This is an announcement for the paper “Fourier multiplier theorems involving type and cotype” by Jan Rozendaal.

Abstract: In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^{d}\to\mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980's and is reasonably well-understood. Far less is known for $p<q$. In the scalar case one can deduce results for $p<q$ from the case $p=q$ and additional arguments such as interpolation and Sobolev embedding techniques. However, in the vector-valued case this leads to restrictions both on the smoothness of the multiplier and the class of Banach spaces. For example, one often needs that $X$ and $Y$ are UMD spaces, and that $m$ satisfies a smoothness condition. For $p<q$ it turns out that the notions of type and cotype and other geometric conditions on $X$ and $Y$ are important to study Fourier multipliers. Moreover, we obtain boundedness results for $T_{m}$ without any smoothness properties of $m$. Under additional smoothness conditions we show that boundedness results can be extrapolated to other values of $p$ and $q$ as long as $\tfrac{1}{p}-\tfrac{1}{q}$ remains constant. Here we even extend classical results in the scalar case.

The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.09340