Fourier multiplier theorems on Besov spaces under type and cotype conditions

TL;DR

This paper establishes optimal Fourier multiplier theorems on vector-valued Besov spaces, extending previous results by removing smoothness constraints and considering the influence of Banach space type and cotype.

Contribution

It provides new Fourier multiplier results on Besov spaces with optimal integrability exponents, including sharp $L^p$-$L^q$ bounds, without requiring multiplier smoothness.

Findings

Optimal multiplier theorems on Besov spaces with type and cotype conditions

Sharp $L^p$-$L^q$ multiplier results

Extension of multiplier boundedness via Fourier support properties

Abstract

In this paper we consider Fourier multiplier operators between vector-valued Besov spaces with different integrability exponents $p$ and $q$, which depend on the type $p$ and cotype $q$ of the underlying Banach spaces. In a previous paper we considered $L^p$-$L^q$-multiplier theorems. In the current paper we show that in the Besov scale one can obtain results with optimal integrability exponents. Moreover, we derive a sharp result in the $L^p$-$L^q$-setting as well. We consider operator-valued multipliers without smoothness assumptions. The results are based on a Fourier multiplier theorem for functions with compact Fourier support. If the multiplier has smoothness properties then the boundedness of the multiplier operator extrapolates to other values of $p$ and $q$ for which $\frac1p - \frac1q$ remains constant.