ﻻ يوجد ملخص باللغة العربية
We show that discrete singular Radon transforms along a certain class of polynomial mappings $P:mathbb{Z}^dto mathbb{Z}^n$ satisfy sparse bounds. For $n=d=1$ we can handle all polynomials. In higher dimensions, we pose restrictions on the admissible polynomial mappings stemming from a combination of interacting geometric, analytic and number-theoretic obstacles.
The purpose of this paper is to study the sparse bound of the operator of the form $f mapsto psi(x) int f(gamma_t(x))K(t)dt$, where $gamma_t(x)$ is a $C^infty$ function defined on a neighborhood of the origin in $(x, t) in mathbb R^n times mathbb R^k
Consider the discrete cubic Hilbert transform defined on finitely supported functions $f$ on $mathbb{Z}$ by begin{eqnarray*} H_3f(n) = sum_{m ot = 0} frac{f(n- m^3)}{m}. end{eqnarray*} We prove that there exists $r <2$ and universal constant $
Consider the discrete quadratic phase Hilbert Transform acting on $ell^{2}$ finitely supported functions $$ H^{alpha} f(n) : = sum_{m eq 0} frac{e^{2 pi ialpha m^2} f(n - m)}{m}. $$ We prove that, uniformly in $alpha in mathbb{T}$, there is a sparse
We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and
We exhibit a range of $ell ^{p}(mathbb{Z}^d)$-improving properties for the discrete spherical maximal average in every dimension $dgeq 5$. The strategy used to show these improving properties is then adapted to establish sparse bounds, which extend t