WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON $\mathbb R^n$

doi:10.1007/s10473-022-0404-5

Abstract

Abstract: Let $L=-\mathrm{div}(A\nabla)$ be a second order divergence form elliptic operator with bounded measurable coefficients in ${\Bbb R}^n$. We establish weighted $L^p$ norm inequalities for commutators generated by $\sqrt{L}$ and Lipschitz functions, where the range of $p$ is different from $(1,\infty)$, and we isolate the right class of weights introduced by Auscher and Martell. In this work, we use good-$\lambda$ inequality with two parameters through the weighted boundedness of Riesz transforms $\nabla L^{-1/2}$. Our result recovers, in some sense, a previous result of Hofmann.

Key words: Muckenhoupt weights, commutator, Kato square root, Lipschitz function, elliptic operators

CLC Number:

42B20

Yanping CHEN, Yong DING, Kai ZHU. WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON $\mathbb R^n$[J].Acta mathematica scientia,Series B, 2022, 42(4): 1310-1332.

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