arXiv subjects

Youssef Elmadani

Publications and source records attributed to Youssef Elmadani.

4 recordsLinked to original sources

arXiv2026

Douglas Weighted Dirichlet Spaces: Analytic and Probabilistic Aspects

We introduce a class of weighted Dirichlet spaces on the unit disk, called Douglas weighted Dirichlet spaces, characterized by the existence of a Douglas type boundary representation of the weighted Dirichlet integral. This representation naturally induces a nonlocal Dirichlet form on the unit circle in the sense of Beurling Deny and Fukushima. Our main result shows that the reproducing kernel of the weighted Dirichlet space associated with a regular Douglas weighted Dirichlet form is the image of the Szego kernel under the corresponding 1 resolvent. This resolvent representation is new for superharmonic weights. We also develop the potential theory associated with Douglas weighted Dirichlet spaces. We characterize the capacity in terms of reproducing kernels. Finally, as an application of the fact that every Douglas weight induces a regular nonlocal Dirichlet form, together with the general theory of Dirichlet forms, we obtain an associated symmetric pure jump Hunt process. In the classical Dirichlet case, this process is the wrapped Cauchy process.

math.FA↗

arXiv2026

Schauder--Orlicz-Type Estimates for Divergence-Form Elliptic Equations with Lower-Order Terms

Schauder Orlicz-type estimates are derived for weak solutions to second-order linear elliptic equations in divergence form with lower-order terms. The Orlicz setting $X=L^\psi$ is treated first. Under suitable assumptions on the Young function $\psi$ and on the coefficients, the optimal associated space for the lower-order datum is identified. An \textit{a priori} estimate in $W^{1,\psi}$ is then obtained. The discussion is next extended to rearrangement-invariant Banach function spaces. A class $(\mathcal C)$ is introduced to characterize the spaces $X$ for which a corresponding associated space $Y$ yields Schauder-type estimates. Lorentz spaces are finally examined as concrete examples.

math.AP↗

arXiv2024

Stochastic Aggregation Diffusion-Equation : Analysis via Dirichlet Forms

In this article, we study the stochastic aggregation-diffusion equation with a singular drift represented by a monotone radial kernel. We demonstrate the existence and uniqueness of a diffusion process that acts as a weak solution to our equation. This process can be described as a distorted Brownian motion originating from a delocalized point. Utilizing Dirichlet form theory, we prove the existence of a weak solution for a quasi-everywhere point in a state space. However uniqueness is not assured for solutions commencing from points outside polar sets, and explicitly characterizing these sets poses a significant challenge. To address this, we employ the H_2-condition introduced by Albeverio et al.(2003). This condition provides a more thorough understanding of the uniqueness issue within the framework of Dirichlet forms. Consequently the H_2-condition is pivotal in enhancing the analysis of weak solutions, ensuring a more detailed comprehension of the problem. An explicit expression for the generalized Schrödinger operator associated with certain kernels is also provided.

math.PR↗

arXiv2016

Cyclicity and invariant subspaces in the Dirichlet spaces

Let $μ$ be a positive finite measure on the unit circle and $\mathcal{D} (μ)$ the associated Dirichlet space. The generalized Brown-Shields conjecture asserts that an outer function $f \in \mathcal{D} (μ)$ is cyclic if and only if $c\_μ(Z (f))= 0$, where $c\_μ$ is the capacity associated with $\mathcal{D} (μ)$ and $Z(f)$ is the zero set of $f$. In this paper we prove that this conjecture is true for measures with countable support. We also give in this case a complete and explicit characterization of invariant subspaces.

math.CV↗