Weighted Hardy spaces associated with elliptic operators. Part I: Weighted norm inequalities for conical square functions

TL;DR

This paper establishes weighted norm inequalities for conical square functions linked to elliptic operators, identifying natural classes of weights and developing sharp change of angle formulas, advancing harmonic analysis in weighted spaces.

Contribution

It introduces new weighted inequalities and comparison techniques for conical square functions associated with elliptic operators, extending the theory to weighted Lebesgue spaces.

Findings

Identified classes of Muckenhoupt weights where square functions are comparable and bounded.

Developed sharp weighted change of angle formulas for conical square functions.

Generalized Carleson measure condition for estimates below p=2.

Abstract

This is the first part of a series of three articles. In this paper, we obtain weighted norm inequalities for different conical square functions associated with the Heat and the Poisson semigroups generated by a second order divergence form elliptic operator with bounded complex coefficients. We find classes of Muckenhoupt weights where the square functions are comparable and/or bounded. These classes are natural from the point of view of the ranges where the unweighted estimates hold. In doing that, we obtain sharp weighted change of angle formulas which allow us to compare conical square functions with different cone apertures in weighted Lebesgue spaces. A key ingredient in our proofs is a generalization of the Carleson measure condition which is more natural when estimating the square functions below $p=2$.