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arXiv · 1406.2832

Bounds for partial derivatives: necessity of UMD and sharp constants

Alejandro J. Castro·Tuomas P. Hytönen
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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.

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Alejandro J. Castro, Tuomas P. Hytönen. 2014-06-11. Bounds for partial derivatives: necessity of UMD and sharp constants. https://doi.org/10.1007/s00209-015-1556-y

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