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We study maximal operators related to bases on the infinite-dimensional torus $mathbb{T}^omega$. {For the normalized Haar measure $dx$ on $mathbb{T}^omega$ it is known that $M^{mathcal{R}_0}$, the maximal operator associated with the dyadic basis $mathcal{R}_0$, is of weak type $(1,1)$, but $M^{mathcal{R}}$, the operator associated with the natural general basis $mathcal{R}$, is not. We extend the latter result to all $q in [1,infty)$. Then we find a wide class of intermediate bases $mathcal{R}_0 subset mathcal{R} subset mathcal{R}$, for which maximal functions have controlled, but sometimes very peculiar behavior.} Precisely, for given $q_0 in [1, infty)$ we construct $mathcal{R}$ such that $M^{mathcal{R}}$ is of restricted weak type $(q,q)$ if and only if $q$ belongs to a predetermined range of the form $(q_0, infty]$ or $[q_0, infty]$. Finally, we study the weighted setting, considering the Muckenhoupt $A_p^mathcal{R}(mathbb{T}^omega)$ and reverse Holder $mathrm{RH}_r^mathcal{R}(mathbb{T}^omega)$ classes of weights associated with $mathcal{R}$. For each $p in (1, infty)$ and each $w in A_p^mathcal{R}(mathbb{T}^omega)$ we obtain that $M^{mathcal{R}}$ is not bounded on $L^q(w)$ in the whole range $q in [1,infty)$. Since we are able to show that [ bigcup_{p in (1, infty)}A_p^mathcal{R}(mathbb{T}^omega) = bigcup_{r in (1, infty)} mathrm{RH}_r^mathcal{R}(mathbb{T}^omega), ] the unboundedness result applies also to all reverse Holder weights.
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We establish the sharp growth order, up to epsilon losses, of the $L^2$-norm of the maximal directional averaging operator along a finite subset $V$ of a polynomial variety of arbitrary dimension $m$, in terms of cardinality. This is an extension of the works by Cordoba, for one-dimensional manifolds, Katz for the circle in two dimensions, and Demeter for the 2-sphere. For the case of directions on the two-dimensional sphere we improve by a factor of $sqrt{log N}$ on the best known bound, due to Demeter, and we obtain a sharp estimate for our model operator. Our results imply new $L^2$-estimates for Kakeya-type maximal functions with tubes pointing along polynomial directions. Our proof technique is novel and in particular incorporates an iterated scheme of polynomial partitioning on varieties adapted to directional operators, in the vein of Guth, Guth-Katz, and Zahl.
We prove that the maximal functions associated with a Zygmund dilation dyadic structure in three-dimensional Euclidean space, and with the flag dyadic structure in two-dimensional Euclidean space, cannot be bounded by multiparameter sparse operators associated with the corresponding dyadic grid. We also obtain supplementary results about the absence of sparse domination for the strong dyadic maximal function.
In the present paper, we are aiming to study limiting behavior of infinite dimensional Volterra operators. We introduce two classes $tilde {mathcal{V}}^+$ and $tilde{mathcal{V}}^-$of infinite dimensional Volterra operators. For operators taken from the introduced classes we study their omega limiting sets $omega_V$ and $omega_V^{(w)}$ with respect to $ell^1$-norm and pointwise convergence, respectively. To investigate the relations between these limiting sets, we study linear Lyapunov functions for such kind of Volterra operators. It is proven that if Volterra operator belongs to $tilde {mathcal{V}}^+$, then the sets and $omega_V^{(w)}(xb)$ coincide for every $xbin S$, and moreover, they are non empty. If Volterra operator belongs to $tilde {mathcal{V}}^-$, then $omega_V(xb)$ could be empty, and it implies the non-ergodicity (w.r.t $ell^1$-norm) of $V$, while it is weak ergodic.
In this paper we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{Ha}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-tau_y A_rf$, where $tau_yf(x)=f(xy^{-1})$ is the right translation operator.
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.
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