Conical square functions for degenerate elliptic operators

TL;DR

This paper investigates the boundedness of conical square functions related to degenerate elliptic operators in weighted spaces, extending previous results on the Kato problem and functional calculus to new classes of weights.

Contribution

It establishes $L^p(w)$ and $L^p(v dw)$ boundedness of conical square functions for degenerate elliptic operators, identifying new degeneracy weights where $L^2$ estimates hold.

Findings

Identified a class of degeneracy weights for bounded conical square functions.

Extended $L^p$ boundedness results to weighted and degenerate settings.

Opened pathways for Hardy space and boundary value problem analysis.

Abstract

The aim of this paper is to study the boundedness of different conical square functions that arise naturally from second order divergence form degenerate elliptic operators. More precisely, let $L_w=w^{-1}\,{\rm div}(w\,A\,\nabla)$ where $w\in A_2$ and $A$ is an $n\times n$ bounded, complex-valued, uniformly elliptic matrix. D. Cruz-Uribe and C. Rios solved the $L^2(w)$-Kato square root problem obtaining that $\sqrt{L_w}$ is equivalent to the gradient on $L^2(w)$. The same authors in collaboration with the second named author of this paper studied the $L^p(w)$-boundedness of operators that are naturally associated with $L_w$, such as the functional calculus, Riesz transforms, or vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in $L^p(v dw)$ for $v\in A_\infty(w)$), and in particular a class of "degeneracy" weights $w$ was found in such a way that the classical $L^2$-Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on $L^p(w)$ and on $L^p(v dw)$, with $v\in A_\infty(w)$, of the conical square functions that one can construct using the heat or Poisson semigroup associated with $L_w$. As a consequence of our methods, we find a class of degeneracy weights $w$ for which $L^2$-estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with $L_w$.