# Vector Valued Differentiation Theorems for Multiparameter Additive Processes in Lp Spaces

Vector Valued Differentiation Theorems for Multiparameter Additive Processes in Lp Spaces 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). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Vector Valued Differentiation Theorems for Multiparameter Additive Processes in Lp Spaces

, Volume 2 (1) – Oct 7, 2004
18 pages

Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009728507209
Publisher site
See Article on Publisher Site

### 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).

### Journal

PositivitySpringer Journals

Published: Oct 7, 2004

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