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 $Y$, $1leq pleq qleq infty$ and $m:mathbb{R}^dto mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980s, but far less is known for $p<q$. In the scalar setting one can deduce results for $p<q$ from the case $p=q$. However, in the vector-valued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that $X$ and $Y$ are UMD spaces and that $m$ satisfies a smoothness condition. We show that for $p<q$ other geometric conditions on $X$ and $Y$, such as the notions of type and cotype, can be used to study Fourier multipliers. Moreover, we obtain boundedness results for $T_m$ without any smoothness properties of $m$. Under smoothness conditions the boundedness results can be extrapolated to other values of $p$ and $q$ as long as $tfrac{1}{p}-tfrac{1}{q}$ remains constant.
تحميل البحث