The Plemelj-Privalov theorem in variable exponent Clifford analysis

Let Ω be a bounded oriented connected open subset of ℝ ⁿ whose boundary is a compact topological surface Γ. We consider non-standard generalized Hölder spaces, denoted by C ^0,ω(·)(Γ), of Clifford algebra-valued functions u, whose local modulus of continuity ω(u, x, t) in the variable t for each x ∈ Γ has a majorant ω(x, t), 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 C ^0,ω(·)(Γ), when Γ is an Ahlfors-David regular surface of ℝ ⁿ. This can be viewed as the Plemelj-Privalov theorem on generalized Hölder spaces in the variable exponent Clifford analysis setting.