Bounds for partial derivatives: necessity of UMD and sharp constants

摘要

We prove the necessity of the UMD condition, with a quantitative estimate of the UMD constant, for any inequality in a family of bounds between different partial derivatives of . In particular, we show that the estimate 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.