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 $1<p\le \mathrm{\infty }$ there exists a finite constant $C_p$ such that for each $\lambda >0,$ 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 $R^{\ast }(f,g)$ is a.e. finite for pairs of functions $(f,g)\in (L(\log L{)}^{2\alpha },L^1)$.