Related papers: $\ell^p(\mathbb{Z}^d)$-Improving Properties and Sp…

We study maximal averages associated with singular measures on $\rr$. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension $1 - \epsilon$, $0 \leq \epsilon < {1/3}$ for which the…

We prove a conjecture of Ambrus, Ball and Erd\'{e}lyi that equally spaced points maximize the minimum of discrete potentials on the unit circle whenever the potential is of the form \sum_{k=1}^n f(d(z,z_k)), where $f:[0,\pi]\to [0,\infty]$…

In this paper, we study the $L^p(\mathbb{R}^2)$-improving bounds, i.e., $L^p(\mathbb{R}^2)\rightarrow L^q(\mathbb{R}^2)$ estimates, of the maximal function $M_{\gamma}$ along a plane curve $(t,\gamma(t))$, where…

We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions $d\ge 2$. In previous works we studied the model problem of a discrete elliptic equation on $\mathbb{Z}^d$. Under the…

The Bochner-Riesz multipliers $ B_{\delta }$ on $ \mathbb R ^{n}$ are shown to satisfy a range of sparse bounds, for all $0< \delta < \frac {n-1}2 $. The range of sparse bounds increases to the optimal range, as $ \delta $ increases to the…

We prove $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to two endpoints. The results can be applied to improve currently known bounds on sparse…

In this paper we study $L_p$-norm spherical copulas for arbitrary $p \in [1,\infty]$ and arbitrary dimensions. The study is motivated by a conjecture that these distributions lead to a sharp bound for the value of a certain generalized mean…

Fix a number field $k$, integers $\ell, n \geq 2$, and a prime $p$. For all $r \geq 1$, we prove strong unconditional upper bounds on the $r$-th moment of $\ell$-torsion in the ideal class groups of degree $p$ extensions of $k$ and of…

We start by providing a very simple and elementary new proof of the classical bound due to J. Beck which states that the spherical cap $\mathbb{L}_2$-discrepancy of any $N$ points on the unit sphere $\mathbb S^d$ in $\mathbb{R}^{d+1}$,…

We apply Christ's method of refinements to the $\ell^p$-improving problem for discrete averages $\mathcal{A}_N$ along polynomial curves in $\mathbb{Z}^d$. Combined with certain elementary estimates for the number of solutions to certain…

We prove sharp weak type weighted estimates for a class of sparse operators that includes majorants of standard $\alpha$-fractional singular integrals, fractional integral operators, Marcinkiewicz integral operators, and square functions.…

In this work we treat the space-time discretization of the generalized Stokes equations in the case of Dirichlet boundary conditions. We prove error estimates in the case $p\in[\frac{2d}{d+2},\infty)$ that are independent of the degeneracy…

We prove $\ell^p\big(\mathbb Z^d\big)$ bounds for $p\in(1, \infty)$, of $r$-variations $r\in(2, \infty)$, for discrete averaging operators and truncated singular integrals of Radon type. We shall present a new powerful method which allows…

The problem of finding the asymptotic behavior of the maximal density of sphere packings in high Euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry. One century ago, Minkowski obtained a…

The Wasserstein distance $\mathcal{W}_p$ is an important instance of an optimal transport cost. Its numerous mathematical properties as well as applications to various fields such as mathematical finance and statistics have been well…

For a smooth $k$-dimensional submanifold $\Sigma$ of a $d$-dimensional compact Riemannian manifold $M$, we extend the $L^p(\Sigma)$ restriction bounds of Burq-G\'erard-Tzvetkov -- originally proved for individual Laplace--Beltrami…

Let $ \Lambda $ denote von Mangoldt's function, and consider the averages \begin{align*} A_N f (x) &=\frac{1}{N}\sum_{1\leq n \leq N}f(x-n)\Lambda(n) . \end{align*} We prove sharp $ \ell ^{p}$-improving for these averages, and sparse bounds…

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…

The zero bias distribution $W^*$ of $W$, defined though the characterizing equation $\mathit{EW}f(W)=\sigma^2Ef'(W^*)$ for all smooth functions $f$, exists for all $W$ with mean zero and finite variance $\sigma^2$. For $W$ and $W^*$ defined…

In this paper we give an estimate on the asymptotic behavior of eigenvalues of discretized elliptic boundary values problems. We first prove a simple min-max principle for selfadjoint operators on a Hilbert space. Then we show two sided…