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 $L_w:=-w^{-1}\mathrm{div}(A\nabla )$ the degenerate elliptic operator on the Euclidean space ${\mathbb{R}}^n$, $n\ge 2$. 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 $f\in C_c^{\infty }({\mathbb{R}}^n)$, $\Vert L_w^{1/2}(f){\Vert }_{L^p(w,{\mathbb{R}}^n)}\sim \Vert \nabla f{\Vert }_{L^p(w,{\mathbb{R}}^n)}$, where $C_c^{\infty }({\mathbb{R}}^n)$ denotes the set of all infinitely differential functions with compact supports.

40 pages, Submitted

References in corpus (2)

• Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators
• Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators