Spherical fractional and hypersingular integrals of variable order in generalized Hölder spaces with variable characteristic
✍ Scribed by Natasha Samko; Boris Vakulov
- Publisher
- John Wiley and Sons
- Year
- 2011
- Tongue
- English
- Weight
- 174 KB
- Volume
- 284
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Abstract
We consider non‐standard generalized Hölder spaces of functions f on the unit sphere \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}${\mathbb S}^{n-1} $\end{document} in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}${\mathbb R}^n $\end{document}, whose local continuity modulus Ω(f, x, h) at a point \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$x\in {\mathbb S}^{n-1} $\end{document} has a dominant ω(x, h) which may vary from point to point. We establish theorems on the mapping properties of spherical potential operators of variable order α(x), from such a variable generalized Hölder space to another one with a “better” dominant ω~α~(x, h) = h^ℜα(x)^ω(x, h), and similar mapping properties of spherical hypersingular integrals of variable order α(x) from such a space into the space with “worse” dominant ω~−α~(x, h) = h^−ℜα(x)^ω(x, h). We admit variable complex valued orders α(x) which 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). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim