Abstract
We prove the local ergodic theorem inL ∞: Let {T t}t>0 be a strongly continuous semi-group of positive operators onL 1. IfT t is continuous at 0, then ɛ−1∫ F0 T *1 f(x)dt→T *0 f(x) a.e., for everyf∈L ∞. The technique shows how to obtain theL p local ergodic theorems from theL 1-contraction case. It applies also to differentiation ofL p additive processes. Then-dimensional case, which is new, is proved by reduction to then-dimensionalL 1-contraction case, solved by M. Akcoglu and A. del Junco.
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Research carried out during a sabbatical leave at the Ohio State University.
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Lin, M. On local ergodic convergence of semi-groups and additive processes. Israel J. Math. 42, 300–308 (1982). https://doi.org/10.1007/BF02761411
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DOI: https://doi.org/10.1007/BF02761411