New thoughts on the vector-valued Mihlin–Hörmander multiplier theorem

Abstract:

Let $X$ be a UMD space with type $t$ and cotype $q$, and let $T_m$ be a Fourier multiplier operator with a scalar-valued symbol $m$. If $|\partial ^{\alpha }m(\xi )|\lesssim |{\xi }|^{-|\alpha |}$ for all $|\alpha |\leq \lfloor {n/\max (t,q’)\rfloor }+1$, then $T_m$ is bounded on $L^p(\mathbb {R}^n;X)$ for all $p\in (1,\infty )$. 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 $\lfloor {n/r}\rfloor +1$, where $r\leq \min (t,q’)$ 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.