Fourier multiplier theorems on Besov spaces under type and cotype conditions

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 1/p - 1/q remains constant.