A Local Ergodic Theorem on L p

Source abstract

Two general types of pointwise ergodic theorems have been studied: those as t approaches infinity, and those as t approaches zero. This paper deals with the latter case, which is referred to as the local case. Let ( X , , μ ) be a complete, σ -finite measure space. Let { T t } be a strongly continuous one-parameter semi-group of contractions on , defined for t ≧ 0. For T t positive, it was shown independently in [2] and [5] that 1.1 almost everywhere on X , for any f ∊ L 1 . The same result was obtained in [1], with the continuity assumption weakened to having it hold for t > 0.