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O. El-Fallah

Publications and source records attributed to O. El-Fallah.

7 recordsLinked to original sources
arXiv2026

Superharmonically Weighted Dirichlet Spaces

In this paper, we consider weighted Dirichlet spaces $\cD_\omega$, where $\omega$ is a positive superharmonic weight on the unit disc $\DD$. These spaces include the standard weighted Dirichlet spaces $\cD_\alpha$ and appear in the description of their invariant subspaces. Our goal is to study the spaces $\cD_\omega$. We show that an explicit description of invariant subspaces reduces to the description of those generated by a bounded outer function, and then to the problem of describing cyclic functions, known as the Brown--Shields conjecture. We develop tools, analogous to those used in the harmonic case, that are needed to treat this problem for superharmonically weighted Dirichlet spaces $\cD_\omega$. In particular, we obtain a formula for the Dirichlet integral of outer functions of Carleson--Richter--Sundberg type, estimates for the norm of the reproducing kernel of $\cD_\omega$, and several properties on the capacity associated with $\cD_\omega$. Using these tools, we provide a description of invariant subspaces when the measure $\Delta \omega$ is finite measure or if the $\supp(\Delta \omega)\cap \TT$ is countable, where $\TT$ denotes the unit circle. Finally, we prove that a smooth outer function $f \in \cD_\alpha$ such that $\cZ (f) $ is "regular" is cyclic in $\cD_\alpha$ if and only if $c_{\alpha }(\cZ(f))= 0$.

math.FA↗
arXiv2021

On singular values of Hankel operators on Bergman spaces

In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces $A^2_{\omega _\varphi}$, where $\omega _\varphi= e^{-\varphi}$ and $\varphi$ is a subharmonic function. We consider compact Hankel operators $H_{\overline {\phi}}$, with anti-analytic symbols ${\overline {\phi}}$, and give estimates of the trace of $h(|H_{\overline \phi}|)$ for any convex function $h$. This allows us to give asymptotic estimates of the singular values $(s_n(H_{\overline {\phi}}))_n$ in terms of decreasing rearrangement of $|\phi '|/\sqrt{\Delta \varphi}$. For the radial weights, we first prove that the critical decay of $(s_n(H_{\overline {\phi}}))_n$ is achieved by $(s_n (H_{\overline{z}}))_n$. Namely, we establish that if $s_n(H_{\overline {\phi}})= o (s_n(H_{\overline {z}}))$, then $H_{\overline {\phi}} = 0$. Then, we show that if $\Delta \varphi (z) \asymp \frac{1}{(1-|z|^2)^{2+\beta}}$ with $\beta \geq 0$, then $s_n(H_{\overline {\phi}}) = O(s_n(H_{\overline {z}}))$ if and only if $\phi '$ belongs to the Hardy space $H^p$, where $p= \frac{2(1+\beta)}{2+\beta}$. Finally, we compute the asymptotics of $s_n(H_{\overline {\phi}})$ whenever $ \phi ' \in H^{p }$.

math.CA↗
arXiv2015

Constructive approximation in de Branges-Rovnyak spaces

In most classical holomorphic function spaces on the unit disk, a function $f$ can be approximated in the norm of the space by its dilates $f\_r(z):=f(rz)~(r \textless{} 1)$. We show that this is \emph{not} the case for the de Branges--Rovnyak spaces $\cH(b)$. More precisely, we give an example of a non-extreme point $b$ of the unit ball of $H^\infty$ and a function $f\in\cH(b)$ such that $\lim\_{r\to1^-}\|f\_r\|\_{\cH(b)}=\infty$. It is known that, if $b$ is a non-extreme point of the unit ball of $H^\infty$, then polynomials are dense in $\cH(b)$. We give the first constructive proof of this fact.

math.FA↗
arXiv2014

Kernel estimate and capacity in Dirichlet type spaces

Let $μ$ be a positive finite measure on the unit circle. The Dirichlet type space $\mathcal{D}(μ)$, associated to $μ$, consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of $μ$. First, we give an estimate of the norm of the reproducing kernel $k^μ$ of $\mathcal{D}(μ)$. Next, we study the notion of $μ$-capacity associated to $\mathcal{D}(μ)$, in the sense of Beurling--Deny. Namely, we give an estimate of $μ$-capacity of arcs in terms of the norm of $k^μ$. We also provide a new condition on closed sets to be $μ$-polar. Note that in the particular case where $μ$ is the Lebesgue measure, this condition coincides with Carleson's condition \cite{Ca}. Our method is based on sharp estimates of norms of some outer test functions which allow us to transfer these problems to an estimate of the reproducing kernel of an appropriate weighted Sobolev space.

math.CV↗
arXiv2010

Level sets and Composition operators on the Dirichlet space

We consider composition operators in the Dirichlet space of the unit disc in the plane. Various criteria on boundedness, compactness and Hilbert-Schmidt class membership are established. Some of these criteria are shown to be optimal.

math.FA↗
arXiv2008

Ideaux fermes d'algebres de Beurling analytiques sur le bidisque

We study the closed ideal in the Beurling algebras $\mathcal{A}^{+}_{α,β}$ of holomorphic function $f$ in the bidisc such that $\sum_{n,m\geq 0}|\hat{f}(n,m)|(1+n)^α(1+m)^β<+\infty$. We determine the function $f\in\mathcal{A}^{+}_{α,β}$ such that the ideals generated by $f$ coincide with the ideal generated by their zeros set.

math.CV↗