Bounds for partial derivatives: necessity of UMD and sharp constants

Abstract

We prove the necessity of the UMD condition, with a quantitative estimate of the UMD constant, for any inequality in a family of $L^p$ bounds between different partial derivatives ${\partial }^{\beta }u$ of $u\in C_c^{\infty }({\mathbb{R}}^n,X)$. In particular, we show that the estimate $\Vert u_{xy}{\Vert }_p\le K(\Vert u_{xx}{\Vert }_p+\Vert u_{yy}{\Vert }_p)$ characterizes the UMD property, and the best constant $K$ is equal to one half of the UMD constant. This precise value of $K$ seems to be new even for scalar-valued functions.

arXiv commentv2: corrected typo in the references