The \((L^p,L^q)\) bilinear Hardy-Littlewood function for the tail (Q607853)

Results related to the Hardy-Littlewood maximal function are established in an ergodic theoretical framework. Given a measure preserving transformation \((X, \mathcal B, \mu , T)\), consider the maximal function \[ R^*:(f,g) \in L^p\times L^q\to R^*(f,g)(x)=\sup_n \frac{f(T^n x)g(T^{2n} x)}{n}. \] In the paper it is proved, among other results, that if \(p\geq1 \), \(q\geq1 \) and \(\frac {1}{p} +\frac {1}{q}<2\), then \({R^*}\) maps \({L^p} \times {L^q}\) into \({L^r}\) for each \(0<r<\frac {1}{2}\). This implies that \({R^*}(f,g)\) is finite almost everywhere and \(\lim_{n \to \infty} \frac{f(T^n x)g(T^{2n} x)}{n}=0\) for almost each \(x\).