Geodesic
Ergodic properties of contraction semigroups in $L_p$, $1$
Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) no. 2, pp. 337-346

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Let $\{T(t):t>0\}$ be a strongly continuous semigroup of linear contractions in $L_p$, $1$, of a $\sigma $-finite measure space. In this paper we prove that if there corresponds to each $t>0$ a positive linear contraction $P(t)$ in $L_p$ such that $|T(t)f|\leq P(t)|f|$ for all $f\in L_p$, then there exists a strongly continuous semigroup $\{S(t):t>0\}$ of positive linear contractions in $L_p$ such that $|T(t)f|\leq S(t)|f|$ for all $t>0$ and $f\in L_p$. Using this and Akcoglu's dominated ergodic theorem for positive linear contractions in $L_p$, we also prove multiparameter pointwise ergodic and local ergodic theorems for such semigroups.
Classification : 47A35, 47B38, 47D03, 47D06
Keywords: contraction semigroup; semigroup modulus; majorant; pointwise ergodic \newline theorem; pointwise local ergodic theorem 
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     title = {Ergodic properties of contraction semigroups in $L_p$, $1<p<\infty$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {337--346},
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     url = {http://geodesic.mathdoc.fr/item/CMUC_1994__35_2_a13/}
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Sato, Ryotaro. Ergodic properties of contraction semigroups in $L_p$, $1