Kato's square root problem in Banach spaces

Abstract

Let $L$ be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces $L^p(R^n;X)$ of $X$-valued functions on $R^n$. We characterize Kato's square root estimates $\Vert \sqrt{L}u{\Vert }_p\eqsim \Vert \nabla u{\Vert }_p$ and the $H^{\infty }$-functional calculus of $L$ in terms of R-boundedness properties of the resolvent of $L$, when $X$ is a Banach function lattice with the UMD property, or a noncommutative $L^p$ space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case $X=C$, we get a new approach to the $L^p$ theory of square roots of elliptic operators, as well as an $L^p$ version of Carleson's inequality.

44 pages