Related papers: The Discrete Spherical Maximal Function: A new pro…

In this paper, we solve completely the $L^2\to L^r$ extension conjecture for the zero radius sphere over finite fields. We also obtain the sharp $L^p\to L^4$ extension estimate for non-zero radii spheres over finite fields, which improves…

The optimal $L^p \to L^q$ mapping properties for the (local) helical maximal function are obtained, except for endpoints. The proof relies on tools from multilinear harmonic analysis and, in particular, a localised version of the…

In this article, we prove the analogue theorems of Stein-Tomas and Srtichartz on the discrete surface restrictions of Fourier-Hermite transforms associated with the normalized Hermite polynomials and obtain the Strichartz estimate for the…

It is proved that certain discrete analogues of maximally modulated singular integrals of Stein-Wainger type are bounded on $\ell^p(\mathbb{Z}^n)$ for all $p\in (1,\infty)$. This extends earlier work of the authors concerning the case…

We improve the $L^p(\mathbb{R}^n)$ bounds on Stein's square function to the best-known range of the Fourier restriction problem when $n\geq4$. Applications including certain local smoothing estimates are also discussed.

We obtain a necessary and sufficient condition on a polynomial $P(t_1,t_2)$ for the $\ell^{p}$ boundedness of the discrete double Hilbert transforms associated with $P(t)$ for $1 < p < \infty$. The proof is based on the multi-parameter…

Let $ \lambda ^2 \in \mathbb N $, and in dimensions $ d\geq 5$, let $ A_{\lambda } f (x)$ denote the average of $ f \;:\; \mathbb Z ^{d} \to \mathbb R $ over the lattice points on the sphere of radius $\lambda$ centered at $x$. We prove $…

This is the second of two articles in which we prove a sharp $L^p-L^2$ Fourier restriction theorem for a large class of smooth, finite type hypersurfaces in R^3, which includes in particular all real-analytic hypersurfaces.

Consider spherical means on the Heisenberg group with a codimension two incidence relation, and associated spherical local maximal functions $M_Ef$ where the dilations are restricted to a set $E$. We prove $L^p\to L^q$ estimates for these…

The study of high-dimensional distributions is of interest in probability theory, statistics and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The $\ell^p$ spaces…

We prove square function estimates in $L_2$ for general operators of the form $B_1D_1+D_2B_2$, where $D_i$ are partially elliptic constant coefficient homogeneous first order self-adjoint differential operators with orthogonal ranges, and…

We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense -- they cannot be improved by employing polynomials of the same or lower degrees…

We consider perturbed discrete tight-binding models in $\ell^2(\mathbb{Z_h},\mathcal{G})$ describing union of quantum particles with localized interactions, where $\mathbb{Z_h}$ is the 1D lattice $h\mathbb{Z_h}$, $h > 0$, and $\mathcal G$…

We present an elementary approach to prove restriction theorems for particular surfaces for which the Tomas-Stein theorem does not apply, which in turn provide short proofs for well-known Strichartz estimates for associated PDEs. The method…

Let $0<\alpha<1$. We obtain the boundedness of the discrete fractional Hardy-Littlewood maximal operators ${\mathcal M}_\alpha$ on discrete weighted Lebesgue spaces. From this and a discrete version of Whitney decomposition theorem, we…

We establish upper and lower universal bounds for potentials of weighted designs on the sphere $\mathbb{S}^{n-1}$ that depend only on quadrature nodes and weights derived from the design structure. Our bounds hold for a large class of…

We show that the discrete lacunary spherical maximal function is bounded on $l^p(\mathbb{Z}^d)$ for all $p >\frac{d+1}{d-1}$. Our range is new in dimension 4, where it appears that little was previously known for general lacunary radii. Our…

In this article, we study discrete maximal function associated with the Birch-Magyar averages over sparse sequences. We establish sparse domination principle for such operators. As a consequence, we obtain $\ell^p$-estimates for such…

We prove that if a complete connected $n$-dimensional Riemannian manifold $M$ has radial sectional curvature at a base point $p\in M$ bounded from below by the radial curvature function of a two-sphere of revolution $\widetilde M$ belonging…

Let $B = \left\{ B\left( x\right),\, x\in \mathbb{S}^{2}\right\} $ be the fractional Brownian motion indexed by the unit sphere $\mathbb{S}^{2}$ with index $0<H\leq \frac{1}{2}$, introduced by Istas \cite{IstasECP05}. We establish optimal…