Smoothness and Function Spaces Generated by Homogeneous Multipliers

Abstract

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 𝐵^{𝑠𝑝,𝑞} and Triebel-Lizorkin spaces 𝐹^{𝑠𝑝,𝑞} 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., [1–5] 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<𝑝<+∞ and 𝑠>0, that the space 𝐹^𝑠𝑝,2 coincides with the (fractional) Sobolev 𝐻^𝑠𝑝 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 𝐻^𝑠𝑝 consists of periodic functions 𝑓 in 𝐿_𝑝 such that (𝐼−Δ)^𝑠/2𝑓 belongs to 𝐿_𝑝 as well. In other words, we have 𝐻^𝑠𝑝=(𝐼−Δ)^−𝑠/2(𝐿_𝑝) which means that 𝐻^𝑠𝑝 is characterized as the image of 𝐿_𝑝 by the operator (𝐼−Δ)^−𝑠/2. With this exception both Besov spaces 𝐵^{𝑠𝑝,𝑞} and Triebel-Lizorkin spaces 𝐹^{𝑠𝑝,𝑞} are not related to any operator acting on 𝐿_𝑝 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 [7]. 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 𝑋_𝑝(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 𝑋_𝑝(𝜓) which coincides with the fractional Sobolev space 𝐻^𝑠𝑝 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 𝐵^𝑠𝑝,1 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 𝑋_𝑝(𝜓) in further works by(i)finding representation formulae for operators generated by homogeneous functions of degree 𝑠≥1 using higher-order differences, (ii)studying generalized 𝐾-functionals and their realizations, (iii) constructing new moduli of smoothness related to 𝒟(𝜓), (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 𝒟(𝜓) and spaces 𝑋_𝑝(𝜓) are described in Section 3. Some relations between spaces 𝑋_𝑝(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 Preliminaries

2.1. Numbers and Vectors

By the symbols ℕ, ℕ₀, ℤ, ℝ, ℂ, ℕ^𝑑0, ℤ^𝑑, and ℝ^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 𝕋^𝑑 is reserved for the 𝑑-dimensional torus [0,2𝜋)^𝑑. We will also use the notations 𝑥𝑦=𝑥₁𝑦₁+⋯+𝑥_𝑑𝑦_d,|𝑥|≡|𝑥|₂=𝑥²¹+⋯+𝑥^2𝑑^1/2,(2.1) for the scalar product and the 𝑙₂-norm of vectors and 𝐵_𝑟=𝑥∈ℝ^𝑑∶|𝑥|<𝑟,𝐵_𝑟=𝑥∈ℝ^𝑑∶|𝑥|≤𝑟(2.2) for the open and closed balls, respectively.

2.2. 𝐿_𝑝-Spaces

As usual, 𝐿_𝑝≡𝐿_𝑝(𝕋^𝑑), where 1≤𝑝<+∞, 𝕋^𝑑=[0,2𝜋)^𝑑, is the space of measurable real-valued functions 𝑓=𝑓(𝑥), 𝑥=(𝑥₁,…,𝑥_𝑑) which are 2𝜋-periodic with respect to each variable satisfying ‖𝑓‖_𝑝=_𝕋𝑑||𝑓(𝑥)||^𝑝𝑑𝑥^1/𝑝<+∞.(2.3) In the case 𝑝=+∞, we consider the space 𝐶≡𝐶(𝕋^𝑑)(𝑝=+∞) of real-valued 2𝜋-periodic continuous functions equipped with the Chebyshev norm ‖𝑓‖_∞=max_{𝑥∈𝕋𝑑}||𝑓(𝑥)||.(2.4) For 𝐿_𝑝-spaces of non-periodic functions defined on a measurable set Ω⊆ℝ^𝑑 we will use the notation 𝐿_𝑝(Ω).

2.3. Fourier Coefficients and Fourier Transform

The Fourier coefficients of 𝑓∈𝐿₁ are defined by 𝑓^∧(𝑘)=(2𝜋)^−𝑑_𝕋𝑑𝑓(𝑥)𝑒^{−𝑖𝑘𝑥}𝑑𝑥,𝑘∈ℤ^𝑑.(2.5) Let ℰ_per and ℰ^′_per be the space of infinitely differentiable periodic functions and its dual the space of periodic distributions, respectively. The Fourier coefficients of 𝑓∈ℰ^′_per are given by 𝑓^∧(𝑘)=(2𝜋)^−𝑑𝑓,𝑒^{−𝑖𝑘⋅},𝑘∈ℤ^𝑑,(2.6) where, as usual, ⟨𝑓,𝑔⟩ means the value of the functional 𝑓 at 𝑔∈ℰ_per.

The Fourier transform and its inverse are given by 𝑓(𝜉)=_ℝ𝑑𝑓(𝑥)𝑒^{−𝑖𝑥𝜉}𝑑𝑥,𝑓^∨(𝑥)=(2𝜋)^−𝑑_ℝ𝑑𝑓(𝜉)𝑒^𝑖𝑥𝜉𝑑𝜉,𝑓∈𝐿₁ℝ^𝑑.(2.7) For an element 𝑓 belonging to the space of tempered distributions 𝒮^=𝒮′(ℝ^𝑑), which is the dual of the Schwartz space 𝒮=𝒮(ℝ^𝑑) of rapidly decreasing infinitely differentiable functions, the Fourier transform is defined by setting 𝑓,𝑔=⟨𝑓,𝑔̂⟩,𝑔∈𝒮.(2.8)

2.4. Trigonometric Polynomials

Let 𝜎 be a real nonnegative real number. By 𝒯_𝜎 we denote the space of all real-valued trigonometric polynomials of (spherical) order 𝜎. It means 𝒯_𝜎=⎧⎪⎨⎪⎩𝑡(𝑥)=_{||𝑘||≤𝜎}𝑐_𝑘𝑒^𝑖𝑘𝑥∶𝑐_−𝑘=𝑐_𝑘,||𝑘||≤𝜎⎫⎪⎬⎪⎭,(2.9) where 𝑐 is the complex conjugate to 𝑐∈ℂ. Further, 𝒯 stands for the space of all real-valued trigonometric polynomials of arbitrary order. Let 1≤𝑝≤+∞. As usual, we put 𝐸_𝜎(𝑓)_𝑝=inf_{𝑡∈𝒯𝜎}‖𝑓−𝑡‖_𝑝,𝜎>0,(2.10) for the best approximation of 𝑓 in 𝐿_𝑝 (𝑓∈𝐶 if 𝑝=+∞) by trigonometric polynomials of order at most 𝜎 in the metric of 𝐿_𝑝.

2.5. Homogeneous Functions

A complex-valued function 𝜓 defined on ℝ^𝑑⧵{0} is called homogeneous of order 𝑠∈ℝ if 𝜓(𝑡𝜉)=𝑡^𝑠𝜓(𝜉)(2.11) for 𝑡>0 and 𝜉∈ℝ^𝑑⧵{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>0⟨𝜓,𝑔(𝑡⋅)⟩=𝑡^{−(𝑠+𝑑)}⟨𝜓,𝑔⟩,𝑔∈𝒮.(2.12) 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 ℋ_𝑠 we denote the class of functions 𝜓 satisfying the following conditions: (i)𝜓 is continuous on ℝ^𝑑 and complex valued; (ii)𝜓 is infinitely differentiable on ℝ^𝑑⧵{0}; (iii)𝜓 is homogeneous of order 𝑠; (iv)𝜓(−𝜉)=𝜓(𝜉) for each 𝜉∈ℝ^𝑑; (v)𝜓(𝜉)≠0 for 𝜉∈ℝ^𝑑⧵{0}.

It follows that 𝜓∈𝒮′ and that the restriction of 𝜓 to ℝ^𝑑⧵{0} belongs to 𝐶^∞(ℝ^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 [4] for the nonperiodic case). Let 𝜑₀ be a real-valued centrally symmetric (𝜑₀(−𝜉)=𝜑₍𝜉) for all 𝜉∈ℝ^𝑑) infinitely differentiable function satisfying 𝜑₀(𝜉)=01,,||||𝜉𝜉||||≥≤112.,(2.13) We put 𝜃(𝜉)=𝜑₀(𝜉)−𝜑₀(2𝜉),𝜑_𝑗(𝜉)=𝜃2^−𝑗𝜉,𝑗∈ℕ.(2.14) Clearly, these functions are also centrally symmetric and infinitely differentiable with compact support. We have supp𝜃⊂𝐵₁𝐵_1/4;supp𝜑_𝑗⊂𝐵₂^𝑗𝐵₂^𝑗−2,𝑗∈ℕ.(2.15) By (2.14) we obtain 𝜑_𝑗(𝜉)=𝜑₀2^−𝑗𝜉−𝜑₀2^−𝑗+1𝜉,𝑗∈ℕ.(2.16) Combining (2.13) and (2.16), one has ^+∞_𝑗=0𝜑_𝑗(𝜉)=1,𝜉∈ℝ^𝑑.(2.17) In view of (2.15) and (2.17), the system Φ={𝜑_𝑗}^+∞𝑗=0 is called a smooth dyadic resolution of unity.

Let 𝑠>0, 1≤𝑝≤+∞, and 1≤𝑞≤+∞. The periodic Besov space 𝐵^{𝑠𝑝,𝑞} and the periodic Triebel-Lizorkin space 𝐹^{𝑠𝑝,𝑞} are given by (cf., [5, Chapter 3]) 𝐵^{𝑠𝑝,𝑞}=⎧⎪⎨⎪⎩𝑓∈𝐿_𝑝∶‖𝑓‖_𝐵^{𝑠𝑝,𝑞}≡^+∞_𝑗=02^𝑗𝑠𝑞‖‖𝑓_𝑗‖‖^𝑞𝑝^1/𝑞<+∞⎫⎪⎬⎪⎭(2.18) if 𝑞<∞, 𝐵^𝑠𝑝,∞=𝑓∈𝐿_𝑝∶‖𝑓‖_𝐵^𝑠𝑝,∞≡sup_𝑗∈ℕ02^𝑗𝑠‖‖𝑓_𝑗‖‖_𝑝<+∞,(2.19) and by 𝐹^{𝑠𝑝,𝑞}=⎧⎪⎨⎪⎩𝑓∈𝐿_𝑝∶‖𝑔‖_𝐹^{𝑠𝑝,𝑞}≡‖‖‖‖^+∞_𝑗=02^𝑗𝑠𝑞||𝑓_𝑗(𝑥)||^𝑞^1/𝑞‖‖‖‖_𝑝<+∞⎫⎪⎬⎪⎭(2.20) if 𝑝<+∞,𝑞<+∞, 𝐹^𝑠𝑝,∞=𝑓∈𝐿_𝑝∶‖𝑓‖_𝐹^𝑠𝑝,∞≡‖‖‖‖sup_𝑗∈ℕ02^𝑗𝑠||𝑓_𝑗(𝑥)||‖‖‖‖_𝑝<+∞,(2.21) if 𝑝<+∞, where the function𝑓_𝑗, 𝑗∈ℕ₀, are defined by 𝑓_𝑗(𝑥)=_{𝑘∈ℤ𝑑}𝜑_𝑗(𝑘)𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥,𝑥∈ℝ^𝑑,𝑗∈ℕ₀,(2.22) and 𝜑_𝑗, 𝑗∈ℕ₀, 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∈ℝ,0<𝑝,and𝑞≤+∞, 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 ℱ^𝜎(𝜑) which are given by ℱ^𝜎(𝜑)(𝑓;𝑥)=(2𝜋)^−𝑑_𝕋𝑑𝑓(ℎ)𝑊_𝜎(𝜑)(𝑥−ℎ)𝑑ℎ,𝜎≥0,(2.23) for 𝑓∈𝐿₁. The function ℱ^𝜎(𝜑)(𝑓) is called Fourier mean of 𝑓 generated by 𝜑. The functions 𝑊_𝜎(𝜑) in (2.23) are defined as 𝑊₀(𝜑)(ℎ)=1;𝑊_𝜎(𝜑)(ℎ)=_{𝑘∈ℤ𝑑}𝜑𝑘𝜎𝑒^𝑖𝑘𝑥,𝜎>0.(2.24) The Fourier means describe classical methods of trigonometric approximation which are well defined for functions in 𝐿_𝑝, where 1≤𝑝≤+∞. 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 ‖‖ℱ^𝜎(𝜑)(𝑓)‖‖_(𝑝)=sup_{‖𝑓‖𝑝≤1}‖‖ℱ^𝜎(𝜑)‖‖_𝑝(2.25) of operators defined on 𝐿_𝑝 by (2.23) is uniformly bounded, and we have sup_𝜎≥0‖‖ℱ^𝜎(𝜑)‖‖_(𝑝)<+∞,(2.26) for 𝑝=1, 𝑝=+∞ or for all 1≤𝑝≤+∞ if and only if 𝜑 belongs to 𝐿₁(ℝ^𝑑); (ii)for 𝑓∈𝐿₁ it holds that ℱ^𝜎(𝜑)(𝑓;𝑥)=_{𝑘∈ℤ𝑑}𝜑𝑘𝜎𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥,𝜎>0;(2.27)(iii)if 𝜑∈𝐿₁(ℝ^𝑑) and 𝜑(0)=1, then the Fourier means ℱ^𝜎(𝜑) converge in 𝐿_𝑝 for all 1≤𝑝≤+∞, and we have lim_𝜎→+∞‖‖𝑓−ℱ^𝜎(𝜑)(𝑓)‖‖_𝑝=0,𝑓∈𝐿_𝑝;(2.28)(iv)if 𝜑∈𝐿₁(ℝ^𝑑) and 𝜑(𝜉)=10,,||||𝜉𝜉||||≤>𝜌1,,(2.29) for some 𝜌>0, then for 1≤𝑝≤+∞‖‖𝑓−ℱ^𝜎(𝜑)(𝑓)‖‖_𝑝≤𝑐𝐸_𝜌𝜎(𝑓)_𝑝,𝑓∈𝐿_𝑝,𝜎≥0,(2.30) 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 𝑓_𝑗 which are well defined for 𝑓∈𝐿₁ by (2.22) can be represented as 𝑓₀(𝑥)=𝑓^∧(0),𝑓_𝑗(𝑥)=ℱ^2(𝜃)_𝑗(𝑓;𝑥),𝑗∈ℕ.(2.31) Moreover, using (2.16) and (2.27) we get ^𝐽_𝑗=0𝑓_𝑗(𝑥)=_{𝑘∈ℤ𝑑}^𝐽_𝑗=0𝜑_𝑗(𝑘)𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥=_{𝑘∈ℤ𝑑}𝜑₀2^−𝐽𝑘𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥=ℱ^(𝜑0)2_𝐽(𝑓;𝑥)(2.32) for 𝐽∈ℕ. Taking into account that 𝜑₀∈𝐿₁(ℝ^𝑑), we obtain therefrom by the convergence property (2.28) of the Fourier means the decomposition 𝑓=^+∞_𝑗=0𝑓_𝑗,𝑓∈𝐿_𝑝,(2.33) into a series of trigonometric polynomials being convergent in the space 𝐿_𝑝,1≤𝑝≤+∞.

2.8. Inequalities of Multiplier Type for Trigonometric Polynomials

Let 𝜓∈ℋ_𝑠 for some 𝑠>0. It generates the family of operators (𝐴_𝜎(𝜓))_𝜎 defined by 𝐴_𝜎(𝜓)𝑡(𝑥)=_{𝑘∈ℤ𝑑}𝜓𝑘𝜎𝑡^∧(𝑘)𝑒^𝑖𝑘𝑥,𝜎>0,𝑡∈𝒯,(2.34) on the space 𝒯 of trigonometric polynomials. We have proved in [12, 13] that the inequality ‖‖𝐴_𝜎𝜓₁𝑡‖‖_𝑝≤𝑐‖‖𝐴_𝜎𝜓₂𝑡‖‖_𝑝,(2.35) where 𝐴_𝜎(𝜓₁) is generated by 𝜓₁∈ℋ_𝑠1 and 𝐴_𝜎(𝜓₂) is generated by 𝜓₂∈ℋ_𝑠2, is valid for all 𝑡∈𝒯_𝜎 and 𝜎≥1 with a certain constant 𝑐 independent of 𝑡 and 𝜎 for all 1≤𝑝≤+∞ 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. 𝜓-Smoothness and Basic Properties

Let 𝜓∈ℋ_𝑠 for some 𝑠>0. It generates an operator 𝒟(𝜓) by setting 𝒟(𝜓)𝑡(𝑥)=_{𝑘∈ℤ𝑑}𝜓(𝑘)𝑡^∧(𝑘)𝑒^𝑖𝑘𝑥,𝑡∈𝒯,(3.1) which is initially defined on the space of trigonometric polynomials. The domain of definition can be extended within the spaces 𝐿_𝑝, 1≤𝑝≤+∞. To this end we introduce the space 𝑋_𝑝(𝜓) which consists of functions 𝑓 in 𝐿_𝑝 having the property that the set 𝜓(𝑘)𝑓^∧(𝑘),𝑘∈ℤ^𝑑(3.2) is the system of the Fourier coefficients of a certain function in𝐿_𝑝. 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),𝑘∈ℤ^𝑑,(3.3) for the Fourier coefficients of the𝜓-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 𝑓∈𝑋_𝑝(𝜓), then the series in (3.1) with 𝑓 in place of 𝑡 converges in 𝐿_𝑝 (see the proof of Theorem 3.1). In this sense we can reformulate 𝑋_𝑝(𝜓)=𝑓∈𝐿_𝑝∶𝒟(𝜓)𝑓=_{𝑘∈ℤ𝑑}𝜓(𝑘)𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥∈𝐿_𝑝.(3.4)

We give some examples. Let 𝑑=1, 𝑠∈ℕ, 1<𝑝<+∞, and 𝜓(𝜉)=(𝑖𝜉)^𝑠. Then, 𝒟(𝜓) is the operator of the usual derivative of order𝑠. In this case 𝑋_𝑝(𝜓) is the Sobolev space 𝑊^𝑠𝑝 of (𝑠−1)-times differentiable functions 𝑓 such that 𝑓^(𝑠−1) is absolutely continuous, 𝑓^(𝑠) exists almost everywhere and belongs to 𝐿_𝑝. If 𝑑=1, 𝑠∉ℕ, and 𝜓(𝜉)=(𝑖𝜉)^𝑠=|𝜉|^𝑠𝑒^{sgn𝜉⋅(𝑠𝜋𝑖)/2}, then 𝒟(𝜉) is the Weyl derivative (⋅)^(𝑠) of fractional order 𝑠 and 𝑋_𝑝(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 𝑋_𝑝(𝜓) coincides with the space 𝑊^1𝑝 of those functions where both the function itself and its conjugate belong to 𝑊^1𝑝. It is well known that this space coincides with 𝑊^1𝑝 if and only if 1<𝑝<+∞. For more details concerning the derivatives and spaces mentioned above, we refer to [5–8].

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 𝑋_𝑝(𝜓)≡𝑋_𝑝((−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 𝑋_𝑝(𝜓). Recall that we have to replace 𝐿_𝑝(𝕋^𝑑) by 𝐶(𝕋^𝑑) if 𝑝=+∞.

Theorem 3.1. Let 𝜓∈ℋ_𝑠, 𝑠>0, and 1≤𝑝≤+∞. Then, (i)𝑋_𝑝(𝜓) is a Banach space with respect to the norm ‖𝑓‖_{𝑋𝑝(𝜓)}=‖𝑓‖_𝑝+‖𝒟(𝜓)𝑓‖_𝑝;(3.5)(ii)𝒯 is dense in 𝑋_𝑝(𝜓); (iii)all spaces 𝑋_𝑝(𝜓) with 𝜓∈ℋ_𝑠coincide 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 {𝑓_𝑛} be a Cauchy sequence in 𝑋_𝑝(𝜓). In view of (3.5) ‖‖𝑓_𝑛−𝑓_𝑚‖‖_𝑝,‖‖𝒟(𝜓)𝑓_𝑛−𝒟(𝜓)𝑓_𝑚‖‖_𝑝⟶0(𝑛,𝑚⟶+∞).(3.6) By completeness of 𝐿_𝑝 there exist functions𝑓,𝐹∈𝐿_𝑝, such that 𝑓⟶_𝑛𝐿^𝑝𝑓,𝒟(𝜓)𝑓⟶_𝑛𝐿^𝑝𝐹(𝑛⟶+∞).(3.7) Using (3.3) and Hölder's inequality, we get ||𝐹^∧(𝑘)−𝜓(𝑘)𝑓^∧(𝑘)||≤≤(||2𝜋𝐹)−𝒟_−𝑑/𝑝(𝜓)𝑓_𝑛^∧(𝑘)||+||𝜓(𝑘)||⋅||𝑓−𝑓_𝑛^∧(𝑘)||‖‖𝐹−𝒟(𝜓)𝑓_𝑛‖‖_𝑝+||𝜓(𝑘)||⋅‖‖𝑓−𝑓_𝑛‖‖_𝑝(3.8) for any 𝑘∈ℤ^𝑑 and 𝑛∈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),𝑘∈ℤ^𝑑.(3.9) Hence, 𝑓 is 𝜓-differentiable, 𝒟(𝜓)𝑓=𝐹, and completeness is proved.
In order to show part (ii) we consider the Fourier means ℱ^(𝜑0)𝜎 of type (2.23)-(2.24), where 𝜑₀ satisfies the conditions described in Section 2 (see, in particular, (2.13)). Let 𝑓∈𝑋_𝑝(𝜓). Taking into account that 𝒟(𝜓)ℱ^(𝜑0)𝜎(𝑓)=ℱ^(𝜑0)𝜎(𝒟(𝜓)𝑓),𝜎≥0,(3.10) by (2.27) and (3.3) we get ‖‖𝑓−ℱ^(𝜑0)𝜎(𝑓)‖‖_{𝑋𝑝(𝜓)}=‖‖𝑓−ℱ^(𝜑0)𝜎(𝑓)‖‖_𝑝+‖‖𝒟(𝜓)𝑓−𝒟(𝜓)ℱ^(𝜑0)𝜎(𝑓)‖‖_𝑝=‖‖𝑓−ℱ^(𝜑0)𝜎(𝑓)‖‖_𝑝+‖‖𝒟(𝜓)𝑓−ℱ^(𝜑0)𝜎(𝒟(𝜓)𝑓)‖‖_𝑝(3.11) for 𝜎≥0. The terms on the right-hand side tend to 0 if 𝜎→+∞ by (2.28). This yields the desired density of 𝒯 in 𝑋_𝑝(𝜓).
Now we prove part (iii). Let 1<𝑝<+∞, 𝜓₁,𝜓₂∈ℋ_𝑠, and let 𝑓 be an arbitrary function in 𝑋_𝑝(𝜓₂). First we observe that in view of (2.34) and (3.1), inequality (2.35) can be rewritten as ‖‖𝒟(𝜓₁)𝑡‖‖_𝑝≤𝑐‖‖𝒟(𝜓₂)𝑡‖‖_𝑝,𝑡∈𝒯.(3.12) It is valid for 1<𝑝<+∞ (see the comment at the end of Section 2). Therefore, we obtain ‖‖𝒟𝜓₁ℱ^(𝜑0)𝑛(𝑓)−ℱ^(𝜑0)𝑚(𝑓)‖‖_𝑝≤𝑐‖‖𝒟𝜓₂ℱ^(𝜑0)𝑛(𝑓)−ℱ^(𝜑0)𝑚(𝑓)‖‖_𝑝=𝑐‖‖ℱ^(𝜑0)𝑛𝒟𝜓₂𝑓−ℱ^(𝜑0)𝑚𝒟𝜓₂𝑓‖‖_𝑝(3.13) for 𝑛,𝑚∈ℕ. Hence, {𝒟(𝜓₁)(ℱ^(𝜑0)𝑛(𝑓))} is a Cauchy sequence in 𝐿_𝑝 by (2.28), and there exists 𝐹∈𝐿_𝑝 such that 𝒟𝜓₁ℱ^(𝜑0)𝑛(𝑓)⟶^𝐿𝑝𝐹(𝑛⟶+∞).(3.14) By the help of (3.3), (2.27), and (3.14) we get 𝐹^∧(𝑘)=lim_𝑛→+∞𝒟𝜓¹ℱ^(𝜑0)𝑛(𝑓)^∧(𝑘)=𝜓₁(𝑘)𝑓^∧(𝑘).(3.15) For each 𝑘∈ℤ^𝑑. Hence, 𝑓 belongs to 𝑋_𝑝(𝜓₁) and 𝒟(𝜓₁)𝑓=𝐹. In order to prove that the embedding 𝑋_𝑝𝜓₂⊂𝑋_𝑝𝜓₁(3.16) is continuous, it is enough to notice that ‖‖𝒟𝜓₁𝑓‖‖_𝑝=lim_𝑛→+∞‖‖𝒟𝜓₁ℱ^(𝜑0)𝑛(𝑓)‖‖_𝑝≤𝑐lim_𝑛→+∞‖‖𝒟𝜓₂ℱ^(𝜑0)𝑛(𝑓)‖‖_𝑝=‖‖𝒟𝜓₂𝑓‖‖_𝑝(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<𝑝<+∞ there is only one space 𝑋^𝑝(𝑠)of 𝑠-smooth functions. It can be characterized as 𝑋^𝑝(𝑠)=𝑓∈𝐿_𝑝∶𝒟(|⋅|^𝑠)𝑓=_{𝑘∈ℤ𝑑}||𝑘||^𝑠𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥∈𝐿_𝑝(3.18) and may be equipped with the norm ‖𝑓‖_𝑋^𝑝(𝑠)=‖𝑓‖_𝑝+‖𝒟(|⋅|^𝑠)𝑓‖_𝑝.(3.19)

Theorem 3.2. Suppose 1≤𝑝≤+∞, 𝜓₁∈ℋ_𝑠1, and 𝜓₂∈ℋ_𝑠2. If 𝑠₁>𝑠₂>0, then 𝑋_𝑝𝜓₁⊂𝑋_𝑝𝜓₂(3.20) and this embedding is continuous.

Proof. Let 𝑓∈𝑋_𝑝(𝜓₁). In view of (2.15) the function 𝜑=((𝜃𝜓₂)/𝜓₁), where 𝜃 is given by (2.14), is infinitely differentiable on ℝ^𝑑 and has compact support. Hence, its Fourier transform belongs to 𝐿₁(ℝ^𝑑) and the operators ℱ^𝜎(𝜑) are uniformly bounded in 𝐿_𝑝 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 ‖‖‖‖^𝑛_𝑗=𝑚𝒟𝜓₂𝑓_𝑗‖‖‖‖_𝑝≤^𝑛_𝑗=𝑚‖‖𝒟𝜓₂𝑓_𝑗‖‖_𝑝=^𝑛_𝑗=𝑚‖‖‖‖_{𝑘∈ℤ𝑑}𝜃2^−𝑗𝑘𝜓₂(𝑘)𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥‖‖‖‖_𝑝=^𝑛_𝑗=𝑚2^{−(𝑠1−𝑠2)𝑗}‖‖‖‖_𝑘≠0𝜑2^−𝑗𝑘𝜓₁(𝑘)𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥‖‖‖‖_𝑝=^𝑛_𝑗=𝑚2^{−(𝑠1−𝑠2)𝑗}‖‖ℱ^2(𝜑)_𝑗𝒟𝜓₁𝑓‖‖_𝑝≤𝑐‖‖𝒟𝜓₁𝑓‖‖_𝑝𝑛_𝑗=𝑚2^{−(𝑠1−𝑠2)𝑗}(3.21) for 𝑛>𝑚≥1. Here, 𝑓_𝑗,𝑗∈ℕ, has the meaning of (2.22). Because of 𝑠₁>𝑠₂ the sequence of the partial sums of the series ∑^+∞𝑗=1𝒟(𝜓₂)𝑓_𝑗 is fundamental in 𝐿_𝑝 and there exists 𝐹∈𝐿_𝑝 such that 𝐹=^𝐿𝑝+∞_𝑗=1𝒟𝜓₂𝑓_𝑗.(3.22) By (2.13) we have 𝜑₀(𝑘)≠0 only for 𝑘=0. Thus, we obtain ^𝐽_𝑗=1𝒟𝜓₂𝑓_𝑗(𝑥)=_{𝑘∈ℤ𝑑}𝜑₀2𝑘^𝐽𝜓²(𝑘)𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥,𝑥∈ℝ^𝑑(3.23) for any 𝐽∈ℕ in analogy to (2.32). Considering the limit process 𝐽→+∞ we find 𝐹^∧(𝜈)=lim_𝐽→+∞^𝐽_𝑗=1𝒟𝜓₂𝑓_𝑗^∧(𝜈)=𝜓₂(𝜈)𝑓^∧(𝜈)(3.24) for any 𝜈∈ℤ^𝑑 with the help of (2.13) and (3.22). Hence, 𝑓 belongs to 𝑋_𝑝(𝜓₂) and 𝒟(𝜓₂)𝑓=𝐹. In order to prove that embedding (3.20) is continuous, it is enough to put 𝑚=1in estimate (3.21) and to consider 𝑛 to +∞. This completes the proof.

Theorem 3.3. Let 1≤𝑝≤+∞ and 𝜓∈ℋ_𝑠, 𝑠>0. Then, 𝒟(𝜓)∶𝑋_𝑝(𝜓)⟶𝐿^0𝑝≡𝑓∈𝐿_𝑝∶𝑓^∧(0)=0(3.25) is a surjective mapping.

Proof. Let 𝑓∈𝐿^0𝑝. We introduce functions 𝑔_𝑗, 𝑗∈ℕ, by setting 𝑔_𝑗(𝑥)=_𝑘≠0𝜃2^−𝑗𝑘(𝜓(𝑘))⁻¹𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥,𝑗∈ℕ,(3.26) 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 ‖‖‖‖^𝑛_𝑗=𝑚𝑔_𝑗‖‖‖‖_𝑝≤^𝑛_𝑗=𝑚‖‖𝑔_𝑗‖‖_𝑝=^𝑛_𝑗=𝑚2^−𝑗𝑠‖‖ℱ^2(𝜑)_𝑗(𝑓)‖‖_𝑝≤𝑐‖𝑓‖_𝑝𝑛_𝑗=𝑚2^−𝑗𝑠(3.27) for 𝑛>𝑚≥1. Hence, there exists 𝑔∈𝐿_𝑝 such that 𝑔=^𝐿𝑝+∞_𝑗=1𝑔_𝑗.(3.28)
Taking into account that 𝜑₀(𝑘)≠0 only for 𝑘=0 in view of (2.13), we obtain analogously to (2.32) ^𝐽_𝑗=1𝑔_𝑗=_𝑘≠0𝜑₀2^−𝐽𝑘(𝜓(𝑘))⁻¹𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥(3.29) by (3.26). As a consequence of (3.28) and (3.29) we get 𝑔^∧(𝜈)=lim_𝐽→+∞^𝐽_𝑗=1𝑔_𝑗^∧(𝜈)=(𝜓(𝜈))⁻¹𝑓^∧(𝜈)(3.30) for 𝜈≠0. Finally, because of 𝑓^∧(0)=0 we obtain 𝑓^∧(𝜈)=𝜓(𝜈)𝑔^∧(𝜈),𝜈∈ℤ^𝑑.(3.31) From (3.30). Hence, 𝑔∈𝑋_𝑝(𝜓), 𝒟(𝜓)𝑔=𝑓, and the proof is complete.

Theorem 3.3 shows that for any 𝜓∈ℋ_𝑠 an operation which is inverse to 𝒟(𝜓) is well defined on 𝐿^0𝑝. The operator 𝐼(𝜓)=(𝒟(𝜓))⁻¹∶𝐿^0𝑝⟶𝑋_𝑝(𝜓)∩𝐿^0𝑝(3.32) is called operator of 𝜓-integration. The Fourier series of the function 𝐼(𝜓)𝑓, 𝑓∈𝐿^0𝑝, is given by _𝑘_≠0(𝜓(𝑘))⁻¹𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥.(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) 𝑚=1 and considering 𝑛 to +∞, we obtain the boundedness of the operator 𝐼(𝜓) in 𝐿^0𝑝.

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 ℋ_𝑠, 𝑠>0. Then, 𝑋₁𝜓₁⊄𝑋₁𝜓₂,𝑋_∞𝜓₁⊄𝑋_∞𝜓₂.(3.34)

Proof. We consider the case 𝑝=1. For 𝑝=+∞ the proof is similar. Let us assume (to the contrary) that 𝑋₁𝜓₁⊂𝑋₁𝜓₂.(3.35) Then, by Theorem 3.3 the operators ℒ_𝑛=ℱ^(𝜑0)𝑛∘𝒟𝜓₂∘𝐼𝜓₁,𝑛∈ℕ,(3.36) where 𝜑₀ has the meaning of Section 2 (see also (2.13)), are well defined on the space 𝐿⁰¹. The operator ℒ_𝑛 acts in accordance with the following chain of mappings and inclusions: 𝐿⟶^{01𝐼𝜓1}𝑋₁𝜓₁⊂𝑋₁𝜓₂⟶^𝒟(𝜓2)𝐿⟶^{01ℱ0)𝑛(𝜑}𝒯_𝑛.(3.37) By (2.23), (2.27), (3.3), and (3.33) we get ℱ^(𝜑0)𝑛∘𝒟𝜓₂∘𝐼𝜓₁(𝑓;𝑥)=_𝑘≠0𝜑₀𝑘𝑛𝜓²(𝑘)𝜓₁(𝑘)⁻¹𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥=(2𝜋)^−𝑑_𝕋𝑑𝑓(𝑥−ℎ)Φ_𝑛(ℎ)𝑑ℎ(3.38) for 𝑛∈ℕ. Here, Φ_𝑛(ℎ)=_𝑘≠0𝜑₀𝑘𝑛𝜓²(𝑘)𝜓₁(𝑘)⁻¹𝑒^𝑖𝑘𝑥,𝑛∈ℕ.(3.39) In view of (3.38) each operator ℒ_𝑛 is bounded in 𝐿⁰¹. Moreover, the boundedness of ℱ^(𝜑0)𝑛 (see Section 2) yields ‖‖ℒ_𝑛(𝑓)‖‖₁≤𝑐‖‖𝒟(𝜓₂)∘𝐼(𝜓₁)𝑓‖‖₁(3.40) for any 𝑓∈𝐿⁰¹. Applying now the Banach-Steinhaus principle we conclude that the operators ℒ_𝑛 are uniformly bounded in 𝐿⁰¹. This leads to the estimate ‖‖‖‖_𝑘≠0𝜑₀𝑘𝑛𝜓²(𝑘)𝜓₁(𝑘)⁻¹𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥‖‖‖‖₁≤𝑐‖𝑓‖₁,(3.41) where the constant 𝑐 does not depend on 𝑓∈𝐿⁰¹ and 𝑛∈ℕ. Let now 𝑡(𝑥)=_{|𝑘|≤𝑚}𝑐_𝑘𝑒^𝑖𝑘𝑥(3.42) 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𝑡‖‖₁≤𝑐‖‖𝒟𝜓₁𝑡‖‖₁.(3.43) 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. 𝜓-Smoothness and Besov and Triebel-Lizorkin Spaces

The aim of this section is to compare the spaces 𝑋_𝑝(𝜓),𝜓∈ℋ_𝑠, with periodic Besov and Triebel-Lizorkin spaces. As we have seen already in part (iii) of Theorem 3.1 there is a unique space 𝑋^𝑝(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<+∞) 𝐻^𝑠𝑝=𝑓∈𝐿_𝑝∶_{𝑘∈ℤ𝑑}1+||𝑘||²^𝑠/2𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥∈𝐿_𝑝(4.1) and equipped with the norm ‖𝑓‖_𝐻^𝑠𝑝=‖‖‖‖_{𝑘∈ℤ𝑑}1+||𝑘||²^𝑠/2𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥‖‖‖‖_𝑝.(4.2) Note that 𝐻^𝑠𝑝=𝑊^𝑠𝑝=𝑓∈𝐿_𝑝∶𝐷^𝛼𝑓∈𝐿_𝑝for|𝛼|≤𝑠(4.3) if 𝑠∈ℕ (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 𝑋^𝑝(𝑠)=𝐻^𝑠𝑝=𝐹^𝑠𝑝,2(4.4) 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 1≤𝑝≤+∞, and let 𝜓∈ℋ_𝑠 for some 𝑠>0. Then, 𝐵^𝑠𝑝,1⊂𝑋_𝑝(𝜓)⊂𝐵^𝑠𝑝,∞(4.5) and these embeddings are continuous.

Proof. Let 𝜂 be an infinitely differentiable positive function defined on ℝ^𝑑 such that 𝜂(𝜉)=1if||𝜉||≤18,𝜂(𝜉)=0if||𝜉||≥14.(4.6) We put 𝜓_∗=𝜓(1−𝜂). By (2.15) we have 𝜓2^−𝑗𝑘𝜃2^−𝑗𝑘=𝜓_∗2^−𝑗𝑘𝜃2^−𝑗𝑘,𝑘∈ℤ^𝑑,𝑗∈ℕ,(4.7) where 𝜃 is given by (2.14).
Let 𝑓 be an arbitrary function in 𝐵^𝑠𝑝,1, and let 𝑓_𝑗, 𝑗∈ℕ₀, 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 ‖‖‖‖^𝑛_𝑗=𝑚𝒟(𝜓)𝑓_𝑗‖‖‖‖_𝑝≤^𝑛_𝑗=𝑚‖‖𝒟(𝜓)𝑓_𝑗‖‖_𝑝=^𝑛_𝑗_=m‖‖‖‖_{𝑘∈ℤ𝑑}𝜓(𝑘)𝜃2^−𝑗𝑘𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥‖‖‖‖_𝑝=^𝑛_𝑗=𝑚2^𝑗𝑠‖‖‖‖_{𝑘∈ℤ𝑑}𝜓_∗2^−𝑗𝑘𝜃2^−𝑗𝑘𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥‖‖‖‖_𝑝≤𝑐^𝑛_𝑗=𝑚2^𝑗𝑠‖‖ℱ^2(𝜃)_𝑗(𝑓)‖‖_𝑝=^𝑛_𝑗=𝑚2^𝑗𝑠‖‖𝑓_𝑗‖‖_𝑝(4.8) for 𝑛>𝑚≥1. In view of (2.18) and (4.8) we can conclude that the series ∑^+∞𝑗=0𝒟(𝜓)𝑓_𝑗 converges in 𝐿_𝑝. Now, by standard arguments we see as in the proof of Theorem 3.1 that 𝑓 belongs to 𝑋_𝑝(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 ℝ^𝑑. By (2.15) we have 𝜓2^−𝑗𝑘𝜃2^−𝑗𝑘=𝜓^∗2^−𝑗𝑘𝜃2^−𝑗𝑘,𝑘∈ℤ^𝑑,𝑗∈ℕ,(4.9) where 𝜃 has the meaning of (2.14).
Let 𝑓 be an arbitrary function in 𝑋_𝑝(𝜓), and let 𝑓_𝑗, 𝑗∈ℕ₀, be given by (2.22). Applying (4.9), the homogeneity property of 𝜓, and (2.27), we obtain 𝑓_𝑗(𝑥)=_{𝑘∈ℤ𝑑}𝜃2^−𝑗𝑘𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥=_{𝑘∈ℤ𝑑}𝜓^∗2^−𝑗𝑘⁻¹𝜓2^−𝑗𝑘𝜃2^−𝑗𝑘𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥=2^−𝑗𝑠_𝑘_∈ℤ𝑑𝜓^∗2^−𝑗𝑘⁻¹𝜓(𝑘)𝜃2^−𝑗𝑘𝑓^∧(𝑘)𝑒^𝑖𝑘𝑥=2^−𝑗𝑠_𝑘_∈ℤ𝑑𝜓^∗2^−𝑗𝑘⁻¹ℱ^2(𝜃)𝑗(𝒟(𝜓)𝑓)^∧(𝑘)𝑒^𝑖𝑘𝑥(4.10) for any 𝑗∈ℕ. 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 ℱ^2(𝜃)_𝑗 in 𝐿_𝑝, we obtain the inequalities ‖‖𝑓_𝑗‖‖_𝑝≤𝑐2^−𝑗𝑠‖‖ℱ^2(𝜃)_𝑗(𝒟(𝜓)𝑓)‖‖_𝑝≤𝑐^2^−𝑗𝑠‖𝒟(𝜓)𝑓‖_𝑝(4.11) for 𝑗∈ℕ from (4.10). Obviously, ‖‖𝑓₀‖‖_𝑝≤𝑐‖𝑓‖_𝑝.(4.12) Hence, by (4.11) and (3.5) ‖𝑓‖_𝐵^𝑠𝑝,∞≤𝑐‖𝑓‖_{𝑋𝑝(𝜓)}(4.13) for some constant 𝑐>0 and all 𝑓∈𝑋_𝑝(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 𝐵^𝑠𝑝,1⊂𝐵^{𝑠𝑝,min(𝑝,2)}⊂𝐹^𝑠𝑝,2⊂𝐵^{𝑠𝑝,max(𝑝,2)}⊂𝐵^𝑠𝑝,∞(4.14) (see [5, Remark  3.5.1/4, page 164]).

5. A Representation Formula for 𝜓-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 𝜓∈ℋ_𝑠. For the sake of simplicity we restrict ourselves to the case 0<𝑠<1.

Let 𝜓∈ℋ_𝑠. Recall that 𝑔∈ℰ_per if and only if ||𝑔^∧(𝑘)||≤𝐶(𝑚)||𝑘||^−𝑚,𝑘≠0,(5.1) for each 𝑚∈ℕ and note that ||𝜓(𝜉)||≤𝑐||𝜉||^𝑠,𝜉∈ℝ^𝑑,(5.2) by homogeneity. Hence, it makes sense to consider the periodic distribution Ψ(𝑥)=_{𝑘∈ℤ𝑑}𝜓(𝑘)𝑒^{−𝑖𝑘𝑥}(5.3) (convergence in ℰ^′_per). Let 𝑔∈ℰ_per. Expansion into the Fourier series leads to the representation Ψ(𝑥)=_{𝜈∈ℤ𝑑}𝜓(𝑥+2𝜋𝜈)(5.4) (convergence in ℰ^′_per). By definition of the Fourier coefficients, the definition of the 𝜓-derivative and, (5.3) we get 𝒟(𝜓)(𝑔)(𝑥)=(2𝜋)^−𝑑⟨Ψ(⋅),𝑔(𝑥+⋅)−𝑔(𝑥)⟩,𝑔∈ℰ_per.(5.5) Using (5.4) and (5.5), we see, at least formally, that 𝒟(𝜓)(𝑔)(𝑥)=(2𝜋)^−𝑑=(2𝜋)_−𝑑_{𝜈∈ℤ𝑑}𝜓(⋅+2𝜋𝜈),𝑔(𝑥+⋅)−𝑔(𝑥)_𝜈_∈ℤ𝑑_𝕋𝑑=(2𝜋)_−𝑑𝜓(𝑦+2𝜋𝜈)(𝑔(𝑥+𝑦)−𝑔(𝑥))𝑑𝑦_ℝ𝑑(𝑔(𝑥+ℎ)−𝑔(𝑥))𝜓(ℎ)𝑑ℎ,𝑔∈ℰ_per.(5.6) We claim that the integral on the right-hand side of (5.6) exists for all 𝑥∈𝕋^𝑑. To this end we first recall that (see Section 2) 𝜓∈𝒮^ is a homogeneous distribution of order −(𝑑+𝑠). Its restriction to ℝ^𝑑⧵{0} can be identified with a function 𝜓∈𝐶^∞(ℝ^𝑑⧵{0}) by [9, Theorem  7.1.18]. Hence, we have the estimate ||𝜓(ℎ)||≤𝑐||ℎ||^{−(𝑑+𝑠)},ℎ∈ℝ^𝑑⧵{0}.(5.7) For brevity we use the standard notation Δ_ℎ𝑔(𝑥)=𝑔(𝑥+ℎ)−𝑔(𝑥). Now, we split _ℝ𝑑Δ_ℎ𝑔(𝑥)𝜓(ℎ)𝑑ℎ=_𝐵1Δ_ℎ𝑔(𝑥)𝜓(ℎ)𝑑ℎ+_{ℝ𝑑⧵𝐵1}Δ_ℎ𝑔(𝑥)𝜓(ℎ)𝑑ℎ.(5.8) The second summand is finite because of (5.7) and the boundedness of |Δ_ℎ𝑔(𝑥)| on ℝ^𝑑. As for the first one we use the estimate ||Δ_ℎ𝑔(𝑥)||≤𝑐^||ℎ||ℎ∈𝐵₁,𝑥∈𝕋^𝑑,(5.9) and (5.7) to see that ||||_𝐵1Δ_ℎ𝑔(𝑥)𝜓(ℎ)𝑑ℎ||||≤𝑐^_𝐵1≤𝑐_||ℎ||||𝜓(ℎ)||𝑑ℎ_𝐵1||ℎ||^{−𝑑+(1−𝑠)}𝑑ℎ<+∞.(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 𝒟(𝜓)(𝑔) holds pointwise almost everyone under much weaker assumptions with respect to 𝑔.

Theorem 5.1. Suppose 1≤𝑝≤+∞ and 𝜓∈ℋ_𝑠, where 0<𝑠<1. Then, 𝒟(𝜓)(𝑓)(𝑥)=(2𝜋)^−𝑑_ℝ𝑑(𝑓(𝑥+ℎ)−𝑓(𝑥))𝜓(ℎ)𝑑ℎ,𝑓∈𝐵^𝑠𝑝,1,(5.11) for almost all 𝑥∈𝕋^𝑑 (for all, if 𝑝=+∞).

Proof. First we recall that 𝐵^𝑠𝑝,1⊂𝑋_𝑝(𝜓) by Theorem 4.2. Hence, the left-hand side of (5.11) makes sense. Let 𝑓∈𝐵^𝑠𝑝,1. As is well known (see, e.g., [5, Theorem  3.5.4]), ‖𝑓‖^∗𝐵_𝑠𝑝,1=‖𝑓‖_𝑝+_ℝ𝑑||ℎ||^−𝑠‖‖Δ_ℎ𝑓‖‖_𝑝𝑑ℎ||ℎ||^𝑑(5.12) is an equivalent norm in the Besov space 𝐵^𝑠𝑝,1. 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 ‖𝐼‖_𝑝≤_ℝ𝑑‖‖Δ_ℎ𝑓(𝑥)‖‖_𝑝⋅||𝜓(ℎ)||𝑑ℎ≤𝑐_ℝ𝑑‖‖Δ_ℎ𝑓‖‖_𝑝⋅||ℎ||^{−(𝑑+𝑠)}𝑑ℎ≤𝑐‖𝑓‖^∗𝐵_𝛼𝑝,1.(5.13) Hence, the function 𝐼(𝑥) belongs to 𝐿_𝑝. 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)^−𝑑_𝕋𝑑_ℝ𝑑=𝑓_∧(𝑓(𝑥+ℎ)−𝑓(𝑥))𝜓(ℎ)𝑑ℎ𝑒^{−𝑖𝑘𝑥}𝑑𝑥(𝑘)_ℝ𝑑𝑒^𝑖𝑘ℎ−1𝜓(ℎ)𝑑ℎ(5.14) by Fubini's theorem. It remains to show that _ℝ𝑑𝑒^𝑖𝑘ℎ−1𝜓(ℎ)𝑑ℎ=(2𝜋)^−𝑑𝜓(𝑘),𝑘∈ℤ^𝑑.(5.15) This is obvious if 𝑘=0 because of 𝜓(0)=0. If 𝑘≠0, we have to use appropriate limiting arguments to circumvent the difficulty caused by the fact that 𝜓 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 ℝ^𝑑⧵{0} can be identified with (|⋅|^𝑠)^∧(𝑥)=𝑐(𝑑,𝑠)|𝑥|^{−𝑑−𝑠},𝑐(𝑑,𝑠)=2^𝑑+𝑠𝜋^𝑑/2Γ(𝑠/2+𝑑/2)Γ(−𝑠/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]) 𝑓^⟨𝑠⟩(𝑥)=(2𝜋)⁻¹𝑐(1,𝑠)^+∞−∞𝑓(𝑥+ℎ)−𝑓(𝑥)||ℎ||^𝑠+1𝑑ℎ,𝑓∈𝐵^𝑠𝑝,1,(5.17) which is valid for 0<𝑠<1 and 1≤𝑝≤+∞. 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,𝑓∈𝐵^𝑠𝑝,1(5.18) for the fractional power of the Laplace operator.

Let us mention that formulas for 𝜓-derivatives with 𝑠≥1 can be achieved using differences of higher order. Suppose, for example, that 1≤𝑠<2 and, in addition to the previous conditions, that 𝜓 is real valued. Analogously to (5.6) we find 𝒟(𝜓)(𝑔)(𝑥)=(2𝜋)^−𝑑_ℝ𝑑𝑔(𝑥+ℎ)−2𝑔(𝑥)+𝑔(𝑥−ℎ)2𝜓(ℎ)𝑑ℎ,(5.19) and, in particular, (−Δ)^𝑠/2𝑔(𝑥)=(2𝜋)^−𝑑𝑐(𝑑,𝑠)_ℝ𝑑𝑔(𝑥+ℎ)−2𝑔(𝑥)+𝑔(𝑥−ℎ)2||ℎ||^𝑑+𝑠𝑑ℎ(5.20) for 𝑔∈ℰ_per. 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 𝐵^𝑠𝑝,1.

Acknowledgment

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