Weak type inequalities for maximal operators associated to double ergodic sums
The authors study ergodic maximal operators of the form \[ M_\Gamma f(w)=\sup_{(m,n)\in \Gamma}\frac{1}{mn}\sum^{m-1}_{j=0}\sum^{n-1}_{k=0}|f(u^jv^kw)| \] where \(\Gamma\subseteq\mathbb{Z}^2_+\) is an unbounded region. They prove that if \(U,V\) is a commuting pair of non-periodic, measure-preserving transformations then either {\parindent=6mm \begin{itemize}\item[(1)] \(M_\Gamma\) is of weak type \((1,1)\) and the associated ergodic averages \[ \lim_{\substack{ m,n\rightarrow\infty \\ (m,n)\in\Gamma}}\frac{1}{mn}\sum^{m-1}_{j=0}\sum^{n-1}_{k=0}f(u^jv^kw) \] converge a.e.\ for every \(f\in L^1(\Omega),\) or \item[(2)] \(M_\Gamma\) is of weak type \((L\log L,L^1),\) but given any positive, increasing function \(\phi\) that is \(o(\log x)\) as \(x\rightarrow\infty,\) there exists \(g\in L\phi(L)\) with \(M_\Gamma g=\infty\) a.e. and \[ \lim_{\substack{ m,n\rightarrow\infty \\ (m,n)\in\Gamma}} \frac{1}{mn}\sum^{m-1}_{j=0}\sum^{n-1}_{k=0} g(u^jv^kw) \] fails to converge a.e. \end{itemize}} Whether \(M_\Gamma\) is in the first or second case depends on a dyadic, monotonicity condition on \(\Gamma.\)
scientific article; zbMATH DE number 5935405