Oscillation in ergodic theory: Higher dimensional results
This paper extends the studies of [\textit{R. L. Jones, R. Kaufmann, J. M. Rosenblatt}, and \textit{M. Wierdl}, Ergodic Theory Dyn. Syst. 18, No.~4, 889--935 (1998; Zbl 0924.28009)] of square function inequalities to multi-dimensional averages. The basic tool is a comparison of the ergodic averages with various dyadic reversed martingales, but in contrast to the 1-dimensional case, the Fourier transform arguments are now replaced by more geometric almost orthogonality arguments. Let \(X = \mathbb{Z}^d\), and let \(\sigma_n\) be the \(n\)-th dyadic \(\tau\)-algebra, generated by the dyadic cubes \([u,v)\), for which all coordinates of \(u\) and \(v\) are divisible by \((2^n)\). For \(A\subset X\) let \(M_A f(x)\) denote the average of all \(f(x + y)\) with \(y\in A\). Let \(H_n = [-2^{(n+1)}, 2^{(n+1)})^d\), and let \({\mathcal A}_n\) contain all cubes with side length between \(2^{(n-1)}\) and \(2^n\), contained in \(H_n\). If \({\mathcal E}_n\) is the conditional expectation with respect to \(\sigma_n\), it is shown that \(\|(\sum_n\sup_{(A\in{\mathcal A}_n)}| M_Af -\varepsilon_nf|^2)^\frac12 \;\| p\leq c_p\| f\|_p\) \((1 < p \leq 2)\) (the norms are in \(l_p\).) There is also a similar \(l_1\)-inequality. These estimates have several interesting consequences: Hardy-Littlewood inequalities, square function inequalities, oscillation inequalities, jump inequalities, and variation inequalities. The paper also discusses other averages than those over cubes, and one section deals with different results.
scientific article; zbMATH DE number 2053055
scientific article