Vector valued differentiation theorems for multiparameter additive processes in Lp spaces

Let X be a Banach space and (Omega, Sigma, mu) be a sigma-finite measure space. We consider a strongly continuous d-dimensional semigroup T = {T(u) : u = (u(1), ..., u(d)), u(i) > 0, 1 less than or equal to i less than or equal to d} of linear contractions on L-p((Omega, Sigma, mu); X), with 1 less than or equal to p < infinity. In this paper differentiation theorems are proved for d-dimensional bounded processes in L-p((Omega, Sigma, mu); X) which are additive with respect to T. In the theorems below we assume that each T(lc) possesses a contraction majorant P(u) defined on L-p((Omega, Sigma, mu); R), that is, P(u) is a positive linear contraction on L-p((Omega, Sigma, mu); R) such that \\T(u)f(omega)\\ less than or equal to P(u)\\f(.)\\(omega) almost everywhere on Omega for all f is an element of L-p((Omega, Sigma, mu); X).