Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

Abstract

Let $U_{1}, \ldots, U_{d}$ be a non-periodic collection of commuting measure preserving transformations on a probability space $(\Omega, \Sigma, \mu).$ Also let $\Gamma$ be a nonempty subset of $\mathbb{Z}^{d}_{+}$ and $\mathcal B$ the associated collection of rectangular parallelepipeds in $\mathbb R^d$ with sides parallel to the axes and dimensions of the form $n_1\times\cdots\times n_d$ with $(n_1,\ldots,n_d)\in \Gamma.$ The associated multiparameter geometric and ergodic maximal operators $M_{\mathcal{B}}$ and $M_{\Gamma}$ are defined respectively on $L^{1}(\mathbb{R}^{d})$ and $L^{1}(\Omega)$ by $$ M_{\mathcal{B}}g(x) = \sup_{x \in R \in \mathcal{B}}\frac{1}{|R|} \int_{R}|g(y)|\,dy $$ and $$ M_{\Gamma}f(\omega) = \sup_{(n_{1}, \ldots, n_{d}) \in \Gamma} \frac{1}{n_{1}\cdots n_{d}}\sum_{j_{1} = 0}^{n_{1} - 1}\cdots \sum_{j_{d} = 0}^{n_{d}-1}|f(U_{1}^{j_{1}}\cdots U_{d}^{j_{d}}\omega)|. $$ Given a Young function $\Phi,$ it is shown that $M_{\mathcal{B}}$ satisfies the weak type estimate $$ |\{x \in \mathbb{R}^d : M_{\mathcal{B}}g(x) > \alpha \}|\le C_{\mathcal{B}}\int_{\mathbb R^d}\Phi( c_{\mathcal{B}}{|g|}/ \alpha ) $$ for a pair of positive constants $C_{\mathcal{B}}$, $c_{\mathcal{B}}$ if and only if $M_{\Gamma}$ satisfies a corresponding weak type estimate $$ \mu\{\omega \in \Omega : M_{\Gamma} f(\omega) >\alpha \}\le C_{\Gamma}\int_{\Omega}\Phi( c_{\Gamma}{|f|} /\alpha ). $$ for a pair of positive constants $C_{\Gamma}$, $c_{\Gamma}$. Applications of this transference principle regarding the a.e. convergence of multiparameter ergodic averages associated to rare bases are given.