Bounds for partial derivatives: necessity of UMD and sharp constants

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 ${\mathrm{\partial }}^{\beta }u$ of $u\in C^\infty_c($\mathbb{R}$^n,X)$\mathrm{.}Inparticular,weshowthattheestimate$|u_{xy}|p\leq K(|u{xx}|p+|u{yy}|_p)$ 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.