$H^\infty$ functional calculus and maximal inequalities for semigroups of contractions on vector-valued $L_p$-spaces

Let $\{T_t\}_{t>0}$ be a strongly continuous semigroup of positive contractions on $L_p(X,μ)$ with $1<p<\infty$. Let $E$ be a UMD Banach lattice of measurable functions on another measure space $(Ω,ν)$. For $f\in L_p(X; E)$ define $\mathcal M(f)(x, ω)=\sup_{t>0}\frac1t\Big|\int_0^tT_s(f(\cdot,ω))(x)ds\Big|,\quad (x,ω)\in X\timesΩ.$ Then the following maximal ergodic inequality holds $\big\|\mathcal M(f)\big\|_{L_p(X; E)}\lesssim \big\|f\big\|_{L_p(X; E)},\quad f\in L_p(X; E).$ If the semigroup $\{T_t\}_{t>0}$ is additionally assumed to be analytic, then $\{T_t\}_{t>0}$ extends to an analytic semigroup on $L_p(X; E)$ and $\mathcal M(f)$ in the above inequality can be replaced by the following sectorial maximal function $\mathcal T_θ(f)(x, ω)=\sup_{|{\rm arg}(z)|<θ}\big|T_z(f(\cdot,ω))(x)\big|$ for some $θ>0$. Under the latter analyticity assumption and if $E$ is a complex interpolation space between a Hilbert space and a UMD Banach space, then $\{T_t\}_{t>0}$ extends to an analytic semigroup on $L_p(X; E)$ and its negative generator has a bounded $H^\infty(Σ_σ)$ calculus for some $σ<π/2$.

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