Solyanik estimates in ergodic theory

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.

A note on local Holder continuity of weighted Tauberian functions

Let $mathsf M$ and $mathsf M _{mathsf S}$ respectively denote the Hardy-Littlewood maximal operator with respect to cubes and the strong maximal operator on $mathbb{R}^n$, and let $w$ be a nonnegative locally integrable function on $mathbb{R}^n$. We define the associated Tauberian functions $mathsf{C}_{mathsf{HL},w}(alpha)$ and $mathsf{C}_{mathsf{S},w}(alpha)$ on $(0,1)$ by [ mathsf{C}_{mathsf{HL},w}(alpha) :=sup_{substack{E subset mathbb{R}^n 0 < w(E) < infty}} frac{1}{w(E)}w({x in mathbb{R}^n : mathsf M chi_E(x) > alpha}) ] and [ mathsf{C}_{mathsf{S},w}(alpha) := sup_{substack{E subset mathbb{R}^n 0 < w(E) < infty}} frac{1}{w(E)}w({x in mathbb{R}^n : mathsf M _{mathsf S}chi_E(x) > alpha}). ] Utilizing weighted Solyanik estimates for $mathsf M$ and $mathsf M_{mathsf S}$, we show that the function $mathsf{C}_{mathsf{HL},w} $ lies in the local Holder class $C^{(c_n[w]_{A_{infty}})^{-1}}(0,1)$ and $mathsf{C}_{mathsf{S},w} $ lies in the local Holder class $C^{(c_n[w]_{A_{infty}^ast})^{-1}}(0,1)$, where the constant $c_n>1$ depends only on the dimension $n$.

Weighted Solyanik estimates for the strong maximal function

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.