Pointwise Multiplication on Vector-Valued Function Spaces with Power Weights

Abstract

We investigate pointwise multipliers on vector-valued function spaces over \({\mathbb {R}}^d\), equipped with Muckenhoupt weights. The main result is that in the natural parameter range, the characteristic function of the half-space is a pointwise multiplier on Bessel-potential spaces with values in a UMD Banach space. This is proved for a class of power weights, including the unweighted case, and extends the classical result of Shamir and Strichartz. The multiplication estimate is based on the paraproduct technique and a randomized Littlewood–Paley decomposition. An analogous result is obtained for Besov and Triebel–Lizorkin spaces.