相关论文: The $(L^{1},L^{1})$ bilinear Hardy-Littlewood func…

We prove a pointwise convergence result for additive ergodic averages associated with certain multiplicative actions of the Gaussian integers. We derive several applications in dynamics and number theory, including: (i) Wirsing's theorem…

By building some suitable strictly ergodic models, we prove that for an ergodic system $(X,\mathcal{X},\mu, T)$, $d\in{\mathbb N}$, $f_1, \ldots, f_d \in L^{\infty}(\mu)$, the averages $$\frac{1}{N^2} \sum_{(n,m)\in [0,N-1]^2}…

For any measure preserving system $(X,\mathcal{B},\mu,T_1,\ldots,T_d),$ where we assume no commutativity on the transformations $T_i,$ $1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of…

We generalize the respective ``double recurrence'' results of Bourgain and of the second author, which established for pairs of $L^{\infty}$ functions on a finite measure space the a.e. convergence of the discrete bilinear ergodic averages…

In this note besides two abstract versions of the Vitali Covering Lemma an abstract Hardy-Littlewood Maximal Inequality, generalizing weak type (1,1) maximal function inequality, associated to any outer measure and a family of subsets on a…

We show that on a $\sigma$-finite measure preserving system $X = (X,\nu, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} \Lambda(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$,…

We study the problem of whether the centered Hardy--Littlewood maximal function of a singular function is absolutely continuous. For a parameter $d \in (0,1)$ and a closed set $E\subset [0,1]$, let $\mu$ be a $d$-Ahlfors regular measure…

We prove that for each $p\in (1,\infty),$ the norms on $L^p(\mathbb{R}^d)$ of the maximal functions associated to Gaussians (heat semigroup), balls (Hardy-Littlewood averages), and spheres (spherical averages) converge, as the dimension…

We prove a rearrangement inequality for the uncentered Hardy-Littlewood maximal function $M_{\mu}$ associate to general measure $\mu$ on $\mathbb{R}$. This inequality is analogous to the Stein's result $cf^{**}(t)\leq(Mf)^{*}(t)\leq C…

This paper resolves the question of pointwise convergence for ergodic averages of a single function along the set of polynomial values of primes of the form $x^2 + ny^2$. Following the influential paper of Bourgain…

For every $c\in(1,23/22)$ and every probability dynamical system $(X,\mathcal{B},\mu,T)$ we prove that for any $f,g\in L^{\infty}_{\mu}(X)$ the bilinear ergodic averages \[ \frac{1}{N}\sum_{n=1}^Nf(T^{\lfloor n^c\rfloor}x)g(T^{-\lfloor…

For an ergodic map $T$ and a non-constant, real-valued $f \in L^1$, the ergodic averages $\mathbb{A}_N f(x) = \frac{1} {N} \sum_{n=1}^N f(T^n x)$ converge a.e., but the convergence is never monotone. Depending on particular properties of…

It is shown that the homogeneous ergodic bilinear averages with M\"{o}bius or Liouville weight converge almost surely to zero, that is, if $T$ is a map acting on a probability space $(X,\mathcal{A},\mu)$, and $a,b \in \mathbb{Z}$, then for…

Let $M_G$ denotes the centered Hardy-Littlewood maximal function associated to the Carnot-Carath\'eodory distance or to the pseudo-distance associated to the fundamental solution of the Grushin operator on $\R_x^n \times \R_u$, $\Delta_G =…

Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ and assume that the Hardy--Littlewood maximal operator satisfies the Fefferman--Stein vector-valued maximal inequality on $X$, and let $q\in[1,\infty)$ and $d\in(0,\infty)$.…

A remarkable theorem of Besicovitch is that an integrable function $f$ on $\mathbb{R}^2$ is strongly differentiable if and only if its associated strong maximal function $M_S f$ is finite a.e. We provide an analogue of Besicovitch's result…

In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy-Littlewood maximal operator associated to certain families of doubling, radial decreasing…

Given a $\sigma$-finite infinite measure space $(\Omega,\mu)$, it is shown that any Dunford-Schwartz operator $T:\,\mathcal L^1(\Omega)\to\mathcal L^1(\Omega)$ can be uniquely extended to the space $\mathcal L^1(\Omega)+\mathcal…

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\mathbb{R}) \times W^{1,q}(\mathbb{R}) \to W^{1,r}(\mathbb{R})$ with $1 <p,q < \infty$ and $r\geq 1$, boundedly and…

Let $M$ be a manifold with ends constructed in \cite{GS} and $\Delta$ be the Laplace-Beltrami operator on $M$. In this note, we show the weak type $(1,1)$ and $L^p$ boundedness of the Hardy-Littlewood maximal function and of the maximal…