A general differentiation theorem for multiparameter additive processes (Q2773282)

Let \((\Omega,\Sigma,\mu)\) be a \(\sigma\)-finite measure space and \((L,\|.\|_L)\) be a Banach lattice of equivalence classes of real valued measurable functions on \(\Omega\). The author generalizes a theorem of \textit{M. A. Akcoglu} and \textit{A. Del Junco} [Can. J. Math. 33, 749-768 (1981; Zbl 0477.47012)] about a differentiation theorem in the case \(L=L^{1}(\mu)\) for the semigroup of positive linear contractions \(L^1\) and the additive process bounded in \(L^1\), to the \(L\) spaces with locally bounded semigroup. The operators should also be proper, otherwise a counterexample was given by the author himself [Stud. Math. 63, 45-55 (1978; Zbl 0391.47022)]. NEWLINENEWLINENEWLINEThe main result is the almost everywhere convergence of \(\alpha ^{-d}F((0,\alpha]^d)\) as \(\alpha\) tends to 0 through the set \({\mathbb D}\) , and \(F\) a locally bounded \(d\)-dimensional process on \(L\) which is additive with respect to the proper operator \(T\). Moreover, the paper contains an example which shows that the limit of the above average does not belong to \(L\) even in the dimension 1; however, in the case \(L=L^{p}\) the limit is a function in \(L^{p}\), \(1\leq p\leq \infty .\) It will be interesting to examine further the vector case (the function is integrable in the sense of Bochner) as well as the vector operators dominated (in norm) by positive operators, e.g., the surjective isometries in reflexive Banach spaces.