AbstractWe consider non‐standard generalized Hölder spaces of functionsfon 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 ωα(x,h) =hℜα(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