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

摘要

We consider non-standard generalized Hölder spaces of functions f on the unit sphere in , whose local continuity modulus Ω(f, x, h) at a point 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 (x), from such a variable generalized Hölder space to another one with a “better” dominant ω(x, h) = h(x)ω(x, h), and similar mapping properties of spherical hypersingular integrals of variable order (x) from such a space into the space with “worse” dominant ω₋(x, h) = h−(x)ω(x, h). We admit variable complex valued orders (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 (x). © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim