Pointwise convergence for semigroups in vector-valued $L^p$ spaces

TL;DR

This paper establishes pointwise convergence and maximal theorems for vector-valued extensions of symmetric diffusion semigroups in UMD Banach spaces, generalizing classical ergodic results to a broader functional setting.

Contribution

It proves a vector-valued Hopf--Dunford--Schwartz ergodic theorem and extends maximal and pointwise convergence results to semigroups acting on L^p spaces with UMD Banach space targets.

Findings

Proved a vector-valued ergodic theorem for symmetric diffusion semigroups.

Extended maximal theorems to analytic continuations of the semigroup.

Established pointwise convergence of the semigroup extensions in vector-valued L^p spaces.

Abstract

Suppose that T_t is a symmetric diffusion semigroup on L^2(X) and consider its tensor product extension to the Bochner space L^p(X,B), where B belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf--Dunford--Schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of the semigroup's extension to L^p(X,B). As an application, we show that such continuations exhibit pointwise convergence.