Two weight Sobolev norm inequalities for fractional vector Riesz transforms and doubling weights

TL;DR

This paper establishes a T1 theorem for fractional vector Riesz transforms between weighted Sobolev spaces with doubling measures, characterizing boundedness via classical A_2 and testing conditions, and shows norm equivalences in this setting.

Contribution

It introduces a new T1 theorem for fractional vector Riesz transforms on weighted Sobolev spaces with doubling measures, including norm equivalence results.

Findings

Boundedness characterized by A_2 and testing conditions

Equivalence of weighted Sobolev norms for doubling measures

Extension of classical harmonic analysis results to Sobolev spaces

Abstract

We prove a T1 theorem for fractional vector Riesz transforms mapping one weighted Sobolev space to another, where the weights are doubling measures on Euclidean space. Boundedness is characterized by the classical A_2 condition and two dual testing conditions on indicators of cubes. We also show the equivalence of various weighted Sobolev norms when the measure is doubling, something that fails in general.