New thoughts on the vector-valued Mihlin-H\"ormander multiplier theorem

TL;DR

This paper extends the vector-valued Mihlin-H"ormander multiplier theorem to UMD spaces with specific type and cotype, improving conditions for boundedness of Fourier multipliers on Bochner spaces.

Contribution

It provides a refined boundedness criterion for scalar-valued multipliers in UMD spaces, enhancing previous results by Girardi and Weis.

Findings

Improved multiplier boundedness conditions for scalar-valued symbols

Extension of the theorem to UMD spaces with type t and cotype q

Limitations noted for operator-valued multipliers

Abstract

Let X be a UMD space with type t and cotype q, and let T be a Fourier multiplier operator with a scalar-valued symbol m. If the Mihlin multiplier estimate holds for all partial derivatives of m up to the order n/max(t,q')+1, then T is bounded on the X-valued Bochner spaces. For scalar-valued multipliers, this improves the theorem of Girardi and Weis (J. Funct. Anal., 2003) who required similar assumptions for derivatives up to the order n/r+1, where r is a Fourier-type of X. However, the present method does not apply to operator-valued multipliers, which are also covered by the Girardi-Weis theorem.