Fractional integrals and hypersingular integrals in variable order Hölder spaces on homogeneous spaces

Natasha Samko,Stefan Samko,Boris Vakulov

doi:10.1155/2010/659456

Natasha Samko, Stefan Samko + Show 1 more

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https://doi.org/10.1155/2010/659456

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Abstract

We consider non-standard Hölder spacesHλ(⋅)(X)of functionsfon a metric measure space (X, d, μ), whose Hölder exponentλ(x) is variable, depending onx∈X. We establish theorems on mapping properties of potential operators of variable orderα(x), from such a variable exponent Hölder space with the exponentλ(x) to another one with a “better” exponentλ(x) +α(x), and similar mapping properties of hypersingular integrals of variable orderα(x) from such a space into the space with the “worse” exponentλ(x) −α(x) in the caseα(x) <λ(x). These theorems are derived from the Zygmund type estimates of the local continuity modulus of potential and hypersingular operators via such modulus of their densities. These estimates allow us to treat not only the case of the spacesHλ(⋅)(X), but also the generalized Hölder spacesHw(⋅,⋅)(X)of functions whose continuity modulus is dominated by a given functionw(x, h),x∈X, h> 0. We admit variable complex valued ordersα(x), whereℜα(x)may vanish at a set of measure zero. To cover this case, we consider the action of potential operators to weighted generalized Hölder spaces with the weightα(x).

Highlights

Last decade, there was a strong rise of increase of interest to studies of variable spaces, when the parameters defining the space, which are usually constant, may vary from point to point
A typical example is a generalized Lebesgue space with variable exponent defined by the modular |f (x)|p(x) dx, or more generally, Musielak-Orlicz spaces with the Young function varying from point to point
Within the frameworks of the Holder spaces Hλ(·)(Ω) with a variable exponent λ(x) and more general spaces Hw(·,·)(X) with a given variable dominant of continuity modulus of functions, we study mapping properties of potential operators of the form

Summary

Introduction

There was a strong rise of increase of interest to studies of variable spaces, when the parameters defining the space, which are usually constant, may vary from point to point. In the general setting of quasimetric measure spaces (X, , μ) with growth condition, mapping properties of the operators Iα and Dα in Holder spaces Hλ(X) were studied, in the case of constant λ and constant real α , in [4], [5], [6], [7].

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