Transference of weak type bounds of multiparameter ergodic and geometric maximal operators

Paul Hagelstein¹ ; Alexander Stokolos²

¹ Department of Mathematics Baylor University Waco, TX 76798, U.S.A.
² Department of Mathematical Sciences Georgia Southern University 203 Georgia Avenue Statesboro, GA 30460-8093, U.S.A.

Fundamenta Mathematicae, Tome 218 (2012) no. 3, pp. 269-283

Résumé

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.

Détail

DOI : 10.4064/fm218-3-4

Keywords: ldots non periodic collection commuting measure preserving transformations probability space omega sigma gamma nonempty subset mathbb mathcal associated collection rectangular parallelepipeds mathbb sides parallel axes dimensions form times cdots times ldots gamma associated multiparameter geometric ergodic maximal operators mathcal gamma defined respectively mathbb omega mathcal sup mathcal frac int gamma omega sup ldots gamma frac cdots sum cdots sum cdots omega given young function phi shown mathcal satisfies weak type estimate mathbb mathcal alpha mathcal int mathbb phi mathcal alpha pair positive constants mathcal mathcal only gamma satisfies corresponding weak type estimate omega omega gamma omega alpha gamma int omega phi gamma alpha pair positive constants gamma gamma applications transference principle regarding convergence multiparameter ergodic averages associated rare bases given