Embedding vector-valued Besov spaces into spaces of $\gamma$-radonifying operators

Abstract

It is shown that a Banach space $E$ has type $p$ if and only for some (all) $d\ge 1$ the Besov space $B_{p,p}^{(\frac{1}{p}-\frac{1}{2})d}({\mathbb{R}}^d;E)$ embeds into the space $\g(L^2(\R^d),E)$ of $\g$-radonifying operators $L^2({\mathbb{R}}^d)\to E$. A similar result characterizing cotype $q$ is obtained. These results may be viewed as $E$-valued extensions of the classical Sobolev embedding theorems.

To appear in Mathematische Nachrichten