‘—=0๐‘“๐‘—,๐‘“โˆˆ๐ฟ๐‘,(2.33) into a series of trigonometric polynomials being convergent in the space ๐ฟ๐‘,1โ‰ค๐‘โ‰ค+โˆž.

2.8. Inequalities of Multiplier Type for Trigonometric Polynomials

Let ๐œ“โˆˆโ„‹๐‘  for some ๐‘ >0. It generates the family of operators (๐ด๐œŽ(๐œ“))๐œŽ defined by ๐ด๐œŽ๎“(๐œ“)๐‘ก(๐‘ฅ)=๐‘˜โˆˆโ„ค๐‘‘๐œ“๎‚€๐‘˜๐œŽ๎‚๐‘กโˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ,๐œŽ>0,๐‘กโˆˆ๐’ฏ,(2.34) on the space ๐’ฏ of trigonometric polynomials. We have proved in [12, 13] that the inequality โ€–โ€–๐ด๐œŽ๎€ท๐œ“1๎€ธ๐‘กโ€–โ€–๐‘โ€–โ€–๐ดโ‰ค๐‘๐œŽ๎€ท๐œ“2๎€ธ๐‘กโ€–โ€–๐‘,(2.35) where ๐ด๐œŽ(๐œ“1) is generated by ๐œ“1โˆˆโ„‹๐‘ 1 and ๐ด๐œŽ(๐œ“2) is generated by ๐œ“2โˆˆโ„‹๐‘ 2, is valid for all ๐‘กโˆˆ๐’ฏ๐œŽ and ๐œŽโ‰ฅ1 with a certain constant ๐‘ independent of ๐‘ก and ๐œŽ for all 1โ‰ค๐‘โ‰ค+โˆž if ๐‘ 1>๐‘ 2. If ๐‘ 1=๐‘ 2, then (2.35) is valid for 1<๐‘<+โˆž and for arbitrary generators ๐œ“1,๐œ“2. If ๐‘ 1=๐‘ 2 and ๐‘=1 or ๐‘=+โˆž, then the validity of (2.35) implies that the functions ๐œ“1 and ๐œ“2 are proportional.

3. ๐œ“-Smoothness and Basic Properties

Let ๐œ“โˆˆโ„‹๐‘  for some ๐‘ >0. It generates an operator ๐’Ÿ(๐œ“) by setting ๎“๐’Ÿ(๐œ“)๐‘ก(๐‘ฅ)=๐‘˜โˆˆโ„ค๐‘‘๐œ“(๐‘˜)๐‘กโˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ,๐‘กโˆˆ๐’ฏ,(3.1) which is initially defined on the space of trigonometric polynomials. The domain of definition can be extended within the spaces ๐ฟ๐‘, 1โ‰ค๐‘โ‰ค+โˆž. To this end we introduce the space ๐‘‹๐‘(๐œ“) which consists of functions ๐‘“ in ๐ฟ๐‘ having the property that the set ๎€ฝ๐œ“(๐‘˜)๐‘“โˆง(๐‘˜),๐‘˜โˆˆโ„ค๐‘‘๎€พ(3.2) is the system of the Fourier coefficients of a certain function in๐ฟ๐‘. This function will be called ๐œ“-derivative of ๐‘“ in the following. By this definition we have (๐’Ÿ(๐œ“)๐‘“)โˆง(๐‘˜)=๐œ“(๐‘˜)๐‘“โˆง(๐‘˜),๐‘˜โˆˆโ„ค๐‘‘,(3.3) for the Fourier coefficients of the๐œ“-derivative of ๐‘“. Its uniqueness follows from the well-known fact that each ๐ฟ1-function is uniquely determined by the set of its Fourier coefficients. If ๐‘“โˆˆ๐‘‹๐‘(๐œ“), then the series in (3.1) with ๐‘“ in place of ๐‘ก converges in ๐ฟ๐‘ (see the proof of Theorem 3.1). In this sense we can reformulate ๐‘‹๐‘๎ƒฏ(๐œ“)=๐‘“โˆˆ๐ฟ๐‘๎“โˆถ๐’Ÿ(๐œ“)๐‘“=๐‘˜โˆˆโ„ค๐‘‘๐œ“(๐‘˜)๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅโˆˆ๐ฟ๐‘๎ƒฐ.(3.4)

We give some examples. Let ๐‘‘=1, ๐‘ โˆˆโ„•, 1<๐‘<+โˆž, and ๐œ“(๐œ‰)=(๐‘–๐œ‰)๐‘ . Then, ๐’Ÿ(๐œ“) is the operator of the usual derivative of order๐‘ . In this case ๐‘‹๐‘(๐œ“) is the Sobolev space ๐‘Š๐‘ ๐‘ of (๐‘ โˆ’1)-times differentiable functions ๐‘“ such that ๐‘“(๐‘ โˆ’1) is absolutely continuous, ๐‘“(๐‘ ) exists almost everywhere and belongs to ๐ฟ๐‘. If ๐‘‘=1, ๐‘ โˆ‰โ„•, and ๐œ“(๐œ‰)=(๐‘–๐œ‰)๐‘ =|๐œ‰|๐‘ ๐‘’sgn๐œ‰โ‹…(๐‘ ๐œ‹๐‘–)/2, then ๐’Ÿ(๐œ‰) is the Weyl derivative (โ‹…)(๐‘ ) of fractional order ๐‘  and ๐‘‹๐‘(๐œ“) is the corresponding Weyl class. For ๐‘‘=1 and ๐œ“(๐œ‰)=|๐œ‰|, the operator ๐’Ÿ(๐œ“) is the Riesz derivative (โ‹…)โŸจโ€ฒโŸฉ. 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 ๐‘‹๐‘(๐œ“) coincides with the space ๎‚‹๐‘Š1๐‘ of those functions where both the function itself and its conjugate belong to ๐‘Š1๐‘. It is well known that this space coincides with ๐‘Š1๐‘ if and only if 1<๐‘<+โˆž. For more details concerning the derivatives and spaces mentioned above, we refer to [5โ€“8].

In the multivariate case (๐‘‘>1) the classical Laplace operator ฮ” and its (fractional) power (โˆ’ฮ”)๐‘ /2, ๐‘ >0, are operators of type ๐’Ÿ(๐œ“) associated with ๐œ“(๐œ‰)=โˆ’|๐œ‰|2 and ๐œ“(๐œ‰)=|๐œ‰|๐‘ , respectively. In this case ๐‘‹๐‘(๐œ“)โ‰ก๐‘‹๐‘((โˆ’ฮ”)๐‘ /2) coincides with the periodic version of the Bessel-potential space (the (fractional) Sobolev space if 1<๐‘<+โˆž) considered in [14, Chapter 5] (see also the next subsection).

Now we study the properties of the spaces ๐‘‹๐‘(๐œ“). Recall that we have to replace ๐ฟ๐‘(๐•‹๐‘‘) by ๐ถ(๐•‹๐‘‘) if ๐‘=+โˆž.

Theorem 3.1. Let ๐œ“โˆˆโ„‹๐‘ , ๐‘ >0, and 1โ‰ค๐‘โ‰ค+โˆž. Then, (i)๐‘‹๐‘(๐œ“) is a Banach space with respect to the norm โ€–๐‘“โ€–๐‘‹๐‘(๐œ“)=โ€–๐‘“โ€–๐‘+โ€–๐’Ÿ(๐œ“)๐‘“โ€–๐‘;(3.5)(ii)๐’ฏ is dense in ๐‘‹๐‘(๐œ“); (iii)all spaces ๐‘‹๐‘(๐œ“) with ๐œ“โˆˆโ„‹๐‘ coincide and their norms are equivalent if 1<๐‘<+โˆž.

Proof. In order to prove (i) we only have to check completeness because all other properties of Banach space are obviously fulfilled. Let {๐‘“๐‘›} be a Cauchy sequence in ๐‘‹๐‘(๐œ“). In view of (3.5) โ€–โ€–๐‘“๐‘›โˆ’๐‘“๐‘šโ€–โ€–๐‘,โ€–โ€–๐’Ÿ(๐œ“)๐‘“๐‘›โˆ’๐’Ÿ(๐œ“)๐‘“๐‘šโ€–โ€–๐‘โŸถ0(๐‘›,๐‘šโŸถ+โˆž).(3.6) By completeness of ๐ฟ๐‘ there exist functions๐‘“,๐นโˆˆ๐ฟ๐‘, such that ๐‘“๐‘›๐ฟ๐‘โŸถ๐‘“,๐’Ÿ(๐œ“)๐‘“๐‘›๐ฟ๐‘โŸถ๐น(๐‘›โŸถ+โˆž).(3.7) Using (3.3) and Hรถlder's inequality, we get ||๐นโˆง(๐‘˜)โˆ’๐œ“(๐‘˜)๐‘“โˆง||โ‰ค||๎€ท(๐‘˜)๐นโˆ’๐’Ÿ(๐œ“)๐‘“๐‘›๎€ธโˆง||+||||โ‹…||๎€ท(๐‘˜)๐œ“(๐‘˜)๐‘“โˆ’๐‘“๐‘›๎€ธโˆง||โ‰ค(๐‘˜)(2๐œ‹)โˆ’๐‘‘/๐‘๎€ทโ€–โ€–๐นโˆ’๐’Ÿ(๐œ“)๐‘“๐‘›โ€–โ€–๐‘+||๐œ“||โ‹…โ€–โ€–(๐‘˜)๐‘“โˆ’๐‘“๐‘›โ€–โ€–๐‘๎€ธ(3.8) for any ๐‘˜โˆˆโ„ค๐‘‘ and ๐‘›โˆˆโ„•. In view of (3.7) the right-hand side of this inequality tends to 0 if ๐‘›โ†’+โˆž. This implies ๐นโˆง(๐‘˜)=๐œ“(๐‘˜)๐‘“โˆง(๐‘˜),๐‘˜โˆˆโ„ค๐‘‘.(3.9) Hence, ๐‘“ is ๐œ“-differentiable, ๐’Ÿ(๐œ“)๐‘“=๐น, and completeness is proved.
In order to show part (ii) we consider the Fourier means โ„ฑ(๐œ‘0)๐œŽ of type (2.23)-(2.24), where ๐œ‘0 satisfies the conditions described in Section 2 (see, in particular, (2.13)). Let ๐‘“โˆˆ๐‘‹๐‘(๐œ“). Taking into account that ๎‚€โ„ฑ๐’Ÿ(๐œ“)(๐œ‘0)๐œŽ๎‚(๐‘“)=โ„ฑ(๐œ‘0)๐œŽ(๐’Ÿ(๐œ“)๐‘“),๐œŽโ‰ฅ0,(3.10) by (2.27) and (3.3) we get โ€–โ€–๐‘“โˆ’โ„ฑ(๐œ‘0)๐œŽโ€–โ€–(๐‘“)๐‘‹๐‘(๐œ“)=โ€–โ€–๐‘“โˆ’โ„ฑ(๐œ‘0)๐œŽโ€–โ€–(๐‘“)๐‘+โ€–โ€–๎‚€โ„ฑ๐’Ÿ(๐œ“)๐‘“โˆ’๐’Ÿ(๐œ“)(๐œ‘0)๐œŽ๎‚โ€–โ€–(๐‘“)๐‘=โ€–โ€–๐‘“โˆ’โ„ฑ(๐œ‘0)๐œŽโ€–โ€–(๐‘“)๐‘+โ€–โ€–๐’Ÿ(๐œ“)๐‘“โˆ’โ„ฑ(๐œ‘0)๐œŽโ€–โ€–(๐’Ÿ(๐œ“)๐‘“)๐‘(3.11) for ๐œŽโ‰ฅ0. The terms on the right-hand side tend to 0 if ๐œŽโ†’+โˆž by (2.28). This yields the desired density of ๐’ฏ in ๐‘‹๐‘(๐œ“).
Now we prove part (iii). Let 1<๐‘<+โˆž, ๐œ“1,๐œ“2โˆˆโ„‹๐‘ , and let ๐‘“ be an arbitrary function in ๐‘‹๐‘(๐œ“2). First we observe that in view of (2.34) and (3.1), inequality (2.35) can be rewritten as โ€–โ€–๐’Ÿ(๐œ“1โ€–โ€–)๐‘ก๐‘โ€–โ€–โ‰ค๐‘๐’Ÿ(๐œ“2โ€–โ€–)๐‘ก๐‘,๐‘กโˆˆ๐’ฏ.(3.12) It is valid for 1<๐‘<+โˆž (see the comment at the end of Section 2). Therefore, we obtain โ€–โ€–๐’Ÿ๎€ท๐œ“1๎€ธ๎‚€โ„ฑ(๐œ‘0)๐‘›(๐‘“)โˆ’โ„ฑ(๐œ‘0)๐‘š๎‚โ€–โ€–(๐‘“)๐‘โ€–โ€–๐’Ÿ๎€ท๐œ“โ‰ค๐‘2๎€ธ๎‚€โ„ฑ(๐œ‘0)๐‘›(๐‘“)โˆ’โ„ฑ(๐œ‘0)๐‘š๎‚โ€–โ€–(๐‘“)๐‘โ€–โ€–โ„ฑ=๐‘(๐œ‘0)๐‘›๎€ท๐’Ÿ๎€ท๐œ“2๎€ธ๐‘“๎€ธโˆ’โ„ฑ(๐œ‘0)๐‘š๎€ท๐’Ÿ๎€ท๐œ“2๎€ธ๐‘“๎€ธโ€–โ€–๐‘(3.13) for ๐‘›,๐‘šโˆˆโ„•. Hence, {๐’Ÿ(๐œ“1)(โ„ฑ(๐œ‘0)๐‘›(๐‘“))} is a Cauchy sequence in ๐ฟ๐‘ by (2.28), and there exists ๐นโˆˆ๐ฟ๐‘ such that ๐’Ÿ๎€ท๐œ“1๎€ธ๎‚€โ„ฑ(๐œ‘0)๐‘›๎‚(๐‘“)๐ฟ๐‘โŸถ๐น(๐‘›โŸถ+โˆž).(3.14) By the help of (3.3), (2.27), and (3.14) we get ๐นโˆง(๐‘˜)=lim๐‘›โ†’+โˆž๎‚ƒ๐’Ÿ๎€ท๐œ“1๎€ธ๎‚€โ„ฑ(๐œ‘0)๐‘›(๐‘“)๎‚๎‚„โˆง(๐‘˜)=๐œ“1(๐‘˜)๐‘“โˆง(๐‘˜).(3.15) For each ๐‘˜โˆˆโ„ค๐‘‘. Hence, ๐‘“ belongs to ๐‘‹๐‘(๐œ“1) and ๐’Ÿ(๐œ“1)๐‘“=๐น. In order to prove that the embedding ๐‘‹๐‘๎€ท๐œ“2๎€ธโŠ‚๐‘‹๐‘๎€ท๐œ“1๎€ธ(3.16) is continuous, it is enough to notice that โ€–โ€–๐’Ÿ๎€ท๐œ“1๎€ธ๐‘“โ€–โ€–๐‘=lim๐‘›โ†’+โˆžโ€–โ€–๐’Ÿ๎€ท๐œ“1๎€ธ๎‚€โ„ฑ(๐œ‘0)๐‘›๎‚โ€–โ€–(๐‘“)๐‘โ‰ค๐‘lim๐‘›โ†’+โˆžโ€–โ€–๐’Ÿ๎€ท๐œ“2๎€ธ๎‚€โ„ฑ(๐œ‘0)๐‘›๎‚โ€–โ€–(๐‘“)๐‘=โ€–โ€–๐’Ÿ๎€ท๐œ“2๎€ธ๐‘“โ€–โ€–๐‘(3.17) as a consequence of (2.35). This completes the proof.

In other words part (iii) of the theorem means that in the case 1<๐‘<+โˆž there is only one space ๐‘‹๐‘(๐‘ )of ๐‘ -smooth functions. It can be characterized as ๐‘‹๐‘(๐‘ )=๎ƒฏ๐‘“โˆˆ๐ฟ๐‘โˆถ๐’Ÿ(|โ‹…|๐‘ ๎“)๐‘“=๐‘˜โˆˆโ„ค๐‘‘||๐‘˜||๐‘ ๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅโˆˆ๐ฟ๐‘๎ƒฐ(3.18) and may be equipped with the norm โ€–๐‘“โ€–๐‘‹๐‘(๐‘ )=โ€–๐‘“โ€–๐‘+โ€–๐’Ÿ(|โ‹…|๐‘ โ€–)๐‘“๐‘.(3.19)

Theorem 3.2. Suppose 1โ‰ค๐‘โ‰ค+โˆž, ๐œ“1โˆˆโ„‹๐‘ 1, and ๐œ“2โˆˆโ„‹๐‘ 2. If ๐‘ 1>๐‘ 2>0, then ๐‘‹๐‘๎€ท๐œ“1๎€ธโŠ‚๐‘‹๐‘๎€ท๐œ“2๎€ธ(3.20) and this embedding is continuous.

Proof. Let ๐‘“โˆˆ๐‘‹๐‘(๐œ“1). In view of (2.15) the function ๐œ‘=((๐œƒ๐œ“2)/๐œ“1), where ๐œƒ is given by (2.14), is infinitely differentiable on โ„๐‘‘ and has compact support. Hence, its Fourier transform belongs to ๐ฟ1(โ„๐‘‘) and the operators โ„ฑ๐œŽ(๐œ‘) are uniformly bounded in ๐ฟ๐‘ as stated in Section 2. Using this fact as well as (3.3), (2.31), and the homogeneity property of ๐œ“1 and ๐œ“2, we get โ€–โ€–โ€–โ€–๐‘›๎“๐‘—=๐‘š๐’Ÿ๎€ท๐œ“2๎€ธ๐‘“๐‘—โ€–โ€–โ€–โ€–๐‘โ‰ค๐‘›๎“๐‘—=๐‘šโ€–โ€–๐’Ÿ๎€ท๐œ“2๎€ธ๐‘“๐‘—โ€–โ€–๐‘=๐‘›๎“๐‘—=๐‘šโ€–โ€–โ€–โ€–๎“๐‘˜โˆˆโ„ค๐‘‘๐œƒ๎€ท2โˆ’๐‘—๐‘˜๎€ธ๐œ“2(๐‘˜)๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅโ€–โ€–โ€–โ€–๐‘=๐‘›๎“๐‘—=๐‘š2โˆ’(๐‘ 1โˆ’๐‘ 2)๐‘—โ€–โ€–โ€–โ€–๎“๐‘˜โ‰ 0๐œ‘๎€ท2โˆ’๐‘—๐‘˜๎€ธ๐œ“1(๐‘˜)๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅโ€–โ€–โ€–โ€–๐‘=๐‘›๎“๐‘—=๐‘š2โˆ’(๐‘ 1โˆ’๐‘ 2)๐‘—โ€–โ€–โ„ฑ2(๐œ‘)๐‘—๎€ท๐’Ÿ๎€ท๐œ“1๎€ธ๐‘“๎€ธโ€–โ€–๐‘โ€–โ€–๐’Ÿ๎€ท๐œ“โ‰ค๐‘1๎€ธ๐‘“โ€–โ€–๐‘๐‘›๎“๐‘—=๐‘š2โˆ’(๐‘ 1โˆ’๐‘ 2)๐‘—(3.21) for ๐‘›>๐‘šโ‰ฅ1. Here, ๐‘“๐‘—,๐‘—โˆˆโ„•, has the meaning of (2.22). Because of ๐‘ 1>๐‘ 2 the sequence of the partial sums of the series โˆ‘+โˆž๐‘—=1๐’Ÿ(๐œ“2)๐‘“๐‘— is fundamental in ๐ฟ๐‘ and there exists ๐นโˆˆ๐ฟ๐‘ such that ๐น๐ฟ๐‘=+โˆž๎“๐‘—=1๐’Ÿ๎€ท๐œ“2๎€ธ๐‘“๐‘—.(3.22) By (2.13) we have ๐œ‘0(๐‘˜)โ‰ 0 only for ๐‘˜=0. Thus, we obtain ๐ฝ๎“๐‘—=1๐’Ÿ๎€ท๐œ“2๎€ธ๐‘“๐‘—๎“(๐‘ฅ)=๐‘˜โˆˆโ„ค๐‘‘๐œ‘0๎‚ต๐‘˜2๐ฝ๎‚ถ๐œ“2(๐‘˜)๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ,๐‘ฅโˆˆโ„๐‘‘(3.23) for any ๐ฝโˆˆโ„• in analogy to (2.32). Considering the limit process ๐ฝโ†’+โˆž we find ๐นโˆง(๐œˆ)=lim๐ฝโ†’+โˆž๎ƒฉ๐ฝ๎“๐‘—=1๐’Ÿ๎€ท๐œ“2๎€ธ๐‘“๐‘—๎ƒชโˆง(๐œˆ)=๐œ“2(๐œˆ)๐‘“โˆง(๐œˆ)(3.24) for any ๐œˆโˆˆโ„ค๐‘‘ with the help of (2.13) and (3.22). Hence, ๐‘“ belongs to ๐‘‹๐‘(๐œ“2) and ๐’Ÿ(๐œ“2)๐‘“=๐น. In order to prove that embedding (3.20) is continuous, it is enough to put ๐‘š=1in estimate (3.21) and to consider ๐‘› to +โˆž. This completes the proof.

Theorem 3.3. Let 1โ‰ค๐‘โ‰ค+โˆž and ๐œ“โˆˆโ„‹๐‘ , ๐‘ >0. Then, ๐’Ÿ(๐œ“)โˆถ๐‘‹๐‘(๐œ“)โŸถ๐ฟ0๐‘โ‰ก๎€ฝ๐‘“โˆˆ๐ฟ๐‘โˆถ๐‘“โˆง๎€พ(0)=0(3.25) is a surjective mapping.

Proof. Let ๐‘“โˆˆ๐ฟ0๐‘. We introduce functions ๐‘”๐‘—, ๐‘—โˆˆโ„•, by setting ๐‘”๐‘—(๎“๐‘ฅ)=๐‘˜โ‰ 0๐œƒ๎€ท2โˆ’๐‘—๐‘˜๎€ธ(๐œ“(๐‘˜))โˆ’1๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ,๐‘—โˆˆโ„•,(3.26) where ๐œƒ has the meaning of (2.14), and we put ๐œ‘=๐œƒ/๐œ“. By the same arguments as in the proof of Theorem 3.2 (see, in particular, (3.21)) we get โ€–โ€–โ€–โ€–๐‘›๎“๐‘—=๐‘š๐‘”๐‘—โ€–โ€–โ€–โ€–๐‘โ‰ค๐‘›๎“๐‘—=๐‘šโ€–โ€–๐‘”๐‘—โ€–โ€–๐‘=๐‘›๎“๐‘—=๐‘š2โˆ’๐‘—๐‘ โ€–โ€–โ„ฑ2(๐œ‘)๐‘—โ€–โ€–(๐‘“)๐‘โ‰ค๐‘โ€–๐‘“โ€–๐‘๐‘›๎“๐‘—=๐‘š2โˆ’๐‘—๐‘ (3.27) for ๐‘›>๐‘šโ‰ฅ1. Hence, there exists ๐‘”โˆˆ๐ฟ๐‘ such that ๐‘”๐ฟ๐‘=+โˆž๎“๐‘—=1๐‘”๐‘—.(3.28)
Taking into account that ๐œ‘0(๐‘˜)โ‰ 0 only for ๐‘˜=0 in view of (2.13), we obtain analogously to (2.32) ๐ฝ๎“๐‘—=1๐‘”๐‘—=๎“๐‘˜โ‰ 0๐œ‘0๎€ท2โˆ’๐ฝ๐‘˜๎€ธ(๐œ“(๐‘˜))โˆ’1๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ(3.29) by (3.26). As a consequence of (3.28) and (3.29) we get ๐‘”โˆง(๐œˆ)=lim๐ฝโ†’+โˆž๎ƒฉ๐ฝ๎“๐‘—=1๐‘”๐‘—๎ƒชโˆง(๐œˆ)=(๐œ“(๐œˆ))โˆ’1๐‘“โˆง(๐œˆ)(3.30) for ๐œˆโ‰ 0. Finally, because of ๐‘“โˆง(0)=0 we obtain ๐‘“โˆง(๐œˆ)=๐œ“(๐œˆ)๐‘”โˆง(๐œˆ),๐œˆโˆˆโ„ค๐‘‘.(3.31) From (3.30). Hence, ๐‘”โˆˆ๐‘‹๐‘(๐œ“), ๐’Ÿ(๐œ“)๐‘”=๐‘“, and the proof is complete.

Theorem 3.3 shows that for any ๐œ“โˆˆโ„‹๐‘  an operation which is inverse to ๐’Ÿ(๐œ“) is well defined on ๐ฟ0๐‘. The operator ๐ผ(๐œ“)=(๐’Ÿ(๐œ“))โˆ’1โˆถ๐ฟ0๐‘โŸถ๎€ท๐‘‹๐‘๎€ธ(๐œ“)โˆฉ๐ฟ0๐‘(3.32) is called operator of ๐œ“-integration. The Fourier series of the function ๐ผ(๐œ“)๐‘“, ๐‘“โˆˆ๐ฟ0๐‘, is given by ๎“๐‘˜โ‰ 0(๐œ“(๐‘˜))โˆ’1๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ.(3.33) 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) ๐‘š=1 and considering ๐‘› to +โˆž, we obtain the boundedness of the operator ๐ผ(๐œ“) in ๐ฟ0๐‘.

With the help of Theorem 3.3 we will see that in contrast to the case 1<๐‘<+โˆž, where in view of part (iii) of Theorem 3.1 only one space of ๐‘ -smooth functions exists, in the cases ๐‘=1,+โˆž there are infinitely many ways to define smoothness spaces of order ๐‘  associated with homogeneous generators by means of operators of multiplier type.

Theorem 3.4. Suppose that ๐œ“1 and ๐œ“2 are linear independent homogeneous functions belonging to โ„‹๐‘ , ๐‘ >0. Then, ๐‘‹1๎€ท๐œ“1๎€ธโŠ„๐‘‹1๎€ท๐œ“2๎€ธ,๐‘‹โˆž๎€ท๐œ“1๎€ธโŠ„๐‘‹โˆž๎€ท๐œ“2๎€ธ.(3.34)

Proof. We consider the case ๐‘=1. For ๐‘=+โˆž the proof is similar. Let us assume (to the contrary) that ๐‘‹1๎€ท๐œ“1๎€ธโŠ‚๐‘‹1๎€ท๐œ“2๎€ธ.(3.35) Then, by Theorem 3.3 the operators โ„’๐‘›=โ„ฑ(๐œ‘0)๐‘›๎€ท๐œ“โˆ˜๐’Ÿ2๎€ธ๎€ท๐œ“โˆ˜๐ผ1๎€ธ,๐‘›โˆˆโ„•,(3.36) where ๐œ‘0 has the meaning of Section 2 (see also (2.13)), are well defined on the space ๐ฟ01. The operator โ„’๐‘› acts in accordance with the following chain of mappings and inclusions: ๐ฟ01๐ผ๎€ท๐œ“1๎€ธโŸถ๐‘‹1๎€ท๐œ“1๎€ธโŠ‚๐‘‹1๎€ท๐œ“2๎€ธ๐’Ÿ(๐œ“2)โŸถ๐ฟ01โ„ฑ0)๐‘›(๐œ‘โŸถ๐’ฏ๐‘›.(3.37) By (2.23), (2.27), (3.3), and (3.33) we get โ„ฑ(๐œ‘0)๐‘›๎€ท๐œ“โˆ˜๐’Ÿ2๎€ธ๎€ท๐œ“โˆ˜๐ผ1๎€ธ๎“(๐‘“;๐‘ฅ)=๐‘˜โ‰ 0๐œ‘0๎‚€๐‘˜๐‘›๎‚๐œ“2๎€ท๐œ“(๐‘˜)1๎€ธ(๐‘˜)โˆ’1๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ=(2๐œ‹)โˆ’๐‘‘๎€œ๐•‹๐‘‘๐‘“(๐‘ฅโˆ’โ„Ž)ฮฆ๐‘›(โ„Ž)๐‘‘โ„Ž(3.38) for ๐‘›โˆˆโ„•. Here, ฮฆ๐‘›๎“(โ„Ž)=๐‘˜โ‰ 0๐œ‘0๎‚€๐‘˜๐‘›๎‚๐œ“2๎€ท๐œ“(๐‘˜)1๎€ธ(๐‘˜)โˆ’1๐‘’๐‘–๐‘˜๐‘ฅ,๐‘›โˆˆโ„•.(3.39) In view of (3.38) each operator โ„’๐‘› is bounded in ๐ฟ01. Moreover, the boundedness of โ„ฑ(๐œ‘0)๐‘› (see Section 2) yields โ€–โ€–โ„’๐‘›โ€–โ€–(๐‘“)1โ€–โ€–โ‰ค๐‘๐’Ÿ(๐œ“2)โˆ˜๐ผ(๐œ“1โ€–โ€–)๐‘“1(3.40) for any ๐‘“โˆˆ๐ฟ01. Applying now the Banach-Steinhaus principle we conclude that the operators โ„’๐‘› are uniformly bounded in ๐ฟ01. This leads to the estimate โ€–โ€–โ€–โ€–๎“๐‘˜โ‰ 0๐œ‘0๎‚€๐‘˜๐‘›๎‚๐œ“2๎€ท๐œ“(๐‘˜)1๎€ธ(๐‘˜)โˆ’1๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅโ€–โ€–โ€–โ€–1โ‰ค๐‘โ€–๐‘“โ€–1,(3.41) where the constant ๐‘ does not depend on ๐‘“โˆˆ๐ฟ01 and ๐‘›โˆˆโ„•. Let now ๎“๐‘ก(๐‘ฅ)=|๐‘˜|โ‰ค๐‘š๐‘๐‘˜๐‘’๐‘–๐‘˜๐‘ฅ(3.42) be an arbitrary trigonometric polynomial. We choose ๐‘›>2๐‘š. Then, it holds that ๐œ‘0(๐‘˜/๐‘›)๐œ“2(๐‘˜)=๐œ“2(๐‘˜) for |๐‘˜|โ‰ค๐‘š in view of (2.13). Applying (3.41) to ๐‘“=๐’Ÿ(๐œ“1)๐‘ก, we obtain โ€–โ€–๐’Ÿ๎€ท๐œ“2๎€ธ๐‘กโ€–โ€–1โ€–โ€–๐’Ÿ๎€ท๐œ“โ‰ค๐‘1๎€ธ๐‘กโ€–โ€–1.(3.43) This contradicts the statement on the nonvalidity of inequality (2.35) for ๐‘=1 pointed out at the end of Section 2. Changing the roles of ๐œ“1 and ๐œ“2 completes the proof.

4. ๐œ“-Smoothness and Besov and Triebel-Lizorkin Spaces

The aim of this section is to compare the spaces ๐‘‹๐‘(๐œ“),๐œ“โˆˆโ„‹๐‘ , with periodic Besov and Triebel-Lizorkin spaces. As we have seen already in part (iii) of Theorem 3.1 there is a unique space ๐‘‹๐‘(๐‘ ) if 1<๐‘<+โˆž 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 (๐‘ >0,1<๐‘<+โˆž) ๐ป๐‘ ๐‘=๎ƒฏ๐‘“โˆˆ๐ฟ๐‘โˆถ๎“๐‘˜โˆˆโ„ค๐‘‘๎‚€||๐‘˜||1+2๎‚๐‘ /2๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅโˆˆ๐ฟ๐‘๎ƒฐ(4.1) and equipped with the norm โ€–๐‘“โ€–๐ป๐‘ ๐‘=โ€–โ€–โ€–โ€–๎“๐‘˜โˆˆโ„ค๐‘‘๎‚€||๐‘˜||1+2๎‚๐‘ /2๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅโ€–โ€–โ€–โ€–๐‘.(4.2) Note that ๐ป๐‘ ๐‘=๐‘Š๐‘ ๐‘=๎€ฝ๐‘“โˆˆ๐ฟ๐‘โˆถ๐ท๐›ผ๐‘“โˆˆ๐ฟ๐‘๎€พfor|๐›ผ|โ‰ค๐‘ (4.3) if ๐‘ โˆˆโ„• (all derivatives in the sense of periodic distributions, see [5, Subsection 3.5.4]).

Theorem 4.1. Let 1<๐‘<+โˆž, and let ๐‘ >0. Then, one has ๐‘‹๐‘(๐‘ )=๐ป๐‘ ๐‘=๐น๐‘ ๐‘,2(4.4) with equivalence of norms.

Proof. Both statements are known. The second identity can be found in [5, Theoremโ€‰โ€‰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-Hรถrmander (see, e.g., [15, Theoremโ€‰โ€‰5.2.7, page 367], for the non-periodic version) which can be transferred to the periodic case using [16, Chapter 7, Theoremโ€‰โ€‰3.1]. The arguments are similar to [17, Subsectionsโ€‰โ€‰6.2.2, 6.2.3], or [18, Theoremโ€‰โ€‰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 ๐‘=1 and ๐‘=+โˆž.

Theorem 4.2. Let 1โ‰ค๐‘โ‰ค+โˆž, and let ๐œ“โˆˆโ„‹๐‘  for some ๐‘ >0. Then, ๐ต๐‘ ๐‘,1โŠ‚๐‘‹๐‘(๐œ“)โŠ‚๐ต๐‘ ๐‘,โˆž(4.5) and these embeddings are continuous.

Proof. Let ๐œ‚ be an infinitely differentiable positive function defined on โ„๐‘‘ such that ||๐œ‰||โ‰ค1๐œ‚(๐œ‰)=1if8||๐œ‰||โ‰ฅ1,๐œ‚(๐œ‰)=0if4.(4.6) We put ๐œ“โˆ—=๐œ“(1โˆ’๐œ‚). By (2.15) we have ๐œ“๎€ท2โˆ’๐‘—๐‘˜๎€ธ๐œƒ๎€ท2โˆ’๐‘—๐‘˜๎€ธ=๐œ“โˆ—๎€ท2โˆ’๐‘—๐‘˜๎€ธ๐œƒ๎€ท2โˆ’๐‘—๐‘˜๎€ธ,๐‘˜โˆˆโ„ค๐‘‘,๐‘—โˆˆโ„•,(4.7) where ๐œƒ is given by (2.14).
Let ๐‘“ be an arbitrary function in ๐ต๐‘ ๐‘,1, and let ๐‘“๐‘—, ๐‘—โˆˆโ„•0, be given by (2.22). Using (4.7), the homogeneity property of ๐œ“, and (2.27) and applying a Fourier multiplier theorem which can be found in [5, Theoremโ€‰โ€‰3.3.4, page 150], we obtain โ€–โ€–โ€–โ€–๐‘›๎“๐‘—=๐‘š๐’Ÿ(๐œ“)๐‘“๐‘—โ€–โ€–โ€–โ€–๐‘โ‰ค๐‘›๎“๐‘—=๐‘šโ€–โ€–๐’Ÿ(๐œ“)๐‘“๐‘—โ€–โ€–๐‘=๐‘›๎“๐‘—=mโ€–โ€–โ€–โ€–๎“๐‘˜โˆˆโ„ค๐‘‘๎€ท2๐œ“(๐‘˜)๐œƒโˆ’๐‘—๐‘˜๎€ธ๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅโ€–โ€–โ€–โ€–๐‘=๐‘›๎“๐‘—=๐‘š2๐‘—๐‘ โ€–โ€–โ€–โ€–๎“๐‘˜โˆˆโ„ค๐‘‘๐œ“โˆ—๎€ท2โˆ’๐‘—๐‘˜๎€ธ๐œƒ๎€ท2โˆ’๐‘—๐‘˜๎€ธ๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅโ€–โ€–โ€–โ€–๐‘โ‰ค๐‘๐‘›๎“๐‘—=๐‘š2๐‘—๐‘ โ€–โ€–โ„ฑ2(๐œƒ)๐‘—โ€–โ€–(๐‘“)๐‘=๐‘›๎“๐‘—=๐‘š2๐‘—๐‘ โ€–โ€–๐‘“๐‘—โ€–โ€–๐‘(4.8) for ๐‘›>๐‘šโ‰ฅ1. In view of (2.18) and (4.8) we can conclude that the series โˆ‘+โˆž๐‘—=0๐’Ÿ(๐œ“)๐‘“๐‘— converges in ๐ฟ๐‘. Now, by standard arguments we see as in the proof of Theorem 3.1 that ๐‘“ belongs to ๐‘‹๐‘(๐œ“) and that the first embedding in (4.5) is continuous.
In order to prove the second embedding we introduce an infinitely differentiable function ๐œ“โˆ—(๐œ‰) which coincides with ๐œ“(๐œ‰) for |๐œ‰|โ‰ฅ1/4 and which is not equal to 0 on โ„๐‘‘. By (2.15) we have ๐œ“๎€ท2โˆ’๐‘—๐‘˜๎€ธ๐œƒ๎€ท2โˆ’๐‘—๐‘˜๎€ธ=๐œ“โˆ—๎€ท2โˆ’๐‘—๐‘˜๎€ธ๐œƒ๎€ท2โˆ’๐‘—๐‘˜๎€ธ,๐‘˜โˆˆโ„ค๐‘‘,๐‘—โˆˆโ„•,(4.9) where ๐œƒ has the meaning of (2.14).
Let ๐‘“ be an arbitrary function in ๐‘‹๐‘(๐œ“), and let ๐‘“๐‘—, ๐‘—โˆˆโ„•0, be given by (2.22). Applying (4.9), the homogeneity property of ๐œ“, and (2.27), we obtain ๐‘“๐‘—(๎“๐‘ฅ)=๐‘˜โˆˆโ„ค๐‘‘๐œƒ๎€ท2โˆ’๐‘—๐‘˜๎€ธ๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ=๎“๐‘˜โˆˆโ„ค๐‘‘๎€ท๐œ“โˆ—๎€ท2โˆ’๐‘—๐‘˜๎€ธ๎€ธโˆ’1๐œ“๎€ท2โˆ’๐‘—๐‘˜๎€ธ๐œƒ๎€ท2โˆ’๐‘—๐‘˜๎€ธ๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ=2โˆ’๐‘—๐‘ ๎“๐‘˜โˆˆโ„ค๐‘‘๎€ท๐œ“โˆ—๎€ท2โˆ’๐‘—๐‘˜๎€ธ๎€ธโˆ’1๐œ“๎€ท2(๐‘˜)๐œƒโˆ’๐‘—๐‘˜๎€ธ๐‘“โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ=2โˆ’๐‘—๐‘ ๎“๐‘˜โˆˆโ„ค๐‘‘๎€ท๐œ“โˆ—๎€ท2โˆ’๐‘—๐‘˜๎€ธ๎€ธโˆ’1๎‚€โ„ฑ2(๐œƒ)๐‘—๎‚(๐’Ÿ(๐œ“)๐‘“)โˆง(๐‘˜)๐‘’๐‘–๐‘˜๐‘ฅ(4.10) for any ๐‘—โˆˆโ„•. The right-hand side can be estimated again by means of the Fourier multiplier theorem which can be found in [5, Theoremโ€‰โ€‰3.3.4, page 150]. Using in addition the uniform boundedness of the Fourier means โ„ฑ2(๐œƒ)๐‘— in ๐ฟ๐‘, we obtain the inequalities โ€–โ€–๐‘“๐‘—โ€–โ€–๐‘โ‰ค๐‘2โˆ’๐‘—๐‘ โ€–โ€–โ„ฑ2(๐œƒ)๐‘—โ€–โ€–(๐’Ÿ(๐œ“)๐‘“)๐‘โ‰ค๐‘๎…ž2โˆ’๐‘—๐‘ โ€–๐’Ÿ(๐œ“)๐‘“โ€–๐‘(4.11) for ๐‘—โˆˆโ„• from (4.10). Obviously, โ€–โ€–๐‘“0โ€–โ€–๐‘โ‰ค๐‘โ€–๐‘“โ€–๐‘.(4.12) Hence, by (4.11) and (3.5) โ€–๐‘“โ€–๐ต๐‘ ๐‘,โˆžโ‰ค๐‘โ€–๐‘“โ€–๐‘‹๐‘(๐œ“)(4.13) for some constant ๐‘>0 and all ๐‘“โˆˆ๐‘‹๐‘(๐œ“). This completes the proof.

Having in mind Theorem 4.1 the embeddings (4.5) are well known in the case 1<๐‘<+โˆž. Even a better result holds. This follows also from Theorem 4.1 and the elementary embeddings ๐ต๐‘ ๐‘,1โŠ‚๐ต๐‘ ๐‘,min(๐‘,2)โŠ‚๐น๐‘ ๐‘,2โŠ‚๐ต๐‘ ๐‘,max(๐‘,2)โŠ‚๐ต๐‘ ๐‘,โˆž(4.14) (see [5, Remarkโ€‰โ€‰3.5.1/4, page 164]).

5. A Representation Formula for ๐œ“-Derivatives

The following observations give some motivation and pave the way to find explicit representations of the operator ๐’Ÿ(๐œ“) in terms of the Fourier transform of its generator ๐œ“โˆˆโ„‹๐‘ . For the sake of simplicity we restrict ourselves to the case 0<๐‘ <1.

Let ๐œ“โˆˆโ„‹๐‘ . Recall that ๐‘”โˆˆโ„ฐper if and only if ||๐‘”โˆง||||๐‘˜||(๐‘˜)โ‰ค๐ถ(๐‘š)โˆ’๐‘š,๐‘˜โ‰ 0,(5.1) for each ๐‘šโˆˆโ„• and note that ||||||๐œ‰||๐œ“(๐œ‰)โ‰ค๐‘๐‘ ,๐œ‰โˆˆโ„๐‘‘,(5.2) by homogeneity. Hence, it makes sense to consider the periodic distribution ๎“ฮจ(๐‘ฅ)=๐‘˜โˆˆโ„ค๐‘‘๐œ“(๐‘˜)๐‘’โˆ’๐‘–๐‘˜๐‘ฅ(5.3) (convergence in โ„ฐโ€ฒper). Let ๐‘”โˆˆโ„ฐper. Expansion into the Fourier series leads to the representation ๎“ฮจ(๐‘ฅ)=๐œˆโˆˆโ„ค๐‘‘๎๐œ“(๐‘ฅ+2๐œ‹๐œˆ)(5.4) (convergence in โ„ฐโ€ฒper). By definition of the Fourier coefficients, the definition of the ๐œ“-derivative and, (5.3) we get ๐’Ÿ(๐œ“)(๐‘”)(๐‘ฅ)=(2๐œ‹)โˆ’๐‘‘โŸจฮจ(โ‹…),๐‘”(๐‘ฅ+โ‹…)โˆ’๐‘”(๐‘ฅ)โŸฉ,๐‘”โˆˆโ„ฐper.(5.5) Using (5.4) and (5.5), we see, at least formally, that ๐’Ÿ(๐œ“)(๐‘”)(๐‘ฅ)=(2๐œ‹)โˆ’๐‘‘๎“๐œˆโˆˆโ„ค๐‘‘๎ซ๎ฌ๎๐œ“(โ‹…+2๐œ‹๐œˆ),๐‘”(๐‘ฅ+โ‹…)โˆ’๐‘”(๐‘ฅ)=(2๐œ‹)โˆ’๐‘‘๎“๐œˆโˆˆโ„ค๐‘‘๎€œ๐•‹๐‘‘๎๐œ“(๐‘ฆ+2๐œ‹๐œˆ)(๐‘”(๐‘ฅ+๐‘ฆ)โˆ’๐‘”(๐‘ฅ))๐‘‘๐‘ฆ=(2๐œ‹)โˆ’๐‘‘๎€œโ„๐‘‘(๐‘”(๐‘ฅ+โ„Ž)โˆ’๐‘”(๐‘ฅ))๎๐œ“(โ„Ž)๐‘‘โ„Ž,๐‘”โˆˆโ„ฐper.(5.6) We claim that the integral on the right-hand side of (5.6) exists for all ๐‘ฅโˆˆ๐•‹๐‘‘. To this end we first recall that (see Section 2) ๎๐œ“โˆˆ๐’ฎ๎…ž is a homogeneous distribution of order โˆ’(๐‘‘+๐‘ ). Its restriction to โ„๐‘‘โงต{0} can be identified with a function ๎๐œ“โˆˆ๐ถโˆž(โ„๐‘‘โงต{0}) by [9, Theoremโ€‰โ€‰7.1.18]. Hence, we have the estimate ||||||โ„Ž||๎๐œ“(โ„Ž)โ‰ค๐‘โˆ’(๐‘‘+๐‘ ),โ„Žโˆˆโ„๐‘‘โงต{0}.(5.7) For brevity we use the standard notation ฮ”โ„Ž๐‘”(๐‘ฅ)=๐‘”(๐‘ฅ+โ„Ž)โˆ’๐‘”(๐‘ฅ). Now, we split ๎€œโ„๐‘‘ฮ”โ„Ž๎€œ๐‘”(๐‘ฅ)๎๐œ“(โ„Ž)๐‘‘โ„Ž=๐ต1ฮ”โ„Ž๎€œ๐‘”(๐‘ฅ)๎๐œ“(โ„Ž)๐‘‘โ„Ž+โ„๐‘‘โงต๐ต1ฮ”โ„Ž๐‘”(๐‘ฅ)๎๐œ“(โ„Ž)๐‘‘โ„Ž.(5.8) The second summand is finite because of (5.7) and the boundedness of |ฮ”โ„Ž๐‘”(๐‘ฅ)| on โ„๐‘‘. As for the first one we use the estimate ||ฮ”โ„Ž||๐‘”(๐‘ฅ)โ‰ค๐‘๎…ž||โ„Ž||โ„Žโˆˆ๐ต1,๐‘ฅโˆˆ๐•‹๐‘‘,(5.9) and (5.7) to see that ||||๎€œ๐ต1ฮ”โ„Ž||||๐‘”(๐‘ฅ)๎๐œ“(โ„Ž)๐‘‘โ„Žโ‰ค๐‘๎…ž๎€œ๐ต1||โ„Ž||||||๎๐œ“(โ„Ž)๐‘‘โ„Žโ‰ค๐‘๎…ž๎…ž๎€œ๐ต1||โ„Ž||โˆ’๐‘‘+(1โˆ’๐‘ )๐‘‘โ„Ž<+โˆž.(5.10) 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 ๐’Ÿ(๐œ“)(๐‘”) holds pointwise almost everyone under much weaker assumptions with respect to ๐‘”.

Theorem 5.1. Suppose 1โ‰ค๐‘โ‰ค+โˆž and ๐œ“โˆˆโ„‹๐‘ , where 0<๐‘ <1. Then, ๐’Ÿ(๐œ“)(๐‘“)(๐‘ฅ)=(2๐œ‹)โˆ’๐‘‘๎€œโ„๐‘‘(๐‘“(๐‘ฅ+โ„Ž)โˆ’๐‘“(๐‘ฅ))๎๐œ“(โ„Ž)๐‘‘โ„Ž,๐‘“โˆˆ๐ต๐‘ ๐‘,1,(5.11) for almost all ๐‘ฅโˆˆ๐•‹๐‘‘ (for all, if ๐‘=+โˆž).

Proof. First we recall that ๐ต๐‘ ๐‘,1โŠ‚๐‘‹๐‘(๐œ“) by Theorem 4.2. Hence, the left-hand side of (5.11) makes sense. Let ๐‘“โˆˆ๐ต๐‘ ๐‘,1. As is well known (see, e.g., [5, Theoremโ€‰โ€‰3.5.4]), โ€–๐‘“โ€–โˆ—๐ต๐‘ ๐‘,1=โ€–๐‘“โ€–๐‘+๎€œโ„๐‘‘||โ„Ž||โˆ’๐‘ โ€–โ€–ฮ”โ„Ž๐‘“โ€–โ€–๐‘๐‘‘โ„Ž||โ„Ž||๐‘‘(5.12) is an equivalent norm in the Besov space ๐ต๐‘ ๐‘,1. Using (5.7), (5.12), and the generalized Minkowski inequality we obtain for the integral ๐ผ(๐‘ฅ) at the right-hand side of (5.11) the estimates โ€–๐ผโ€–๐‘โ‰ค๎€œโ„๐‘‘โ€–โ€–ฮ”โ„Žโ€–โ€–๐‘“(๐‘ฅ)๐‘โ‹…||||๎€œ๎๐œ“(โ„Ž)๐‘‘โ„Žโ‰ค๐‘โ„๐‘‘โ€–โ€–ฮ”โ„Ž๐‘“โ€–โ€–๐‘โ‹…||โ„Ž||โˆ’(๐‘‘+๐‘ )๐‘‘โ„Žโ‰ค๐‘โ€–๐‘“โ€–โˆ—๐ต๐›ผ๐‘,1.(5.13) Hence, the function ๐ผ(๐‘ฅ) belongs to ๐ฟ๐‘. To prove (5.11) it is sufficient to show that the Fourier coefficients of the functions on both sides coincide. We have (๐ผ(โ‹…))โˆง(๐‘˜)=(2๐œ‹)โˆ’๐‘‘๎€œ๐•‹๐‘‘๎€œโ„๐‘‘(๐‘“(๐‘ฅ+โ„Ž)โˆ’๐‘“(๐‘ฅ))๎๐œ“(โ„Ž)๐‘‘โ„Ž๐‘’โˆ’๐‘–๐‘˜๐‘ฅ๐‘‘๐‘ฅ=๐‘“โˆง๎€œ(๐‘˜)โ„๐‘‘๎€ท๐‘’๐‘–๐‘˜โ„Ž๎€ธโˆ’1๎๐œ“(โ„Ž)๐‘‘โ„Ž(5.14) by Fubini's theorem. It remains to show that ๎€œโ„๐‘‘๎€ท๐‘’๐‘–๐‘˜โ„Ž๎€ธโˆ’1๎๐œ“(โ„Ž)๐‘‘โ„Ž=(2๐œ‹)โˆ’๐‘‘๐œ“(๐‘˜),๐‘˜โˆˆโ„ค๐‘‘.(5.15) This is obvious if ๐‘˜=0 because of ๐œ“(0)=0. If ๐‘˜โ‰ 0, we have to use appropriate limiting arguments to circumvent the difficulty caused by the fact that ๎๐œ“ is not integrable in a neighbourhood of 0. We do not go into details.

We give some remarks. It is known (see [15, Theoremโ€‰โ€‰2.4.6, page 128]) that for any ๐‘ ,0<๐‘ <2, the restriction of the Fourier transform of ๐œ“(๐œ‰)=|๐œ‰|๐‘  to โ„๐‘‘โงต{0} can be identified with (|โ‹…|๐‘ )โˆง(๐‘ฅ)=๐‘(๐‘‘,๐‘ )|๐‘ฅ|โˆ’๐‘‘โˆ’๐‘ ,๎‚ต๐‘(๐‘‘,๐‘ )=2๐‘‘+๐‘ ๐œ‹๐‘‘/2ฮ“(๐‘ /2+๐‘‘/2)๎‚ถฮ“(โˆ’๐‘ /2),(5.16) combining (5.7) with (5.16) in the one-dimensional case, we obtain the well-known formula for the Riesz derivative (see, e.g., [6]) ๐‘“โŸจ๐‘ โŸฉ(๐‘ฅ)=(2๐œ‹)โˆ’1๐‘๎€œ(1,๐‘ )+โˆžโˆ’โˆž๐‘“(๐‘ฅ+โ„Ž)โˆ’๐‘“(๐‘ฅ)||โ„Ž||๐‘ +1๐‘‘โ„Ž,๐‘“โˆˆ๐ต๐‘ ๐‘,1,(5.17) which is valid for 0<๐‘ <1 and 1โ‰ค๐‘โ‰ค+โˆž. In the multivariate case we get under the same conditions with respect to ๐‘  and ๐‘ the representation formula (โˆ’ฮ”)๐‘ /2๐‘“(๐‘ฅ)=(2๐œ‹)โˆ’๐‘‘๎€œ๐‘(๐‘‘,๐‘ )โ„๐‘‘๐‘“(๐‘ฅ+โ„Ž)โˆ’๐‘“(๐‘ฅ)||โ„Ž||๐‘‘+๐‘ ๐‘‘โ„Ž,๐‘“โˆˆ๐ต๐‘ ๐‘,1(5.18) for the fractional power of the Laplace operator.

Let us mention that formulas for ๐œ“-derivatives with ๐‘ โ‰ฅ1 can be achieved using differences of higher order. Suppose, for example, that 1โ‰ค๐‘ <2 and, in addition to the previous conditions, that ๐œ“ is real valued. Analogously to (5.6) we find ๐’Ÿ(๐œ“)(๐‘”)(๐‘ฅ)=(2๐œ‹)โˆ’๐‘‘๎€œโ„๐‘‘๐‘”(๐‘ฅ+โ„Ž)โˆ’2๐‘”(๐‘ฅ)+๐‘”(๐‘ฅโˆ’โ„Ž)2๎๐œ“(โ„Ž)๐‘‘โ„Ž,(5.19) and, in particular, (โˆ’ฮ”)๐‘ /2๐‘”(๐‘ฅ)=(2๐œ‹)โˆ’๐‘‘๎€œ๐‘(๐‘‘,๐‘ )โ„๐‘‘๐‘”(๐‘ฅ+โ„Ž)โˆ’2๐‘”(๐‘ฅ)+๐‘”(๐‘ฅโˆ’โ„Ž)2||โ„Ž||๐‘‘+๐‘ ๐‘‘โ„Ž(5.20) for ๐‘”โˆˆโ„ฐper. 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 ๐ต๐‘ ๐‘,1.


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