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Let $U_1, ldots, U_n$ be a collection of commuting measure preserving transformations on a probability space $(Omega, Sigma, mu)$. Associated with these measure preserving transformations is the ergodic strong maximal operator $mathsf M ^ast _{mathsf S}$ given by [ mathsf M ^ast _{mathsf S} f(omega) := sup_{0 in R subset mathbb{R}^n}frac{1}{#(R cap mathbb{Z}^n)}sum_{(j_1, ldots, j_n) in Rcap mathbb{Z}^n}big|f(U_1^{j_1}cdots U_n^{j_n}omega)big|, ] where the supremum is taken over all open rectangles in $mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. For $0 < alpha < 1$ we define the sharp Tauberian constant of $mathsf M ^ast _{mathsf S}$ with respect to $alpha$ by [ mathsf C ^ast _{mathsf S} (alpha) := sup_{substack{E subset Omega mu(E) > 0}}frac{1}{mu(E)}mu({omega in Omega : mathsf M ^ast _{mathsf S} chi_E (omega) > alpha}). ] Motivated by previous work of A. A. Solyanik and the authors regarding Solyanik estimates for the geometric strong maximal operator in harmonic analysis, we show that the Solyanik estimate [ lim_{alpha rightarrow 1}mathsf C ^ast _{mathsf S}(alpha) = 1 ] holds, and that in particular we have [mathsf C ^ast _{mathsf S}(alpha) - 1 lesssim_n (1 - frac{1}{alpha})^{1/n}] provided that $alpha$ is sufficiently close to $1$. Solyanik estimates for centered and uncentered ergodic Hardy-Littlewood maximal operators associated with $U_1, ldots, U_n$ are shown to hold as well. Further directions for research in the field of ergodic Solyanik estimates are also discussed.
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Let $mathsf M_{mathsf S}$ denote the strong maximal operator on $mathbb R^n$ and let $w$ be a non-negative, locally integrable function. For $alphain(0,1)$ we define the weighted sharp Tauberian constant $mathsf C_{mathsf S}$ associated with $mathsf M_{mathsf S}$ by $$ mathsf C_{mathsf S} (alpha):= sup_{substack {Esubset mathbb R^n 0<w(E)<+infty}}frac{1}{w(E)}w({xinmathbb R^n:, mathsf M_{mathsf S}(mathbf{1}_E)(x)>alpha}). $$ We show that $lim_{alphato 1^-} mathsf C_{mathsf S} (alpha)=1$ if and only if $win A_infty ^*$, that is if and only if $w$ is a strong Muckenhoupt weight. This is quantified by the estimate $mathsf C_{mathsf S}(alpha)-1lesssim_{n} (1-alpha)^{(cn [w]_{A_infty ^*})^{-1}}$ as $alphato 1^-$, where $c>0$ is a numerical constant; this estimate is sharp in the sense that the exponent $1/(cn[w]_{A_infty ^*})$ can not be improved in terms of $[w]_{A_infty ^*}$. As corollaries, we obtain a sharp reverse Holder inequality for strong Muckenhoupt weights in $mathbb R^n$ as well as a quantitative imbedding of $A_infty^*$ into $A_{p}^*$. We also consider the strong maximal operator on $mathbb R^n$ associated with the weight $w$ and denoted by $mathsf M_{mathsf S} ^w$. In this case the corresponding sharp Tauberian constant $mathsf C_{mathsf S} ^w$ is defined by $$ mathsf C_{mathsf S} ^w alpha) := sup_{substack {Esubset mathbb R^n 0<w(E)<+infty}}frac{1}{w(E)}w({xinmathbb R^n:, mathsf M_{mathsf S} ^w (mathbf{1}_E)(x)>alpha}).$$ We show that there exists some constant $c_{w,n}>0$ depending only on $w$ and the dimension $n$ such that $mathsf C_{mathsf S} ^w (alpha)-1 lesssim_{w,n} (1-alpha)^{c_{w,n}}$ as $alphato 1^-$ whenever $win A_infty ^*$ is a strong Muckenhoupt weight.
This paper concerns the smoothness of Tauberian constants of maximal operators in the discrete and ergodic settings. In particular, we define the discrete strong maximal operator $tilde{M}_S$ on $mathbb{Z}^n$ by [ tilde{M}_S f(m) := sup_{0 in R subset mathbb{R}^n}frac{1}{#(R cap mathbb{Z}^n)}sum_{ jin R cap mathbb{Z}^n} |f(m+j)|,qquad min mathbb{Z}^n, ] where the supremum is taken over all open rectangles in $mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. We show that the associated Tauberian constant $tilde{C}_S(alpha)$, defined by [ tilde{C}_S(alpha) := sup_{substack{E subset mathbb{Z}^n 0 < #E < infty} } frac{1}{#E}#{m in mathbb{Z}^n:, tilde{M}_Schi_E(m) > alpha}, ] is Holder continuous of order $1/n$. Moreover, letting $U_1, ldots, U_n$ denote a non-periodic collection of commuting invertible transformations on the non-atomic probability space $(Omega, Sigma, mu)$ we define the associated maximal operator $M_S^ast$ by [ M^ast_{S}f(omega) := sup_{0 in R subset mathbb{R}^n}frac{1}{#(R cap mathbb{Z}^n)}sum_{(j_1, ldots, j_n)in R}|f(U_1^{j_1}cdots U_n^{j_n}omega)|,qquad omegainOmega. ] Then the corresponding Tauberian constant $C^ast_S(alpha)$, defined by [ C^ast_S(alpha) := sup_{substack{E subset Omega mu(E) > 0}} frac{1}{mu(E)}mu({omega in Omega :, M^ast_Schi_E(omega) > alpha}), ] also satisfies $C^ast_S in C^{1/n}(0,1).$ We will also see that, in the case $n=1$, that is in the case of a single invertible, measure preserving transformation, the smoothness of the corresponding Tauberian constant is characterized by the operator enabling arbitrarily long orbits of sets of positive measure.
We explore the properties of non-piecewise syndetic sets with positive upper density, which we call discordant, in countable amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerdens theorem and Szemeredis theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Here is a small sample of our results. $bullet$ We connect discordant sets to recurrence in dynamical systems, and in this setting we exhibit an intimate analogy between discordant sets and nowhere dense sets having positive measure. $bullet$ We introduce a wide-ranging generalization of the squarefree numbers, producing many examples of discordant sets in $mathbb{Z}$, $mathbb{Z}^d$, and the Heisenberg group. We develop a unified method to compute densities of these discordant sets. $bullet$ We show that, for any countable abelian group $G$, any F{o}lner sequence $Phi$ in $G$, and any $c in (0, 1)$, there exists a discordant set $A subseteq G$ with $d_Phi(A) = c$. Here $d_Phi$ denotes density along $Phi$. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.
We survey the impact of the Poincare recurrence principle in ergodic theory, especially as pertains to the field of ergodic Ramsey theory.
We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator $T$ satisfies a bound like $$ |T|_{L^{p}(w)}le c, [w]^{beta}_{A_p} qquad w in A_{p}, $$ then the optimal lower bound for $beta$ is closely related to the asymptotic behaviour of the unweighted $L^p$ norm $|T|_{L^p(mathbb{R}^n)}$ as $p$ goes to 1 and $+infty$, which is related to Yanos classical extrapolation theorem. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a wide class of operators including maximal-type, Calderon--Zygmund and fractional operators. In particular, we obtain a lower bound for the best possible exponent for Bochner-Riesz multipliers. We also present a new result concerning a continuum family of maximal operators on the scale of logarithmic Orlicz functions. Further, our method allows to consider in a unified way maximal operators defined over very general Muckenhoupt bases.
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