Fourier multiplier theorems on Besov spaces under type and cotype conditions

Abstract

In this article, 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 article, we considered $L^p$-$L^q$ multiplier theorems. In the current article, 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 $\frac{1}{p}-\frac{1}{q}$ remains constant.