Localization of a Class of Muckenhoupt Weights by Using Mellin Pseudo-Differential Operators

Abstract

Let Γ be a finite or infinite interval of \(\mathbb R\), p ∈ (1, ∞), and let w ∈ Ap( Γ) be a Muckenhoupt weight. Relations of the weighted singular integral operator \(wS_{\mathbb{R}_+}w^{-1}I\) on the space \(L^p(\mathbb{R}_{+})\) and Mellin pseudo-differential operators with non-regular symbols are studied. A localization of a class of Muckenhoupt weights to power weights at finite endpoints of Γ, which is related to the Allan-Douglas local principle, is obtained by using quasicontinuous functions and Mellin pseudo-differential operators with non-regular symbols.