Two weighted inequalities for operators associated to a critical radius function

Abstract

In the general framework of ${\mathbb{R}}^d$ equipped with Lebesgue measure and a critical radius function, we introduce several Hardy–Littlewood type maximal operators and related classes of weights. We prove appropriate two weighted inequalities for such operators as well as a version of Lerner’s inequality for a product of weights. With these tools we are able to prove factored weight inequalities for certain operators associated to the critical radius function. As it is known, the harmonic analysis arising from the Schrödinger operator $L=-\Delta +V$, as introduced by Shen, is based on the use of a related critical radius function. When our previous result is applied to this case, it allows to show some inequalities with factored weights for all first and second order Schrödinger–Riesz transforms.