Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of $H^1_L(w)$

TL;DR

This paper explores various characterizations of weighted Hardy spaces associated with elliptic operators, establishing their equivalence and molecular structure, extending unweighted theories to weighted contexts.

Contribution

It introduces new isomorphism results and molecular characterizations for weighted Hardy spaces linked to elliptic operators, broadening the scope of existing unweighted theories.

Findings

All definitions of $H^1_L(w)$ are isomorphic.

$H^1_L(w)$ admits a molecular characterization.

Weighted norm inequalities are established for maximal functions.

Abstract

Given a Muckenhoupt weight $w$ and a second order divergence form elliptic operator $L$, we consider different versions of the weighted Hardy space $H^1_L(w)$ defined by conical square functions and non-tangential maximal functions associated with the heat and Poisson semigroups generated by $L$. We show that all of them are isomorphic and also that $H^1_L(w)$ admits a molecular characterization. One of the advantages of our methods is that our assumptions extend naturally the unweighted theory developed by S. Hofmann and S. Mayboroda and we can immediately recover the unweighted case. Some of our tools consist in establishing weighted norm inequalities for the non-tangential maximal functions, as well as comparing them with some conical square functions in weighted Lebesgue spaces.