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.

Minor revision. Accepted for publication in J. Fourier Anal. Appl

References in corpus (2)

• On the trace problem for Triebel--Lizorkin spaces with mixed norms
• A Representation Theorem for Singular Integral Operators on Spaces of Homogeneous Type