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Keywords:
contraction semigroup; semigroup modulus; majorant; pointwise ergodic \newline theorem; pointwise local ergodic theorem
contraction semigroup; semigroup modulus; majorant; pointwise ergodic \newline theorem; pointwise local ergodic theorem
Summary:
Let $\{T(t):t>0\}$ be a strongly continuous semigroup of linear contractions in $L_p$, $1<p<\infty$, 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.
References:
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