Reduced inequalities for vector-valued functions

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 f (x) g (y) instead of the product of lengths | f (x) | | g (y) | that would result from a na\"ive 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 Lᵖ 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.