Vector Valued Differentiation Theorems for Multiparameter Additive Processes in Lp Spaces

Abstract

Let X be a Banach space and (Ω,Σ,µ) be a Σ-finite measure space. We consider a strongly continuous d-dimensional semigroup T={T(u):u=(u1,..., ud, ui >0, 1≤ i≤ d} of linear contractions on Lp((Ω,Σ,µ); X), with 1≤ p<∞. In this paper differentiation theorems are proved for d-dimensional bounded processes in Lp((Ω,Σ,µ); X) which are additive with respect to T. In the theorems below we assume that each T(u) possesses a contraction majorant P(u) defined on Lp((Ω,Σ,µ); R), that is, P(u) is a positive linear contraction on Lp((Ω,Σ,µ); R) such that ‖T(u)f(w)‖≤ P(u)‖f(·)‖(Ω) almost everywhere on Ω for all f ∈ Lp((Ω,Σ,µ); X).