Estimates for partial derivatives of vector-valued functions

Abstract

An upper bound for $\|D^{\beta}u\|_q$ in terms of other similar norms $\|D^{\alpha}u\|_p$ is derived for vector-valued test functions $u\in C_c^{\infty}(\mathbf{R}^n,X)$, where $X$ is a Banach space with the UMD property. This gives a new proof and an extension of a classical result of Besov-Il'in-Nikol'skiĭ for scalar functions.