A family of interpolation inequalities involving products of low-order derivatives

Abstract

Gagliardo-Nirenberg interpolation inequalities relate Lebesgue norms of iterated derivatives of a function. We present a generalization of these inequalities in which the low-order interpolation factor of the right-hand side is replaced by a Lebesgue norm of a pointwise product of derivatives of the function. We provide a short, independent proof, recovering the usual one-dimensional inequality as a particular case. We give examples of applications to control theory and PDE analysis, and discuss related open problems.

Major simplification of the proof