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

Abstract

Given a Muckenhoupt weight $w$ and a second order divergence form elliptic operator $L$, we consider different versions of the weighted Hardy space $H_L^1(w)$ 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 $H_L^1(w)$ 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.

The introduction has been updated to include some relevant references describing the development of the topic

References in corpus (3)

• Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates
• Weighted Hardy spaces associated with elliptic operators. Part I: Weighted norm inequalities for conical square functions
• Weighted Hardy spaces associated with elliptic operators. Part III: Characterizations of $H_L^p(w)$ and the weighted Hardy space associated with the Riesz transform

Cited by in corpus (7)

• Weighted Hardy spaces associated with elliptic operators. Part I: Weighted norm inequalities for conical square functions
• Weighted Hardy spaces associated with elliptic operators. Part III: Characterizations of $H_L^p(w)$ and the weighted Hardy space associated with the Riesz transform
• A PDE Characterization of Anisotropic Hardy Spaces
• Weak and strong types estimates for square functions associated with operators
• The regularity problem for uniformly elliptic operators in weighted spaces
• Vertical square functions and other operators associated with an elliptic operator
• The regularity problem for degenerate elliptic operators in weighted spaces