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Hicham Arroussi

Publications and source records attributed to Hicham Arroussi.

6 recordsLinked to original sources
arXiv2026

Bergman Projections, Kernel $p$-Norm Estimates, and Toeplitz Operators with Békollé and Bonami weights

In this paper, we establish entirely new $p$-norm estimates for reproducing kernels to characterize the bounded and compact Toeplitz operators $T_μ$ acting between weighted Békollé--Bonami Bergman spaces $A^p_u(\mathbb{D})$ and $A^q_u(\mathbb{D})$ for all positive exponents $0 < p, q < \infty$. These operator-theoretic properties are completely described in terms of generalized Berezin transforms, averaging functions, and Carleson measures. We introduce two explicit conditions on the weights to ensure the boundedness of the weighted Bergman projection $P_u$, generalizing results from Hilbert spaces to Banach spaces.Our work generalizes the main results of Tong, Li, and Arroussi \cite{TLA} from Hilbert spaces to the more general setting of Banach spaces.

math.CV↗
arXiv2025

Toeplitz operators on large vector-valued Fock spaces

We characterize boundedness and compactness of Toeplitz operators on large vector-valued Fock spaces with Dall'Ara's weights [Adv.\ Math., 285 (2015) 1706--1740] in terms of generalized Berezin transforms, averaging functions, and Carleson measures. To determine Schatten class Toeplitz operators, we introduce the operator-valued Berezin transform and averaging functions.

math.FA↗
arXiv2024

A new class of Carleson measures and integral operators on Bergman spaces

Let $n$ be a positive integer and $\mathbf{g}=(g_0,g_1,\cdots,g_{n-1})$, with $g_k\in H(\mathbb{D})$ for $k=0,1,\cdots,n-1$. Let $I_{\mathbf{g}}^{(n)}$ be the generalized Volterra-type operators on $H(\mathbb{C})$, which is represented as $$ I_{\mathbf{g}}^{(n)}f=I^n\left(fg_0+f'g_1+\cdots+f^{(n-1)}g_{n-1}\right), $$ where $I$ denotes the integration operator $$(If)(z)=\int_0^zf(w)dw,$$ and $I^n$ is the $n$th iteration of $I$. This operator is a generalization of the operator that was introduced by Chalmoukis in \cite{Cn}. In this paper, we study the boundedness and compactness of the operator $I_{\mathbf{g}}^{(n)}$ acting on Bergman spaces to another. As a consequence of these characterizations, we obtain conditions for certain linear differential equations to have solutions in Bergman spaces. Moreover, we study the boundedness, compactness and Hilbert-Schmidtness of the following sums of generalized weighted composition operators: Let $\mathbf{u}=(u_0,u_1,\cdots,u_n)$ with $u_k\in H(\mathbb{D})$ for $0\leq k\leq n$ and $φ$ be an analytic self-map of $\mathbb{D}.$ The sums of generalized weighted composition operators is defined by $$L_{\mathbf{u},φ}^{(n)}=\sum_{k=0}^nW_{u_k,φ}^{(k)},$$ where $$W_{u_k,φ}^{(k)}f=u_k\cdot f^{(k)}\circφ.$$ Our approach involves the study of new class of Sobolev-Carleson measures for classical Bergman spaces on unit disk which appears in the first main Theorems \ref{Theorem1.1} and \ref{Theorem1.2}.

math.CV↗