Reduced inequalities for vector-valued functions

Abstract

Building on the notion of convex body domination introduced by Nazarov, Petermichl, Treil, and Volberg, we provide a general principle of bootstrapping bilinear estimates for scalar-valued functions into vector-valued versions with a reduced right-hand side involving iterated norms of a pointwise dot product $\vec f(x)\cdot\vec g(y)$ instead of the product of lengths $|\vec f(x)| |\vec g(y)|$ that would result from a naïve extension of the scalar inequality. On the way, we study connections between convex body domination and tensor norms. In order to cover the full regime of $L^p$ norms, also with $p<1$, that naturally arise in bilinear harmonic analysis, we develop a framework in general quasi-normed spaces. A key application is a vector-valued Kato-Ponce inequality (or fractional Leibnitz rule) with a reduced right-hand side, which we obtain as a soft corollary of the known scalar-valued version and our general bootstrapping method.