Weighted Lp spaces and pointwise ergodic theorems
- Autores: Ryotaro Sato
- Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 39, Nº 2, 1995, págs. 273-283
- Idioma: inglés
- DOI: 10.5565/publmat_39295_05
- Títulos paralelos:
- Espacios Lp ponderados y teoremas ergódicos puntuales
- Enlaces
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Resumen
In this paper we give an operator theoretic version of a recent result of F. J. Mart¹n-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 $p$, $1\le p<\infty$.
