Differentiation of superadditive processes in \(L_p\) (Q1103851)

Let \(\{T_t\}\) be a semigroup of positive linear operators on \(L_p(X)\), \(1<p<\infty\) \((X\) is a \(\sigma\)-finite measure space) which is locally integrable. If \(\{F_t\}\) is a process (i.e. a family of functions) in \(L_p(X)\) which is positive and superadditive (i.e. \(F_{t+s}\ge F_ t+T_tF_s)\) then the authors prove that there exists \(\lim (1/t) F(t)\) where \(t\) approaches to zero through a dense countable set in \((0,1)\): moreover the limit is finite a.e. on a subset \(P\) of \(X\) (such that if \(f\in L^+_p(P)\) and \(\| f\|_p>0\) then \(\| (T_tf)1_P\|_p>0\) for some \(t>0\), where \(1_P\) is the indicator function of \(P)\). Similar results are given when \(p=1\) or \(p=\infty\) by means of the adjoint semigroups. These results generalize theorems of \textit{R. Sato} [Stud. Math. 63, 45--55 (1978; Zbl 0391.47022)] and of \textit{M. A. Akcoglu} and \textit{U. Krengel} [Math. Z. 163, 199--210 (1978; Zbl 0386.60055); 169, 31--40 (1979; Zbl 0412.47022)].