Revistes Catalanes amb Accés Obert (RACO)

On $L^p$ estimates for square roots of second order elliptic operators on $\mathbb{R}^n$

Pascal Auscher

Resum


We prove that the square root of a uniformly complex elliptic operator $L=-\operatorname{div}(A\nabla)$ with bounded measurable coefficients in $\mathbb{R}^n$ satisfies the estimate $\| L^{1/2} f\|_p \lesssim \| \nabla f\|_p$ for $\sup(1,\frac{2n}{n+4}- \varepsilon) < p < \frac{2n}{n-2} + \varepsilon $, which is new for $n \ge 5$ and $p < 2$ or for $n \ge 3$ and $p > \frac{2n}{n-2}$. One feature of our method is a Calderóon-Zygmund decomposition for Sobolev functions. We make some further remarks on the topic of the converse $L^p$ inequalities (i.e. Riesz transforms bounds), pushing the recent results of [BK2] and [HM] for $\frac{2n}{n+2} < p < 2$ when $n\ge 3$ to the range $\sup(1,\frac{2n}{n+2}- \varepsilon) < p < 2+\varepsilon'$. In particular, we obtain that $L^{1/2}$ extends to an isomorphism from $\dot W^{1,p} (\mathbb{R}^n)$ to $L^p (\mathbb{R}^n)$ for $p$ in this range. We also generalize our method to higher order operators.

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