Search

We prove two transference theorems for maximal convolution operators on vector-valued $L^p$-spaces. We then present applications of these results towards ergodic theory. In particular, let $R$ be a distributionally controlled representation of $G$, a locally compact abelian group, acting on $L^1(\Omega,X) \cap L^\infty(\Omega,X)$ where $X$ is a Banach space while $(\Omega,F,\mu)$ is an abstract measure space. We show that, for $p\in [1,\infty)$, if the associated representation $R^{(p)}$ acting on $L^p(\Omega,X)$ is strongly continuous, then $R^{(p)}$ transfers strong-type and weaktype bounds for maximal convolution operators from $L^p(G,X)$ to $L^p(\Omega,X)$. The transference theorems hold for any Banach space $X$; however when seeking ergodic theorems related to singular integral kernels we need to require that $X$ satisfy the UMD condition introduced by D. Burkholder.