Journal of Function Spaces

Journal of Function Spaces / 2010 / Article

Open Access

Volume 8 |Article ID 659456 | 30 pages | https://doi.org/10.1155/2010/659456

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

Academic Editor: Vakhtang Kokilashvili
Received01 Feb 2009

Abstract

We consider non-standard Hölder spaces Hλ()(X) of functions f on a metric measure space (X, d, μ), whose Hölder exponent λ(x) is variable, depending on xX. We establish theorems on mapping properties of potential operators of variable order α(x), from such a variable exponent Hölder space with the exponent λ(x) to another one with a “better” exponent λ(x) + α(x), and similar mapping properties of hypersingular integrals of variable order α(x) from such a space into the space with the “worse” exponent λ(x) − α(x) in the case α(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 spaces Hλ()(X), but also the generalized Hölder spaces Hw(,)(X) of functions whose continuity modulus is dominated by a given function w(x, h), xX, h > 0. We admit variable complex valued orders α(x), where α(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 weight α(x).

Copyright © 2010 Hindawi Publishing Corporation. 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.

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