arXiv subjects

Roberto Bramati

Publications and source records attributed to Roberto Bramati.

8 recordsLinked to original sources

arXiv2026

Jittered sampling and probability measures

This paper investigates the discrepancy of a family of random sampling methods obtained by perturbing the grid $\frac{1}{M}\mathbb{Z}^{d}\cap\left[ -1/2,1/2\right)^{d}$, where $M$ is a large positive integer. Parameterized by an arbitrary probability measure $\mu$, this family encompasses several classical methods for evaluating the quality of an $N$-point set in $\mathbb{T}^{d}$, where $N=M^{d}$. We show that all probability measures, except for Dirac measures, behave like the Lebesgue measure in the Monte Carlo discrepancy. This represents a limiting case where the measure $\mu$ depends on $M$. In this latter context, we prove that, up to a constant, the lowest possible discrepancy is achieved when the support of $\mu$ has diameter $\leq c/M$, and that this upper bound is sharp.

math.NT↗

arXiv2026

Discrepancy estimates for multi-dimensional non-smooth convex bodies: a case study

We study $L^2$-averaged discrepancies of finite sequences of points in the torus $\mathbb{T}^d$ with respect to translated and dilated copies of convex bodies with non-smooth boundary. Under suitable anisotropic assumptions on the decay of the Fourier transform of the body, we prove matching lower and upper bounds for the averaged discrepancy, obtaining the rate $ N^{1 - \frac{d+1}{d^2+d-1}}$. This yields an intermediate regime between smooth convex bodies and polytopes and recovers the known exponent $2/5$ in dimension $d=2$. The argument relies on harmonic analysis techniques combined with averaging procedures adapted to the anisotropic setting. As an application, we analyze a class of convex bodies exhibiting mixed geometric features, including flat regions, curved parts, and edges.

math.CA↗

arXiv2024

Periodic layer potentials and domain perturbations

In this paper, we review the construction of periodic fundamental solutions and periodic layer potentials for various differential operators. Specifically, we focus on the Laplace equation, the Helmholtz equation, the Lam\'e system, and the heat equation. We then describe how these layer potentials can be applied to analyze domain perturbation problems. In particular, we present applications to the asymptotic behavior of quasi-periodic solutions for a Dirichlet problem for the Helmholtz equation in an unbounded domain with small periodic perforations. Additionally, we investigate the dependence of spatially periodic solutions of an initial value Dirichlet problem for the heat equation on regular perturbations of the base of a parabolic cylinder.

math.AP↗

arXiv2023

Continuous harmonic functions on a ball that are not in $H^s$ for $s>1/2$

We show that there are harmonic functions on a ball ${\mathbb{B}_n}$ of $\mathbb{R}^n$, $n\ge 2$, that are continuous up to the boundary (and even Hölder continuous) but not in the Sobolev space $H^s(\mathbb{B}_n)$ for any $s$ sufficiently big. The idea for the construction of these functions is inspired by the two-dimensional example of a harmonic continuous function with infinite energy presented by Hadamard in 1906. To obtain examples in any dimension $n\ge 2$ we exploit certain series of spherical harmonics. As an application, we verify that the regularity of the solutions that was proven for a class of boundary value problems with nonlinear transmission conditions is, in a sense, optimal.

math.AP↗

arXiv2022

The Functional Analytic Approach for quasi-periodic boundary value problems for the Helmholtz equation

We lay down the preliminary work to apply the Functional Analytic Approach to quasi-periodic boundary value problems for the Helmholtz equation. This consists in introducing a quasi-periodic fundamental solution and the related layer potentials, showing how they are used to construct the solutions of quasi-periodic boundary value problems, and how they behave when we perform a singular perturbation of the domain. To show an application, we study a nonlinear quasi-periodic Robin problem in a domain with a set of holes that shrink to points.

math.AP↗

arXiv2022

The resonances of the Capelli operators for small split orthosymplectic dual pairs

Let $(G,G')$ be a reductive dual pair in $Sp(W)$ with rank $G\leq$ rank $G'$ and $G'$ semisimple. The image of the Casimir element of the universal enveloping algebra of $G'$ under the Weil representation $ω$ is a Capelli operator. It is a hermitian operator acting on the smooth vectors of the representation space of $ω$. We compute the resonances of a natural multiple of a translation of this operator for small split orthosymplectic dual pairs. The corresponding resonance representations turn out to be $GG'$-modules in Howe's correspondence. We determine them explicitly.

math.RT↗

arXiv2021

A family of sharp inequalities on real spheres

We prove a family of sharp multilinear integral inequalities on real spheres involving functions that possess some symmetries that can be described by annihilation by certain sets of vector fields. The Lebesgue exponents involved are seen to be related to the combinatorics of such sets of vector fields. Moreover we derive some Euclidean Brascamp--Lieb inequalities localized to a ball of radius $R$, with a blow-up factor of type $R^δ$, where the exponent $δ>0$ is related to the aforementioned Lebesgue exponents, and prove that in some cases $δ$ is optimal.

math.CA↗

arXiv2020

Oscillating spectral multipliers on groups of Heisenberg type

We establish endpoint estimates for a class of oscillating spectral multipliers on Lie groups of Heisenberg type. The analysis follows an earlier argument due to the second and fourth author but requires the detailed analysis of the wave equation on these groups due to Müller and Seeger. We highlight and develop the connection between sharp bounds for oscillating multipliers and the problem of determining the minimal amount of smoothness required for Mihlin-Hörmander multipliers, a problem that was solved for groups of Heisenberg type but remains open for other groups.

math.FA↗