This is an announcement for the paper "$H^\infty$ functional calculus and maximal inequalities for semigroups of contractions on vector-valued $L_p$-spaces" by Quanhua Xu.

Abstract: Let ${T_t}_{t>0}$ be a strongly continuous semigroup of positive contractions on $L_p(X,\mu)$ with $1<p<\infty$. Let $E$ be a UMD Banach lattice of measurable functions on another measure space $(\Omega,\nu)$. For $f\in L_p(X; E)$ define $$\mathcal M(f)(x, \omega)=\sup_{t>0}\frac1t\Big|\int_0^tT_s(f(\cdot,\omega))(x)ds\Big|,\quad (x,\omega)\in X\times\Omega.$$ 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 $$\mathcal T_\theta(f)(x, \omega)=\sup_{|{\rm arg}(z)|<\theta}\big|T_z(f(\cdot,\omega))(x)\big|$$ for some $\theta>0$. 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 $H^\infty(\Sigma_\sigma)$ calculus for some $\sigma<\pi/2$.

Archive classification: math.FA

Mathematics Subject Classification: Primary: 47A35, 47A60. Secondary: 46B20, 42B25

Submitted from: quanhua.xu@univ-fcomte.fr

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1402.2344

or

http://arXiv.org/abs/1402.2344