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2017 Weighted $L^p$ Estimates of Kato Square Roots Associated to Degenerate Elliptic Operators
Dachun Yang, Junqiang Zhang
Publ. Mat. 61(2): 395-444 (2017). DOI: 10.5565/PUBLMAT6121704


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 $f$.


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Dachun Yang. Junqiang Zhang. "Weighted $L^p$ Estimates of Kato Square Roots Associated to Degenerate Elliptic Operators." Publ. Mat. 61 (2) 395 - 444, 2017.


Received: 3 September 2015; Revised: 18 January 2016; Published: 2017
First available in Project Euclid: 29 June 2017

zbMATH: 06781947
MathSciNet: MR3677867
Digital Object Identifier: 10.5565/PUBLMAT6121704

Primary: 47B06
Secondary: 35J70 , 42B30 , 42B35 , 46E30

Keywords: degenerate elliptic operator , Hardy space , Kato square root , Lebesgue space , Muckenhoupt weight , Riesz transform , square function

Rights: Copyright © 2017 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.61 • No. 2 • 2017
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