arXiv subjects

Pouriya Torkinejad Ziarati

Publications and source records attributed to Pouriya Torkinejad Ziarati.

4 recordsLinked to original sources

arXiv2026

Cyclicity in Poletsky-Stessin Weighted Bergman Spaces

We study the cyclicity of polynomials in Poletsky-Stessin weighted Bergman spaces on various domains in $\mathbb{C}^2$, including the unit ball, the bidisk, and the complex ellipsoid. To this end, we introduce a natural extension of the parameter range for Poletsky-Stessin weighted Bergman spaces on complete Reinhardt domains, yielding a family of spaces that resemble Dirichlet-type spaces on the unit ball. We highlight the differences in the cyclicity behavior of polynomials in these spaces on the bidisk compared to those studied by Bénéteau et al. Finally, we propose several open problems concerning the structure of cyclic polynomials in these spaces.

math.CV↗

arXiv2026

Critical cyclicity in Dirichlet-type spaces on the bidisk

Consider the Dirichlet-type spaces on the bidisc defined by $$\mathcal{D}_β(\mathbb{D}^2)=\Bigg\{f(z_1,z_2)=\sum_{k,l}a_{kl}z_1^kz_2^l\in\mathcal{O}(\mathbb{D}^2): \sum_{k,l\ge 0}|a_{kl}|^2(k+l+1)^β<+\infty\Bigg\}.$$ Given $β_c\in(0,2],$ we construct a function $f$ that belongs to the Dirichlet-type space $\mathcal D_{2}(\mathbb{D}^2)$ of the bidisk and is cyclic in $\mathcal D_β(\mathbb{D}^2)$ if and only if $β\leq β_{c}.$ We also show that the critical index satisfies $β_c=2-\mathrm{dim}_H(\mathcal{Z}(f)\cap \mathbb{T}^2),$ where $\mathrm{dim_H}(\mathcal{Z}(f)\cap \mathbb{T}^2)$ is the Hausdorff dimension of the zero set of the function $f$ on the two-torus $\mathbb{T}^2.$

math.CV↗

arXiv2026

Cyclicity of Multipliers on the Unit Ball of $\mathbb{C}^n$: A Corona-Based Approach

We study the cyclicity of multipliers in Dirichlet-type spaces $ D_α(\mathbb{B}_n) $. Specifically, we show that a multiplier $ f $ analytic on a neighborhood of $\overline{\mathbb{B}}_n$, whose zero set on the unit sphere is a compact, smooth, complex tangential submanifold of real dimension $ m \leq n - 1 $, is cyclic in $ D_α(\mathbb{B}_n) $ if and only if $ α\leq \frac{2n - m}{2} $. Our approach combines classical results on peak sets in $ A^\infty(\mathbb{B}_n) $ due to Chaumat and Chollet with a Corona-type theorem with two generators for the multiplier algebra.

math.CV↗

arXiv2026

Examples of critically cyclic functions in the Dirichlet spaces of the ball

In this work, we construct examples of holomorphic functions in $D_2(\B_2)$, the Dirichlet space on $\B_2$, for which there exists an index $α_c \in [\frac12,2]$ such that the function is cyclic in $D_α(\B_2)$ if and only if $α\leq α_c$. To this end, we use the notion of \emph{interpolation sets} in smooth ball algebras, as studied by Bruna, Ortega, Chaumat, and Chollet.

math.CV↗