H-infinity 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< p0) 1/t vertical bar integral(t)(0) T-s(f(., omega)) (x)ds vertical bar, (x,omega) is an element of X x Omega. Then the following maximal ergodic inequality holds: parallel to M(f)parallel to(L) (p(X;E)) less than or similar to parallel to f parallel to(L) (p(X;E)), f is an element of L (p)(X;E). If the semigroup {T-t}(t>0) is additionally assumed to be analytic, then {Tt}(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. T-theta(f)(x,omega) = sup(vertical bar arg(z)vertical bar 0. Under the latter analyticity assumption and if E is a complex interpolation space between a Hilbert space and a UMD Banach space, then {Tt}(t> 0) extends to an analytic semigroup on L (p)(X; E) and its negative generator has a bounded H-infinity(Sigma(sigma)) calculus for some sigma < pi/2.