On the weighted generalized H\"older property of a hypersingular integral on a metric space

Abstract

The weighted Zygmund-type estimates for a hypersingular integral on an almost homogeneous metric space are obtained and, on their basis, theorems on the action of this operator in the weighted generalized variable Hölder space are proved. It is shown that the hypersingular integral "worsens" the characteristic of the generalized Hölder space by the order of the hypersingular integral. The conditions of the presented theorems are formulated in terms of the Zygmund--Bary--Stechkin
class and a special analogue of the Dini condition.