Pointwise multiplication on vector-valued function spaces with power weights

摘要

We investigate pointwise multipliers on vector-valued function spaces over$\mathbb{R}^d$, equipped with Muckenhoupt weights. The main result is that inthe natural parameter range, the characteristic function of the half-space is apointwise multiplier on Bessel-potential spaces with values in a UMD Banachspace. This is proved for a class of power weights, including the unweightedcase, and extends the classical result of Shamir and Strichartz. Themultiplication estimate is based on the paraproduct technique and a randomizedLittlewood-Paley decomposition. An analogous result is obtained for Besov andTriebel-Lizorkin spaces.