Fractional integrals and hypersingular integrals in variable order Hölder spaces on homogeneous spaces - Natasha Samko; Stefan Samko; Boris Vakulov (Journal of Function Spaces and Applications)
We consider non-standard Hölder spaces Hλ(⋅)(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 Hλ(⋅)(X), but also the generalized Hölder spaces Hw(⋅,⋅)(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).