Revistes Catalanes amb Accés Obert (RACO)

Weighted $L_p$ spaces and pointwise ergodic theorems

R. Sato

Resum


In this paper we give an operator theoretic version of a recent result of F. J. Martín-Reyes and A. de la Torre concerning the problem of finding necessary and sufficient conditions for a nonsingular point transformation to satisfy the Pointwise Ergodic Theorem in $L_p$. We consider a positive conservative contraction $T$ on $L_1$ of a $\sigma$-finite measure space $(X,\Cal F,\mu)$, a fixed function $e$ in $L_1$ with $e > 0$ on $X$, and two positive measurable functions $V$ and $W$ on $X$. We then characterize the pairs $(V,W)$ such that for any $f$ in $L_p(V\,d\mu)$ the averages
$$
R_0^n(f,e)=\left.\left(\sum^n_{k=0}T^kf\right)\right/\left(\sum^n_{k=0}T^ke\right)
$$
converge almost everywhere to a function in $L_p(W\,d\mu)$. The characterizations are given for all $p$, $1\le p < \infty$.

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