# $L^p$ bounds for Riesz transforms and square roots associated to second order elliptic operators

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## Abstract

We consider the Riesz transforms $\nabla L^{-1/2}$, where $L \equiv - \operatorname{div} A(x) \nabla$, and $A$ is an accretive, $n \times n$ matrix with bounded measurable complex entries, defined on $\mathbb{R}^n$. We establish boundedness of these operators on $L^p(\mathbb{R}^n)$, for the range $p_n < p \leq 2$, where $p_n = 2n/(n+2)$, $n\geq 2$, and we obtain a weak-type estimate at the endpoint $p_n$. The case $p=2$ was already known: it is equivalent to the solution of the square root problem of T. Kato.

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