$H^{\mathrm{\infty }}$ 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 $L_p(X,\mu )$ with $1<p<\mathrm{\infty }$. Let $E$ be a UMD Banach lattice of measurable functions on another measure space $(\mathrm{\Omega },\nu )$. For $f\in L_p(X; E)$define$$\mathcal M(f)(x, \omega)=\sup_{t>0}\frac1t\Big|\int_0^tT_s(f(\cdot,\omega))(x)ds\Big|,\quad (x,\omega)\in X\times\Omega.$$Thenthefollowingmaximalergodicinequalityholds$$\big|\mathcal M(f)\big|{L_p(X; E)}\lesssim \big|f\big|{L_p(X; E)},\quad f\in L_p(X; E).If the semigroup $\{T_t\}_{t>0}$ is additionally assumed to be analytic, then $\{T_t\}_{t>0}$ extends to an analytic semigroup on $L_p(X; E)$ and $\mathcal M(f)$ in the above inequality can be replaced by the following sectorial maximal function\mathcal T_\theta(f)(x, \omega)=\sup_{|{\rm arg}(z)|<\theta}\big|T_z(f(\cdot,\omega))(x)\big|$$forsome$\theta>0$. 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 $H^{\mathrm{\infty }}({\mathrm{\Sigma }}_{\sigma })$ calculus for some $\sigma <\pi \mathrm{/}2$.