H^∞ 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<∞. Let E be a unconditional martingale differences (UMD) Banach lattice of measurable functions on another measure space (Ω, ν). For f ε L_p (X; E), define (equation required). Then the following maximal ergodic inequality holds:(equation required). 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 M(f) in the above inequality can be replaced by the following sectorial maximal function. (equation required) 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∞ (Σσ ) calculus for some σ <π/2.