Spherical fractional and hypersingular integrals of variable order in generalized Hölder spaces with variable characteristic

Abstract

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