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

Abstract

Let $w$ be a Muckenhoupt $A_2({\mathbb R}^n)$ weight and $L_w:=-w^{-1}\operatorname{div}(A\nabla)$ 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 and the implicit equivalent positive constants are independent of $f$.