arXiv subjects

Soumia Touhami

Publications and source records attributed to Soumia Touhami.

5 recordsLinked to original sources

arXiv2024

A phase space localization operator in negative binomial states

We are dealing with some spectral properties of a phase space localization operator PR corresponding to the indicator function of a disk of radius R < 1. The localization procedure is achieved with respect to a set of negative binomial states (NBS) labeled by points of the complex unit disk D and depending on a parameter 2B > 1. We derive a formula expressing PR as function of the pseudo harmonic oscillator whose potential function depends on B. The phase space content outside the localization domain is estimated in terms of the photon counting probability distribution associated with the NBS. By using the coherent states transform attached to NBS, we transfer the action of the operator PR to a Bergman space AB(D) of analytic functions on D satisfying a growth condition depending on B and we explicitly give its integral kernel whose limit as R goes to 1 coincides with the reproducing kernel of AB(D). This leads to a natural generalization of this Hilbert space with respect to the parameter R.

math-ph↗

arXiv2019

A boundary formula for reproducing kernel Hilbert spaces of real harmonic functions in Lipschitz domains

This paper develops a new Hilbert space method to characterize a family of reproducing kernel Hilbert spaces of real harmonic functions in a bounded Lipschitz domain $Ω\subset \mathbb R^d, d\geq 2$ involving some families of positive self-adjoint operators and making use of characterizations of their trace data and of a special inner product on $H^1(Ω).$ We also establish boundary representation results for this family in terms of the $L^2-$ Bergman kernel. In particular, a boundary integral representation for the very weak solution of the Dirichlet problem for Laplace's equation with $L^2- $ boundary data is provided. Reproducing kernels and orthonormal bases for the harmonic spaces are also found.

math.AP↗

arXiv2018

Functional characterizations of trace spaces in Lipschitz domains

Using a factorization theorem of Douglas, we prove functional characterizations of trace spaces $H^s(\partial Ω)$ involving a family of positive self-adjoint operators. Our method is based on the use of a suitable operator by taking the trace on the boundary $\partial Ω$ of a bounded Lipschitz domain $Ω\subset \mathbb R^d$ and applying Moore--Penrose pseudo-inverse properties together with a special inner product on $H^1(Ω)$. Moreover, generalized results of the Moore--Penrose pseudo-inverse are also established.

math.FA↗

arXiv2018

Harmonic and Trace Inequalities in Lipschitz Domains

We prove boundary inequalities in arbitrary bounded Lipschitz domains on the trace space of Sobolev spaces. For that, we make use of the trace operator, its Moore-Penrose inverse, and of a special inner product. We show that our trace inequalities are particularly useful to prove harmonic inequalities, which serve as powerful tools to characterize the harmonic functions on Sobolev spaces of non-integer order.

math.FA↗

arXiv2018

Riesz bases for $L^2(\partial Ω)$ and regularity for the Laplace equation in Lipschitz domains

In a paper from 1996, D. Jerison and C. Kenig among other results provided a $H^{1/2}$ regularity result for the Dirichlet problem for the Laplace equation in Lipschitz domains. In this article, we adopt a Hilbertian approach to construct two Riesz bases for $L^2(\partial Ω),$ which will allow to find in a different way some of the results of D. Jerison and C. Kenig, and G. Savaré (1998) about the regularity issue of the Laplace equation.

math.AP↗