Fractional integrals and hypersingular integrals in variable order Hölder spaces on homogeneous spaces

Abstract

We consider non-standard Hölder spaces H λ(̇)(X) of functions f on a metric measure space (X, d, μ), whose Hölder exponent λ(x) is variable, depending on x ∈ X. We establish theorems on mapping properties of potential operators of variable order α(x), from such a variable exponent Hölder space with the exponent λ(x) to another one with a "better" exponent λ(x) + α(x), and similar mapping properties of hypersingular integrals of variable order α(x) from such a space into the space with the "worse" exponent λ(x) - α(x) in the case α(x) < λ(x). These theorems are derived from the Zygmund type estimates of the local continuity modulus of potential and hypersingular operators via such modulus of their densities. These estimates allow us to treat not only the case of the spaces H λ(̇)(X), but also the generalized Hölder spaces H w(̇,̇)(X) of functions whose continuity modulus is dominated by a given function w(x, h), x ∈ X, h > 0. We admit variable complex valued orders α(x), where Rα(x) 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). Copyright © 2010 Hindawi Publishing Corporation.