The Plemelj–Privalov theorem in variable exponent Clifford analysis

Abstract

Let be a bounded oriented connected open subset of whose boundary is a compact topological surface . We consider non-standard generalized Hölder spaces, denoted by , of Clifford algebra-valued functions u, whose local modulus of continuity in the variable t for each has a majorant , which may vary from point to point. The main purpose of this paper is to prove the boundedness of the Clifford singular integral operator in the spaces , when is an Ahlfors–David regular surface of ℝn. This can be viewed as the Plemelj–Privalov theorem on generalized Hölder spaces in the variable exponent Clifford analysis setting.