A maximal inequality for the tail of the bilinear Hardy-Littlewood function

Let \((X,\mathcal{B}, \mu, T)\) be an ergodic dynamical system on a non-atomic finite measure space. We assume without loss of generality that \(\mu(X)=1.\) Consider the maximal function \(\dis R^*:(f, g) \in L^p\times L^q \to R^*(f, g)(x) = \sup_{n\geq 1} \frac{f(T^nx)g(T^{2n}x)}{n}.\) We obtain the following maximal inequality. For each \(10,\) and nonnegative functions \(f\in L^p\) and \(g\in L^1\) \mu\{x: R^*(f,g)(x)>\lambda\} \leq C_p \bigg(\frac{\|f\|_p\|g\|_1}{\lambda}\bigg)^{1/2}. We also show that for each \(\alpha>2\) the maximal function \(R^*(f,g)\) is a.e. finite for pairs of functions \((f,g)\in (L(\log L)^{2\alpha}, L^1)\).