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 \(\partial ^\beta u\) of \(u \in C^\infty _c({\mathbb {R}}^n,X)\). 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.