Embedding vector-valued Besov spaces into spaces of γ-radonifying operators

It is shown that a Banach space $E$ has type $p$ if and only for some (all) $d\ge 1$ the Besov space $B_{p,p}^{(\frac1p-\frac12)d}(\R^d;E)$ embeds into the space $\g(L^2(\R^d),E)$ of $\g$-radonifying operators $L^2(\R^d)\to E$. A similar result characterizing cotype $q$ is obtained. These results may be viewed as $E$-valued extensions of the classical Sobolev embedding theorems.