ترغب بنشر مسار تعليمي؟ اضغط هنا

Holder continuity of Tauberian constants associated with discrete and ergodic strong maximal operators

86   0   0.0 ( 0 )
 نشر من قبل Ioannis Parissis
 تاريخ النشر 2016
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

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.



قيم البحث

اقرأ أيضاً

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 $n$.
This paper provides a necessary and sufficient condition on Tauberian constants associated to a centered translation invariant differentiation basis so that the basis is a density basis. More precisely, given $x in mathbb{R}^n$, let $mathcal{B} = cup _{x in mathbb{R}^n} mathcal{B}(x)$ be a collection of bounded open sets in $mathbb{R}^n$ containing $x$. Suppose moreover that these collections are translation invariant in the sense that, for any two points $x$ and $y$ in $mathbb{R}^n$ we have that $mathcal{B}(x + y) = {R + y : R in mathcal{B}(x)}.$ Associated to these collections is a maximal operator $M_{mathcal{B}}$ given by $$M_{mathcal{B}}f(x) :=sup_{R in mathcal{B}(x)} frac{1}{|R|} int_R |f|.$$ The Tauberian constants $C_{mathcal{B}}(alpha)$ associated to $M_{mathcal{B}}$ are given by $$C_{mathcal{B}}(alpha) :=sup_{E subset mathbb{R}^n atop 0 < |E| < infty} frac{1}{|E|}|{x in mathbb{R}^n :, M_{mathcal{B}}chi_E(x) > alpha}|.$$ Given $0 < r < infty$, we set $mathcal{B}_r(x) :={R in mathcal{B}(x) : mathrm{diam } R < r}$, and let $mathcal{B}_r :=cup_{x in mathbb{R}^n} mathcal{B}_r (x).$ We prove that $mathcal{B}$ is a density basis if and only if, given $0 < alpha < infty$, there exists $ r = r(alpha) >0$ such that $C_{mathcal{B}_r}(alpha) < infty$. Subsequently, we construct a centered translation invariant density basis $mathcal{B} = cup_{x in mathbb{R}^n} mathcal{B}(x)$ such that there does not exist any $0 < r$ satisfying $C_{mathcal{B}_{r}}(alpha) < infty$ for all $0 < alpha < 1$.
120 - Ciqiang Zhuo , Dachun Yang 2016
Let $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally $log$-Holder continuous condition and $L$ a non-negative self-adjoint operator on $L^2(mathbb R^n)$ whose heat kernels satisfying the Gaussian upper bound estima tes. Let $H_L^{p(cdot)}(mathbb R^n)$ be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels ${e^{-t^2L}}_{tin (0,infty)}$. In this article, the authors first establish the atomic characterization of $H_L^{p(cdot)}(mathbb R^n)$; using this, the authors then obtain its non-tangential maximal function characterization which, when $p(cdot)$ is a constant in $(0,1]$, coincides with a recent result by Song and Yan [Adv. Math. 287 (2016), 463-484] and further induces the radial maximal function characterization of $H_L^{p(cdot)}(mathbb R^n)$ under an additional assumption that the heat kernels of $L$ have the Holder regularity.
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.
We prove new $ell ^{p} (mathbb Z ^{d})$ bounds for discrete spherical averages in dimensions $ d geq 5$. We focus on the case of lacunary radii, first for general lacunary radii, and then for certain kinds of highly composite choices of radii. In par ticular, if $ A _{lambda } f $ is the spherical average of $ f$ over the discrete sphere of radius $ lambda $, we have begin{equation*} bigllVert sup _{k} lvert A _{lambda _k} f rvert bigrrVert _{ell ^{p} (mathbb Z ^{d})} lesssim lVert frVert _{ell ^{p} (mathbb Z ^{d})}, qquad tfrac{d-2} {d-3} < p leq tfrac{d} {d-2}, dgeq 5, end{equation*} for any lacunary sets of integers $ {lambda _k ^2 }$. We follow a style of argument from our prior paper, addressing the full supremum. The relevant maximal operator is decomposed into several parts; each part requires only one endpoint estimate.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا