Weighted $L^p$ Estimates of Kato Square Roots Associated to Degenerate Elliptic Operators

TL;DR

This paper establishes weighted $L^p$ estimates for the square roots of degenerate elliptic operators associated with Muckenhoupt weights, extending classical results to a weighted setting with degenerate coefficients.

Contribution

The authors prove new weighted $L^p$ bounds for Kato square roots of degenerate elliptic operators with Muckenhoupt weights, covering a range of $p$ values and generalizing previous unweighted results.

Findings

Weighted $L^p$ estimates hold for $p$ in $(rac{2n}{n+1}, 2]$

Square root estimates are equivalent to gradient norms in weighted spaces

Results extend classical Kato square root theory to degenerate elliptic operators

Abstract

Let $w$ be a Muckenhoupt $A_2(\mathbb{R}^n)$ weight and $L_w:=-w^{-1}\mathop\mathrm{div}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$, $n\geq 2$. In this article, the authors establish some weighted $L^p$ estimates of Kato square roots associated to the degenerate elliptic operators $L_w$. More precisely, the authors prove that, for $w\in A_{p}(\mathbb{R}^n)$, $p\in(\frac{2n}{n+1},\,2]$ and any $f\in C^\infty_c(\mathbb{R}^n)$, $\|L_w^{1/2}(f)\|_{L^p(w,\,\mathbb{R}^n)}\sim \|\nabla f\|_{L^p(w,\,\mathbb{R}^n)}$, where $C_c^\infty(\mathbb{R}^n)$ denotes the set of all infinitely differential functions with compact supports.