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Differential operators generated by homogeneous functions

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

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 [

In the present paper, we will introduce and study operators

Let us also mention that it is aimed to extend the approach to generalized smoothness based on the introduction of operators

finding representation formulae for operators generated by homogeneous functions of degree

studying generalized

constructing new moduli of smoothness related to

studying the same concept in nonperiodic case,

investigating generalized differential equations.

The paper is organized as follows. Section

By the symbols

As usual,

The Fourier coefficients of

The Fourier transform and its inverse are given by

Let

A complex-valued function

Let now

It follows that

The Fourier analytical definition is based on dyadic resolutions of unity (see, e.g., [

Let

Let

The set of the norms

for

if

if

If

Let

Let

We give some examples. Let

In the multivariate case (

Now we study the properties of the spaces

Let

all spaces

In order to prove (i) we only have to check completeness because all other properties of Banach space are obviously fulfilled. Let

In order to show part (ii) we consider the Fourier means

Now we prove part (iii). Let

In other words part (iii) of the theorem means that in the case

Suppose

Let

Let

Let

Taking into account that

Theorem

With the help of Theorem

Suppose that

We consider the case

The aim of this section is to compare the spaces

Let

Both statements are known. The second identity can be found in [

Next we consider the cases

Let

Let

Let

In order to prove the second embedding we introduce an infinitely differentiable function

Let

Having in mind Theorem

The following observations give some motivation and pave the way to find explicit representations of the operator

Let

Suppose

First we recall that

We give some remarks. It is known (see [

Let us mention that formulas for

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