Related papers: Variation estimates for averages along primes and …

We prove, in a unified way, $r$-variational estimates, $r>2$, on $\ell^{s}(\mathbb{Z})$ spaces, $s \in (1, \infty)$, for averages and truncated singular integrals along the set of prime numbers.

We find the precise range of $(p,q)$ for which local averages along graphs of a class of two-variable polynomials in $\mathbb{R}^3$ are of restricted weak type $(p,q)$, given the hypersurfaces have Euclidean surface measure. We derive these…

We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our…

We prove long variational estimates for the bilinear ergodic averages \[ A_{N;X}(f,g)(x) = \frac{1}{N} \sum_{n=1}^N f(T^{\lfloor \sqrt{n} \rfloor}x) g(T^nx) \] on an arbitrary measure preserving system $(X,\mu,T)$ for the full expected…

In this paper we give a short proof of the $\ell^p$-improving property of the average operator along the square integers and more general quadratic polynomials. Moreover we obtain a similar result for some higher degree polynomials. We also…

We establish pointwise convergence for nonconventional ergodic averages taken along $\lfloor p^c\rfloor$, where $p$ is a prime number and $c\in(1,4/3)$ on $L^r$, $r\in(1,\infty)$. In fact, we consider averages along more general sequences…

We prove the uniform oscillation and jump inequalities for the polynomial ergodic averages modeled over multi-dimensional subset of primes. These inequalities provide endpoints for the $r$-variational estimates obtained by Trojan…

Let $f\in \ell^2(\mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=\frac{1}{N}\sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ \ell ^{p}$-improving estimate, for $ 3/2 < p \leq 2$:…

We prove the extensions of Birkhoff's and Cotlar's ergodic theorems to multi-dimensional polynomial subsets of prime numbers $\mathbb{P}^k$. We deduce them from $\ell^p$-boundedness of $r$-variational seminorms for the corresponding…

Let $L^2(X,\Sigma,\mu,\tau)$ be a measure-preserving system, with $\tau$ a $\mathbb{Z}$-action. In this note, we prove that the ergodic averages along integer-valued polynomials, $P(n)$, \[ M_N(f):= \frac{1}{N}\sum_{n \leq N} \tau^{P(n)} f…

We establish a higher dimensional counterpart of Bourgain's pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates $V_r$ on $L^p$ spaces for all $1<p<\infty$ and…

We study double averages along orbits for measure preserving actions of $\mathbb{A}^\omega$, the direct sum of countably many copies of a finite abelian group $\mathbb{A}$. In this article we show an $L^p$ norm-variation estimate for these…

We prove $\ell^p(\mathbb{Z}^n)-$estimates for long $r$-variational seminorm of two families of averages: discrete Birch-Magyar averages, for $r>max\{p,p'\}$ with $p>\frac{2c_{\mathfrak{R}}-2}{2c_{\mathfrak{R}}-3}$ and discrete…

We prove sharp $L^p-L^q$ estimates for averaging operators along general polynomial curves in two and three dimensions. These operators are translation-invariant, given by convolution with the so-called affine arclength measure of the curve…

In this paper, equivalence constants between various polynomial norms are calculated. As an application, we also obtain sharp values of the Hardy--Littlewood constants for $2$-homogeneous polynomials on $\ell_p^2$ spaces, $2<p\leq\infty$…

In the combinatorial method proving of $L^p$-improving estimates for averages along curves pioneered by Christ (IMRN, 1998), it is desirable to estimate the average modulus (with respect to some uniform measure on a set) of a…

For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=\frac{1}{N}\sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $\ell^{p}$-improving…

In this paper we study multiple ergodic averages for "good" variable polynomials. In particular, under an additional assumption, we show that these averages converge to the expected limit, making progress related to an open problem posted…

It was shown by S. Kalikow and B. Weiss that, given a measure-preserving action of $\mathbb{Z}^d$ on a probability space $X$ and a nonnegative measurable function $f$ on $X$, the probability that the sequence of ergodic averages $$ \frac 1…

In this paper we establish uniform oscillation estimates on $L^p(X)$ with $p\in(1,\infty)$ for the polynomial ergodic averages. This result contributes to a certain problem about uniform oscillation bounds for ergodic averages formulated by…