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

Abstract

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 $1<p\le \infty$ there exists a finite constant $C_p$ such that for each $\lambda >0,$ and nonnegative functions $f\in L^p$ and $g\in L^1$ μ\{x: R^*(f,g)(x)>λ\} \leq C_p \bigg(\frac{\|f\|_p\|g\|_1}λ\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 $(f,g)\in (L(\log L{)}^{2\alpha },L^1)$.

arXiv commentThis is the final and simplified version of the paper previously uploaded with the same title. The paper has been refereed and will appear in Cont. Math. The maximal inequality has been simplified as stated in this new abstract