Fourier multiplier theorems on Besov spaces under type and cotype conditions

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 $\frac{1}{p}-\frac{1}{q}$ remains constant.

Accepted for publication in Banach journal of mathematical analysis. A large of the paper was part in the 1st version of arXiv:1605.09340, but we decided to present the Besov space result and L^p results in separate papers

References in corpus (2)

• Stability theory for semigroups using $(L^p,L^q)$ Fourier multipliers
• Functional calculus for $C_0$-groups using (co)type

Cited by in corpus (7)

• Optimal rates of decay for operator semigroups on Hilbert spaces
• Sharp growth rates for semigroups using resolvent bounds
• Singular stochastic integral operators
• Stability theory for semigroups using $(L^p,L^q)$ Fourier multipliers
• Operator-valued $(L^p,L^q)$ Fourier multipliers and stability theory for evolution equations
• Functional calculus for $C_0$-groups using (co)type
• Stability properties of stochastic maximal $L^p$-regularity