复变函数

In this note, we show the existence of a solution operator to the $\bar\partial-$equation in the polydisc that preserves H\"older regularity. This solution operator is constructed using Henkin's formula. It is a well-known fact that…

In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we provide $L^2$-Dolbeault fine resolutions and isomorphisms, and $L^2$-estimates, for holomorphic line bundles on…

Let $\Omega \subset \mathbb{R}^n$ be a domain and $f \in W^{1,n}_{\text{loc}} (\Omega,\mathbb{R}^n) $. We say that $f$ has a value of finite distortion at $y_0 \in \mathbb{R}^n$ if there exist measurable functions $K \colon \Omega \to…

Karapetrovi\'c conjectured that the norm of the Hilbert matrix operator on the Bergman space $A^p_\alpha$ is equal to $\pi/\sin((2+\alpha)\pi/p)$ when $-1<\alpha<p-2$. In this paper, we provide a proof of this conjecture for $0\leq \alpha…

Given a smooth, open, oriented surface $X$ endowed with a family of complex structures $\{J_b\}_{b\in B}$ depending continuously on the parameter $b$ in a metrisable space $B$, we construct a continuous family of proper holomorphic maps…

We study discrete period matrices associated with graphs cellularly embedded on closed surfaces, resembling classical period matrices of Riemann surfaces. Defined via integrals of discrete harmonic 1-forms, these period matrices are known…

This paper develops the function and operator theory of Hardy--Carleson--type analytic tent spaces $AT_q^\infty(\omega)$ induced by radial weights $\omega$ satisfying a two-sided doubling condition. We first characterize the positive Borel…

We characterise slice-regularity of functions over a real alternative *-algebra using operators that arise in Dunkl operator theory. We present a unifying perspective on hypercomplex analysis by defining a family of function spaces in the…

Let $\Re_n$ be the set of all rational functions of the type $r(z) = p(z)/w(z),$ where $p(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-a_j)$, $|a_j|>1$ for $1\leq j\leq n$. In this paper, we set up some results…

We study the relation between Pick bodies on Carath\'eodory hyperbolic domains and contractions on finite dimensional Hilbert spaces. We give a condition sufficient to realize Pick bodies on Carath\'eodory hyperbolic domains as a Pick body…

Motivated by W. P. Thurston, we ask: What is the shape of a meromorphic function on a simply connected Riemann surface $\Omega_z$? We consider Speiser functions, i.e. meromorphic functions on a simply connected Riemann surface, that have a…

In this paper, we investigate the geometric properties of complex-valued pluriharmonic mappings defined over convex Reinhardt domains in $\mathbb{C}^n$. We first establish a multidimensional analogue of the Noshiro-Warschawski Theorem,…

We construct a Schauder basis for the space $Hol(\mathbb D)$, the space of holomorphic functions on the closed unit disk, consisting entirely of finite Blaschke products. The expansion coefficients are given explicitly. Our result remains…

We solve the $L^p$-regularity problem of the Bergman projection of two-dimensional Rudin ball quotients.

Generalizing the Zalcman conjecture given by $\vert a_n^2 - a_{2n-1}\vert \leq (n-1)^2$, Ma proposed and proved that the inequality $$\vert a_n a_m-a_{n+m-1}\vert \leq (n-1)(m-1), \quad m,n \in \mathbb{N},$$ holds for functions…

Counterexamples show that many results in the geometric function theory of one complex variable are not applicable for several complex variables. In this paper, we obtain sharp bounds for the Zalcman functional for $n=3$ associated with the…

This paper presents sharp estimates for the second-order Toeplitz determinant whose entries are the coefficients of convex functions defined on the unit disk in $\mathbb{C}$. These estimates are further extended to a subclass of holomorphic…

In this paper, we establish an angle duality and a gap principle for convex combinations of incomplete polynomials, extending two results of Ge and Gonek in [IMRN, 2024].Our approach is geometric: we introduce an ``angle gain'' mechanism…

We characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. We then generalize this result to a characterization of normal functions…

In this article, we give a versal deformation for any transversely holomorphic foliation $\mathcal{F}_0$ given by the intersection of the orbits of a holomorphic vector field $\xi$ defined on a neighborhood of the closure of a bounded…