The $(L^{1},L^{1})$ bilinear Hardy-Littlewood function and Furstenberg averages
| dc.creator | Assani, Idris | |
| dc.creator | Buczolich, Zoltan | |
| dc.date | 2008-04-11 | |
| dc.date.accessioned | 2026-07-07T09:32:02Z | |
| dc.date.available | 2026-07-07T09:32:02Z | |
| dc.description | Let $(X,\mathcal{B}, μ, T)$ be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function $\dis R^*:(f, g) \in L^1\times L^1 \to R^*(f, g)(x) = \sup_{n} \frac{f(T^nx)g(T^{2n}x)}{n}.$ We show that there exist $f$ and $g$ such that $R^*(f, g)(x)$ is not finite almost everywhere. Two consequences are derived. The bilinear Hardy--Littlewood maximal function fails to be a.e. finite for all functions $(f, g)\in L^1\times L^1.$ The Furstenberg averages do not converge for all pairs of $(L^{1},L^{1})$ functions, while by a result of J. Bourgain these averages converge for all pairs of $(L^{p},L^{q})$ functions with $\frac{1}{p}+\frac{1}{q}\leq 1.$ | |
| dc.identifier | https://arxiv.org/abs/0804.1949 | |
| dc.identifier | http://arxiv.org/abs/0804.1949 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/158670 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 37A05 (Primary); 37A50, 28D05 (Secondary) | |
| dc.title | The $(L^{1},L^{1})$ bilinear Hardy-Littlewood function and Furstenberg averages | |
| dc.type | text |