Bounds for partial derivatives: necessity of UMD and sharp constants

TL;DR

This paper establishes that the UMD property is necessary for certain $L^p$ bounds between partial derivatives of smooth functions, providing a precise relation between the best constant and the UMD constant.

Contribution

It demonstrates the necessity of the UMD condition for $L^p$ bounds of partial derivatives and determines the exact constant relating these bounds to the UMD constant.

Findings

The estimate $\|u_{xy}\|_p \leq K(\|u_{xx}\|_p + \\|u_{yy}\|_p)$ characterizes the UMD property.

The best constant $K$ equals half of the UMD constant.

The precise value of $K$ is new even for scalar-valued functions.

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 $u\in C^\infty_c(\mathbb{R}^n,X)$. In particular, we show that the estimate $\|u_{xy}\|_p\leq K(\|u_{xx}\|_p+\|u_{yy}\|_p)$ 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.