Positive operators and maximal operators in a filtered measure space

TL;DR

This paper characterizes weights for positive and maximal operators in filtered measure spaces using discrete Wolff's potential, introduces a refined Carleson embedding theorem, and establishes Sawyer-type and Hytönen-Pérez type inequalities.

Contribution

It provides new characterizations and refined theorems for weighted inequalities involving positive and maximal operators in filtered measure spaces.

Findings

Characterization of weights via discrete Wolff's potential.

Refinement of the Carleson embedding theorem.

Hytönen-Pérez type one-weight norm estimate obtained.

Abstract

In a filtered measure space, a characterization of weights for which the trace inequality of a positive operator holds is given by the use of discrete Wolff's potential. A refinement of the Carleson embedding theorem is also introduced. Sawyer type characterization of weights for which a two-weight norm inequality for a generalized Doob's maximal operator holds is established by an application of our Carleson embedding theorem. Moreover, Hyt\"{o}nen-P\'{e}rez type one-weight norm estimate for Doob's maximal operator is obtained by the use of our two-weight characterization.