Related papers: Quantitative $l^p$-improving for discrete spherica…

We study the worst-case approximation of multivariate periodic functions from the weighted Korobov space $H_{d,\alpha,\gamma}$ with smoothness $\alpha>1/2$ in the Lebesgue norm $L_p([0,1]^d)$ for $1\le p\le\infty$. We analyze a \emph{median…

This expository paper provides a unified and pedagogical introduction to optimal quantization for probability measures supported on spherical curves and discrete subsets of the sphere, emphasizing both continuous and discrete settings. We…

The sharp range of $L^p$-estimates for the class of H\"ormander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of…

It is well-known that Littlewood-Paley operators formed with respect to lacunary sets of finite order are bounded on $L^p (\mathbb{R})$ for all $1<p<\infty$. In this note it is shown that $$ \| S_{\mathcal{I}_{E_2}} \|_{L^p (\mathbb{R})…

We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite…

Let $A_tf(x)=\int f(x+ty)d\sigma(y)$ denote the spherical means in $\Bbb R^d$ ($d\sigma$ is surface measure on $S^{d-1}$, normalized to $1$). We prove sharp estimates for the maximal function $M_E f(x)=\sup_{t\in E}|A_tf(x)|$ where $E$ is a…

We prove $L^{p}$ and weighted $L^{p}$ estimates for bounded functions of a selfadjoint operator satisfying both a pointwise gaussian estimate for its heat kernel and a finite speed of propagation property. As an application, we obtain…

We prove sharp $L^p$ estimates for the Steklov eigenfunctions on compact manifolds with boundary in terms of their $L^2$ norms on the boundary. We prove it by establishing $L^p$ bounds for the harmonic extension operators as well as the…

In 2009, W. D. Banks and I. E. Shparlinski studied the average densities of primes $p \leq x$ for which the reductions of elliptic curves of small height modulo $p$ satisfy certain arithmetic properties, namely cyclicity and divisibility of…

We use Oberlin, Nazarov, and Thiele's Multi-Frequency Calder\'{o}n-Zygmund decomposition to lower estimates on maximal multipliers in $L^p$. We also improve on classical multiplier results of Coifman, Rubio de Francia, and Semmes.

We establish sharp $L^p$ integral mean estimates for $(\alpha,\beta)$-harmonic functions on the unit disk. Explicit bounds for the functions and their partial derivatives are obtained in terms of boundary data, by means of the associated…

Let $E/\mathbb{Q}$ be an elliptic curve that has complex multiplication (CM) by an imaginary quadratic field $K$. For a prime $p$, there exists $\theta_p \in [0, \pi]$ such that $p+1-\#E(\mathbb{F}_p) = 2\sqrt{p} \cos \theta_p$. Let $x>0$…

We study the iterative algorithm proposed by S. Armstrong, A. Hannukainen, T. Kuusi, J.-C. Mourrat to solve elliptic equations in divergence form with stochastic stationary coefficients. Such equations display rapidly oscillating…

We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schr\"odinger equation based on recent time discretization and filtering techniques. For this new scheme, we perform a rigorous error analysis and establish…

We design and analyze a posteriori error estimators for the Stokes system with singular sources in suitable $\mathbf{W}^{1,p}\times \mathrm{L}^p$ spaces. We consider classical low-order inf-sup stable and stabilized finite element…

We prove a dispersive estimate for periodic discrete Schr\"odinger operators on the line with optimal rate of decay. Additionally, by standard methods, we deduce dispersive estimates for the discrete nonlinear Schr\"odinger equation with…

We provide novel linear combination of unitaries decompositions for a class of discrete elliptic differential operators. Specifically, Poisson problems augmented with periodic, Dirichlet, Neumann, Robin, and mixed boundary conditions are…

We find the sharp range for boundedness of the discrete bilinear spherical maximal function for dimensions $d \geq 5$. That is, we show that this operator is bounded on $l^{p}(\mathbb{Z}^d)\times l^{q}(\mathbb{Z}^d) \to l^{r}(\mathbb{Z}^d)$…

We prove quenched~$L^p$--type estimates for the gradient of a solution of a quasilinear elliptic equation with random coefficients.

Based on a proper hypothesis on the noncommutative Fourier integral operators, we establish in this paper the strong-type $(p,p)$ (with $2\leq p\leq \infty$) estimates for the operator-valued Stein's maximal spherical means.