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We prove several new rigidity results for polynomial automorphisms of $\mathbb C^2$ with positive entropy. A first result is that a complex slice of the (forward or backward) Julia set is never a smooth, or even rectifiable, curve. We also show that such an automorphism cannot preserve a global holomorphic foliation, nor a real-analytic foliation with complex leaves. These results are used to show that under mild assumptions, two real-analytically conjugate automorphisms are polynomially conjugate. For mappings defined over a number field, we also study the fields of definition of multipliers of saddle periodic orbits., Comment: Some new results compared to the initially released preprint version

We consider a class of external potentials on the complex plane $\mathbb{C}$ for which the coincidence set to the obstacle problem contains a Jordan curve in the exterior of the droplet. We refer to this curve as a spectral outpost. We study the corresponding Coulomb gas at $\beta=2$. Generalizing recent work in the radially symmetric case, we prove that the number of particles which fall near the spectral outpost has an asymptotic Heine distribution, as the number of particles $n\to\infty$.We also consider a class of potentials with disconnected droplets whose connected components are separated by a ring-shaped spectral gap. We prove that the fluctuations of the number of particles that fall near a given component has an asymptotic discrete normal distribution, which depends on $n$. For the case of disconnected droplets we also consider fluctuations of general smooth linear statistics and show that they tend to distribute as the sum of a Gaussian field and an independent, oscillatory, discrete Gaussian field. Our techniques involve a new asymptotic formula on the norm of monic orthogonal polynomials in the bifurcation regime and a variant of the method of limit Ward identities of Ameur, Hedenmalm, and Makarov., Comment: 37 pages

We prove that the sublevel set $\big\{z\in\mathbb D\colon k_{\mathbb D}\big(z,z_0\big)-k_{\mathbb D}\big(f(z),w_0\big)<\mu\big\}$, ${\mu\in\mathbb R}$, is geodesically convex with respect to the Poincar\'e distance $k_{\mathbb D}$ in the unit disk $\mathbb D$ for every ${z_0,w_0\in\mathbb D}$ and every holomorphic ${f:\mathbb D\to\mathbb D}$ if and only if ${\mu\leqslant0}$. An analogous result is established also for the set $\{z\in\mathbb D \colon 1-|f(z)|^2<\lambda(1-|z|^2)\}$, ${\lambda>0}$. This extends a result of Solynin (2007) and solves a problem posed by Arango, Mej\'{\i}a and Pommerenke (2019). We also propose several open questions aiming at possible extensions to more general settings.

In this paper, we construct a quasiregular mapping $f$ in $\mathbb{R}^3$ that is the first to illustrate several important properties: the quasi-Fatou set contains bounded, hollow components, the Julia set contains bounded components and, moreover, some of these components are genuine round spheres. The key tool to this construction is a new quasiregular interpolation in round rings in $\mathbb{R}^3$ between power mappings of differing degrees on the boundary components. We also exhibit the flexibility of constructions based on these interpolations by showing that we may obtain quasiregular mappings which grow as quickly, or as slowly, as desired.

This paper investigates flat webs on the projective plane. We present two methods for constructing such webs: the first involves taking the product of two convex reduced foliations and invariant lines, while the second consists of taking the product of finitely many convex homogeneous foliations and invariant lines. In both cases, we demonstrate that the dual web is flat.

Let $M$ be a compact complex manifold, and $D\, \subset\, M$ a reduced normal crossing divisor on it, such that the logarithmic tangent bundle $TM(-\log D)$ is holomorphically trivial. Let ${\mathbb A}$ denote the maximal connected subgroup of the group of all holomorphic automorphisms of $M$ that preserve the divisor $D$. Take a holomorphic Cartan geometry $(E_H,\,\Theta)$ of type $(G,\, H)$ on $M$, where $H\, \subset\, G$ are complex Lie groups. We prove that $(E_H,\,\Theta)$ is isomorphic to $(\rho^* E_H,\,\rho^* \Theta)$ for every $\rho\, \in\, \mathbb A$ if and only if the principal $H$--bundle $E_H$ admits a logarithmic connection $\Delta$ singular on $D$ such that $\Theta$ is preserved by the connection $\Delta$., Comment: Final version to appear in Journal of Differential Geometry and its Applications

Here we give a survey of consequences from the theory of the Beltrami equation in the complex plane $\mathbb C$ to generalized Cauchy-Riemann equation in the real plane $\mathbb R^2$ and clarify the relationships of the latter to the $A-$harmonic equation ${\rm div} A(z)\,{\rm grad}\, u(z) = 0$, $z=x+iy$, with matrix valued coefficients $A$ that is one of the main equations of the potential theory, namely, of the hydro\-mechanics (fluid mechanics) in anisotropic and inhomogeneous media. The survey includes various types of results as theorems on existence, representation and regularity of its solutions, in particular, for the main boundary value problems of Hilbert, Dirichlet, Neumann, Poincare and Riemann., Comment: 35 pages

Given a complex compact manifold $X$, we prove a Baum-Bott type formula for one-dimensional foliations on $X$, logarithmic along a hypersurface with isolated singularities. In this case, we show that the residues of the singularities of foliations can be given in terms of the Schwartz index. Furthermore, in this context, we proved that the Schwartz index is positive when $dim(X)$ is even, and that the GSV index is positive when $dim(X)$ is odd. We also obtained some applications in the context of the Poincar\'e's problem.

The purpose of this paper is to make an introduction to univalent function theory for readers of any level, assuming only foundational knowledge in real and complex analysis. In particular, we state and proof (with details) important theorems utilised in proving Bieberbach's conjecture, especially those that were missed or merely sketched by other texts on univalent functions. We will finally prove the conjecture following the proof given by Lenard Weinstein in a comprehensive and self-contained manners, exposing its full technical details in each step. Readers are encouraged to make used of this paper to accompany the preliminary chapters of Duren's Univalent Functions and Lenard Weinstein's the Bieberbach Conjecture., Comment: 74 pages, 0 figures

This paper studies the theoretical construction and analytic error estimation of complex Bessel function-based conformal mappings in regions with randomly perturbed boundaries. First, we construct a conformal mapping applicable to such boundary conditions and prove the existence and uniqueness of the mapping. On this basis, an analytical error estimation method is proposed to quantify the effect of the magnitude of the boundary perturbation on the accuracy of the mapping. By deriving the error formula, we show the stability of the complex Bessel function under perturbed boundary conditions and prove the asymptotic convergence of the mapping error under small perturbation conditions. This study provides new theoretical support for conformal mapping under complex boundary conditions and reveals the potential of complex Bessel functions in dealing with stochastic boundary problems.

We study the action of symmetric PDO on the module of anti-symmetric functions in $z_1, \dots, z_k$. We show that over the Weyl algebra of the elementary symmetric functions, this module is simple.

We prove that, for a spherical Fano threefold not in the Mori-Mukai family 2-29, and a weight function associated with the action of the connected center of a Levi subgroup of its automorphism group, weighted K-polystability is equivalent to vanishing of the weighted Futaki invariant. This is surprising since unlike the case of toric Fano manifold, there exist non-product, special, equivariant test configurations. For the K\"ahler-Einstein Fano threefold 2-29, and for well-chosen torus action on the three dimensional quadric, we show that this property is false and exhibit explicit examples of weighted optimal degenerations. We then generalize this to higher-dimensional quadrics and blowups of quadrics along a codimension 2 subquadric., Comment: 17 pages

We give very simple proofs of the classical results of Magnus and Hill on the spectral properties of the Hilbert matrix $$ H = \left ( {1 \over i+j+ 1 } \right )_{i,j\geq 0} $$ which defines a bounded linear operator on the sequence space $\ell^2$. In particular, we use the Mehler-Fock transform to find the spectrum and the latent eigenfunctions of the Hilbert matrix, that is, we show that the spectrum of $H$ is $[0,\pi]$ with no eigenvalues (Magnus' result) and describe all complex sequences $x$ such that $Hx=\mu x$ for some complex number $\mu$ (Hill's result)., Comment: To appear in Operator Theory: Advances and Applications (IWOTA 2023)

A simple necessary and sufficient condition is given for an absolutely convergent Dirichlet series with imaginary exponents and only real zeros to be a finite product of sines. The proof is based on Meyer's theorem on quasicrystals., Comment: 5 pages, 17 references

The effective freeness in Fujita's conjecture states that, for an ample line bundle $L$ on a complex projective manifold $X$, the adjoint bundle $K_X\otimes L^{\otimes m}$ is globally generated when $m \geq \dim_{\mathbb C} X + 1$. Following the approach of Angehrn and Siu, a solution is provided in this paper via the use of adjoint ideal sheaves, which provide a finer control of the non-integrable loci given by multiplier ideal sheaves, so that one can work directly with the lc singularities and the associated (minimal) lc centres as in the algebraic approaches of Kawamata and Helmke. The substitute for the Nadel or Kawamata-Viehweg vanishing theorem used in previous approaches is an extension theorem based on the techniques developed for the injectivity theorems., Comment: 40 pages

Let $F(z)=\prod\limits_{k=1}^n(z-z_k)$ be a monic complex polynomial of degree $n$. In 1998, Pawlowski [Trans. Amer. Math. Soc. 350 (1998)] studied the radius $\gamma_n$ of the smallest concentric disk with center at $\tfrac{\sum\limits_{k=1}^nz_k}{n}$ contained at least one critical point of $F(z)$. He showed that $\gamma_n\le \tfrac{2n^\frac{1}{n-1}}{n^\frac{2}{n-1}+1}$. In this paper, we refine Pawlowski's result in the spirit of Borcea variance conjectures and classic Schoenberg inequality, specifically, we show that $\gamma_n\le\sqrt{\tfrac{n-2}{n-1}}$ in a very concise manner. Moreover, we obtain various generalizations of Schoenberg inequalities based on classic Schoenberg inequality including refining Lin-Xie-Zhang's result [J. Math. Anal. Appl. 502 (2021)], which is inspired by the author's recent work on Clarkson-McCarthy inequalities [arXiv:2410.21961] . Finally, we make $D$-companion matrix introduced by Cheung-Ng [Linear Algebra Appl. 432 (2010)] and operator inequalities involved Schatten $p$-norm react so that we provide an additional relationship between the zeros of $F(z)$ and its critical points in the case where all $z_k\ge 0$, which can be regarded as complements of Schmeisser's result [Comput. Methods Funct. Theory 3 (2003)]. By an operator 2-norm identity, we also prove Sendov conjecture with a condition that depends only on $\left( \tfrac{1}{n-1}\sum\limits_{k=1}^{n-1}\left|z_k-z_n\right|^2\right)^\frac{1}{2}$., Comment: 16 pages. All comments are welcome!

In this paper we investigate the Gromov hyperbolicity of the classical Kobayashi and Hilbert metrics, and the recently introduced minimal metric. Using the linear isoperimetric inequality characterization of Gromov hyperbolicity, we show if these metrics have an "expanding property" near the boundary, then they are Gromov hyperbolic. This provides a new characterization of the convex domains whose Hilbert metric is Gromov hyperbolic, a new proof of Balogh-Bonk's result that the Kobayashi metric is Gromov hyperbolic on a strongly pseudoconvex domain, a new proof of the second author's result that the Kobayashi metric is Gromov hyperbolic on a convex domain with finite type, and a new proof of Fiacchi's result that the minimal metric is Gromov hyperbolic on a strongly minimally convex domain. We also characterize the smoothly bounded convex domains where the minimal metric is Gromov hyperbolic., Comment: 44 pages. Comments welcome!

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\geq 0}$ with entries $\mu_{n,k}=\mu_{n+k}$, where $\mu_{n}=\int_{[0,1)}t^nd\mu(t)$, induces, formally, the Derivative-Hilbert operator $$\mathcal{DH}_\mu(f)(z)=\sum_{n=0}^\infty\left(\sum_{k=0}^\infty \mu_{n,k}a_k\right)(n+1)z^n , ~z\in \mathbb{D},$$ where $f(z)=\sum_{n=0}^\infty a_nz^n$ is an analytic function in $\mathbb{D}$. We characterize the measures $\mu$ for which $\mathcal{DH}_\mu$ is a bounded operator on $BMOA$ space. We also study the analogous problem from the $\alpha$-Bloch space $\mathcal{B}_\alpha(\alpha>0)$ into the $BMOA$ space., Comment: arXiv admin note: text overlap with arXiv:2410.20435

Let $M$ be a compact complex $n$-manifold. A Gauduchon metric is a Hermitian metric whose fundamental 2-form $\omega$ satisfies the equation $dd^c(\omega^{n-1})=0$. Paul Gauduchon has proven that any Hermitian metric is conformally equivalent to a Gauduchon metric, which is unique (up to a constant multiplier) in its conformal class. Then $d^c(\omega^{n-1})$ is a closed $(2n-1)$-form; the set of cohomology classes of all such forms, called the Lee-Gauduchon cone, is a convex cone, superficially similar to the Kahler cone. We prove that the Lee-Gauduchon cone is a bimeromorphic invariant, and compute it for several classes of non-Kahler manifolds., Comment: version 2.0, 17 pages, we answered Question 5.7 by an example, which is Example 5.7 in this version; no other changes

Very recently, the concept of generalized partial-slice monogenic (or regular) functions has been introduced to unify the theory of monogenic functions and of slice monogenic functions over Clifford algebras. Inspired by the work of A. Perotti, in this paper we provide two analogous versions of the Almansi decomposition in this new setting. Additionally, two enhancements of the Fueter-Sce theorem have been obtained for generalized partial-slice regular functions., Comment: 18 pages

We demonstrate that the weight operator associated with a submultiplicative filtration on the section ring of a polarized complex projective manifold is a Toeplitz operator. We further analyze the asymptotics of the associated weighted Bergman kernel, presenting the local refinement of earlier results on the convergence of jumping measures for submultiplicative filtrations towards the pushforward measure defined by the corresponding geodesic ray., Comment: 31 pages

We review the multivariate holomorphic functional calculus for tuples in a commutative Banach algebra and establish a simple "na\"ive" extension to commuting tuples in a general Banach algebra. The approach is na\"ive in the sense that the na\"ively defined joint spectrum maybe too big. The advantage of the approach is that the functional calculus then is given by a simple concrete formula from which all its continuity properties can easily be derived. We apply this framework to multivariate functions arising as divided differences of a univariate function. This provides a rich set of examples to which our na\"ive calculus applies. Foremost, we offer a natural and straightforward proof of the Connes-Moscovici Rearrangement Lemma in the context of the multivariate holomorphic functional calculus. Secondly, we show that the Daletski-Krein type noncommutative Taylor expansion is a natural consequence of our calculus. Also Magnus' Theorem which gives a nonlinear differential equation for the $\log$ of the solutions to a linear matrix ODE follows naturally and easily from our calculus. Finally, we collect various combinatorial related formulas.

In this paper, we first prove that a compact K\"ahler manifold is projective if it satisfies certain quasi-positive curvature conditions, including quasi-positive $S_2^\perp,\, S_2^+,\,\mbox{Ric}_3^\perp, \,\mbox{Ric}_3^+$ or $2$-quasi-positive $\mbox{Ric}_k$. Subsequently, we prove that a compact K\"ahler manifold with a restricted holonomy group is both projective and rationally conected if it satisfies some non-negative curvature condition, including non-negative $S_2^\perp,\, S_2^+,\,\mbox{Ric}_3^\perp, \,\mbox{Ric}_3^+$ or $2$-non-negative $\mbox{Ric}_k$.

P\'olya in 1926 showed that the hypergeometric function $F(z)=\null_2F_1(a,b;c;z)$ has a totally monotone sequence as its coefficients; that is, $F$ is the generating function of a Hausdorff moment sequence, when $0\le a\le 1$ and $0\le b\le c.$ In this paper, we give a complete characterization of such hypergeometric functions $F$ in terms of complex parameters $a,b,c.$ To this end, we study the class of general properties of generating functions of Hausdorff moment sequences and, in particular, we provide a sufficient condition for the class by making use of a Phragm\`en-Lindel\"of type theorem. As an application, we give also a necessary and sufficient condition for a shifted hypergeometric function to be universally starlike., Comment: 15 pages

Let $\mathcal{H}$ be the space of all functions that are analytic in $\mathbb{D}$. Let $\mathcal{A}$ denote the family of all functions $f\in\mathcal{H}$ and normalized by the conditions $f(0)=0=f'(0)-1$. In 2011, Obradovi\'{c} and Ponnusamy introduced the class $\mathcal{M}(\lambda)$ of all functions $f\in\mathcal{A}$ satisfying the condition $\left|z^2\left(z/f(z)\right)''+f'(z)\left(z/f(z)\right)^2-1\right|\leq \lambda$ for $z\in\mathbb{D}$ with $\lambda>0$. We show that the class $\mathcal{M}(\lambda)$ is preserved under omitted-value transformation, but this class is not preserved under dilation. In this paper, we investigate the largest disk in which the property of preservation under dilation of the class $\mathcal{M}:=\mathcal{M}(1)$ holds. We also address a radius property of the class $\mathcal{M}(\lambda)$ and a number of associated results pertaining to $\mathcal{M}$. Furthermore, we examine the largest disks with sharp radius for which the functions $F$ defined by the relations $g(z)h(z)/z$, $z^2/g(z)$, and $z^2/\int_0^z (t/g(t))dt$ belong to the class $\mathcal{M}$, where $g$ and $h$ belong to some suitable subclasses of $\mathcal{S}$, the class of univalent functions from $\mathcal{A}$. In the final analysis, we obtain the sharp Bohr radius, Bohr-Rogosinski radius and improved Bohr radius for a certain subclass of starlike functions., Comment: 26 Pages, 9 figures, Latex-V1

The primary objective of this paper is to establish several sharp versions of improved Bohr inequality, refined Bohr inequality, and Bohr-Rogosinski inequality for the class of $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\Bbb{D}:=\{z\in\mathbb{C}: |z|<1\}$. In this setting, the analytic component is subordinate to an analytic function $\phi$ in $\Bbb{D}$, where $\phi$ is either a convex or a general univalent function. To do these, we employ the non-negative quantity $S_r(h)$ and the concept of replacing the initial coefficients with the absolute values of the analytic function and its derivative in the majorant series. Furthermore, we obtain the sharp Bohr radius and Bohr-Rogosinski inequality for sense-preserving $K$-quasiconformal harmonic mappings in which the analytic part is subordinate to a function belonging to the family of concave univalent functions with opening angle $\pi \alpha$ at infinity, where $\alpha\in[1,2]$., Comment: 26 Pages, Latex V1

We prove a Baum-Bott type residual formula for one-dimensional holomorphic foliations, with isolated singularities, and logarithmic along free divisors. More precisely, this provides a Baum--Bott theorem for a foliated triple $(X,\mathscr{F}, D)$, where $ \mathscr{F}$ is a foliation by curves and $D $ is a free divisor on a complex manifold $X$. From the local point of view, we show that the log Baum--Bott residues are a generalization of the Aleksandrov logarithmic index for vector fields with isolated singularities on hypersurfaces. We also show how these new indices are related to Poincar\'e's Problem for foliations by curves. In the case of foliated surfaces, we show that the differences between the logarithmic residues and Baum--Bott indices along invariant curves can be expressed in terms of the GSV and Camacho--Sad indices. We also obtain a Baum--Bott type formula for singular varieties via log resolutions. Finally, we prove a weak global version of the Lipman--Zariski conjecture for compact algebraic surfaces., Comment: 31 pages

In this paper, we first establish an improved Bohr inequality for the class of operator-valued holomorphic functions $f$ on a simply connected domain $\Omega$ in $\mathbb{C}$. Next, we establish a generalization of refined version of the Bohr inequality and the Bohr-Rogosinski inequality with the help of the sequence $\varphi=\{\varphi_n(r) \}^{\infty}_{n=0}$ of non-negative continuous functions in $[0,1)$ such that the series $\sum_{n=0}^{\infty}\varphi_n(r)$ converges locally uniformly on the interval $[0,1)$. All the results are proved to be sharp. Moreover, We establish the Bohr inequality and the Bohr-Rogosinski inequality for the class of operator-valued $\nu$-Bloch functions defined in two different simply connected domains, $\Omega$ and $\Omega_{\gamma}$, in $\mathbb{C}$.

We provide a general quantified Ingham-Karamata Tauberian theorem with a flexible one-sided Tauberian condition under several types of boundary behavior for the Laplace transform. Our results in particular improve a theorem by Stahn, removing a vexing restriction on the growth of the Laplace transform. Improving existing optimality results, we also show that the obtained quantified rate is optimal in almost all cases., Comment: 58 pages

We consider mappings satisfying an upper bound for the distortion of families of curves. We establish lower bounds for the distortion of distances under such mappings. As applications, we obtain theorems on the discreteness of the limit mapping of a sequence of mappings converging locally uniformly. We separately consider cases when mappings are defined in Euclidean $n$-dimensional space and in a metric space.

We prove the convergence of layer potential operators for the harmonic transmission problem over a sequence of converging two-sided extension domains. Consequently, the Neumann-Poincar{\'e} operators, Calder{\'o}n projectors, and associated Neumann series converge in this setting. As a result, we generalize the notion of Cauchy integrals and, in a sense, of Hilbert transforms for a class of extension domains. Our approach relies on dyadic approximations of arbitrary open sets, considering convergence in terms of characteristic functions, Hausdorff distance, and compact sets.

The best uniform rational approximation of the Sign function on two intervals separated by zero was explicitly found by E.I. Zolotar\"ev in 1877. The natural extension of this problem to three bands was solved by E.Stiefel in 1961. We indicate the solutions overlooked by the prominent geometer and study their properties., Comment: 13 pages, 5 figures

Exponential actions defined by vector configurations provide a universal framework for several constructions of holomorphic dynamics, non-Kaehler complex geometry, toric geometry and topology. These include leaf spaces of holomorphic foliations, intersections of real and Hermitian quadrics, the quotient construction of toric varieties, LVM and LVMB manifolds, complex-analytic structures on moment-angle manifolds and their partial quotients. In all of these cases, the geometry and topology of the appropriate quotient object can be described by combinatorial data including a pair of Gale dual vector configurations., Comment: 50 pages

The primary objective of this paper is to establish several sharp versions of improved Bohr inequality, refined Bohr-type inequality, and refined Bohr-Rogosinski inequality for the class of $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| < 1\}$. In order to achieve these objectives, we employ the non-negative quantity $S_\rho(h)$ and the concept of replacing the initial coefficients of the majorant series by the absolute values of the analytic function and its derivative, as well as other various settings. Moreover, we obtain the sharp Bohr-Rogosinski radius for harmonic mappings in the unit disk by replacing the bounding condition on the analytic function $h$ with the half-plane condition., Comment: 24 Pages, Latex V1

We study a generalization of the Schramm-Loewner evolution loop measure to pairs of non-intersecting Jordan curves on the Riemann sphere. We also introduce four equivalent definitions for a two-loop Loewner potential: respectively expressing it in terms of normalized Brownian loop measure, zeta-regularized determinants of the Laplacian, an integral formula generalizing universal Liouville action, and Loewner-Kufarev energy of a foliation. Moreover, we prove that the potential is finite if and only if both loops are Weil-Petersson quasicircles, that it is an Onsager-Machlup functional for the two-loop SLE, and a variational formula involving Schwarzian derivatives. Addressing the question of minimization of the two-loop Loewner potential, we find that any such minimizers must be pairs of circles. However, the potential is not bounded, diverging to negative infinity as the circles move away from each other and to positive infinity as the circles merge, thus preventing a definition of two-loop Loewner energy for the prospective large deviations theory for the two-loop SLE. To remedy the divergence, we study a way of generalizing the two-loop Loewner potential by taking into account how conformal field theory (CFT) partition functions depend on the modulus of the annulus between the loops. This generalization is motivated by the correspondence between SLE and CFT, and it also emerges from the geometry of the real determinant line bundle as introduced by Kontsevich and Suhov., Comment: 36 pages, 5 figures

Let $f$ be an entire function of finite exponential type less than or equal to $\sigma$ which is bounded by $1$ on the real axis and satisfies $f(0) = 1$. Under these assumptions H\"ormander showed that $f$ cannot decay faster than $\cos(\sigma x)$ on the interval $(-\pi/\sigma,\pi/\sigma)$. We extend this result to the setting of de Branges spaces with cosine replaced by the real part of the associated Hermite-Biehler function. We apply this result to study the point evaluation functional and associated extremal functions in de Branges spaces (equivalently in model spaces generated by meromorphic inner functions) generalizing some recent results of Brevig, Chirre, Ortega-Cerd\`a, and Seip.

Let $\mathbb H$ be the finite direct sums of $H^2(\mathbb D)$. In this paper, we give a characterization of the closed subspaces of $\mathbb H$ which are invariant under the shift, thus obtaining a concrete Beurling-type theorem for the finite index shift. This characterization presents any such a subspace as the finite intersection, up to an inner function, of pre-images of a closed shift-invariant subspace of $H^2(\mathbb D)$ under ``determinantal operators'' from $\mathbb H$ to $H^2(\mathbb D)$, that is, continuous linear operators which intertwine the shifts and appear as determinants of matrices with entries given by bounded holomorphic functions. With simple algebraic manipulations we provide a direct proof that every invariant closed subspace of codimension at least two sits into a non-trivial closed invariant subspace. As a consequence every bounded linear operator with finite defect has a nontrivial closed invariant subspace., Comment: this new version takes into account some references related to the paper

We prove two isoperimetric inequalities on complex hyperbolic ball. As an application, we prove a contraction property for the holomorphic functions in Hardy and weighted Bergman spaces on the unit ball, thus extending the results of Kulikov (GAFA(2022)) to higher dimensional space.

The classical Bohr theorem and its subsequent generalizations have become active areas of research, with investigations conducted in numerous function spaces. Let $\{\psi_n(r)\}_{n=0}^\infty$ be a sequence of non-negative continuous functions defined on $[0,1)$ such that the series $\sum_{n=0}^\infty \psi_n(r)$ converges locally uniformly on the interval $[0, 1)$. The main objective of this paper is to establish several sharp versions of generalized Bohr inequalities for the class of $K$-quasiconformal sense-preserving harmonic mappings on the unit disk $\D := \{z \in \mathbb{C} : |z| < 1\}$. To achieve these, we employ the sequence of functions $\{\psi_n(r)\}_{n=0}^\infty$ in the majorant series rather than the conventional dependence on the basis sequence $\{r^n\}_{n=0}^\infty$. As applications, we derive a number of previously published results as well as a number of sharply improved and refined Bohr inequalities for harmonic mappings in $\D$. Moreover, we obtain a convolution counterpart of the Bohr theorem for harmonic mapping within the context of the Gaussian hypergeometric function, Comment: 26 pages, AMS-LaTeX v1

We study operators that are products of two orthogonal projections. Our results complement some of the classical results of Crimmins and von Neumann. Particular emphasis has been given to projections associated with inner functions defined on the polydisc., Comment: 21 pages

Famous results due to von Neumann, Sz.-Nagy and Arveson assert that the following four statements are equivalent; a Hilbert space operator $T$ is a contraction; the closed unit disk $\overline{\mathbb D}$ is a spectral set for $T$; $T$ can be dilated to a Hilbert space isometry; $\overline{\mathbb D}$ is a complete spectral set for $T$. In this article, we show by counter examples that no two of them are equivalent for Banach space operators. If $\mathcal F_r$ is the family of all Banach space operators having norm less than or equal to $r$ and if $D_R$ denotes the open disk in the complex plane with centre at the origin and radius $R$, then we prove by an application of Bohr's theorem that $\overline{D}_R$ is the minimal spectral set for $\mathcal F_r$ if and only if $r=\frac{R}{3}$. Also, we prove the equivalence of the following two facts: the Bohr radius of $D_R$ is $\frac{R}{3}$ and $\sup \{ r>0\,:\, \overline{D}_R \text{ is a spectral set for } \mathcal F_r \}=\frac{R}{3}$. We found several new characterizations for a Hilbert space in terms of spectral set and complete spectral set for different operators., Comment: 15 pages

In 1985, W.K.Hayman (Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber, 1984(1985), 1-13.) proved that there do not exist non-constant meromorphic functions $f,$ $g$ and $h$ satisfying the functional equation $f^n+g^n+h^n=1$ for $n\geq 9.$ We prove that there do not exist non-constant meromorphic solutions $f,$ $g,$ $h$ satisfying the functional equation $f^8+g^8+h^8=1.$ In 1971, N. Toda (T\^{o}hoku Math. J. 23(1971), no. 2, 289-299.) proved that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying $f^n+g^n+h^n=1$ for $n\geq 7.$ We prove that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying the functional equation $f^6+g^6+h^6=1.$ Our results answer questions of G. G. Gundersen., Comment: 40 pages

In this paper, we derive the sharp improved versions of Bohr-type inequalities for the Ces\'aro operator acting on the class of bounded analytic functions defined on the unit disk $\D=\left\{z\in\C:\left|z\right|<1\right\}$. In order to achieve these results, we utilize the principle of substituting the initial coefficients of the majorant series with the absolute values of the Ces\'aro operator associated with a bounded analytic function defined on $\D$ and its derivative, as well as for the Schwarz function.

Let $\mathcal{B}(\mathcal{H})$ denote the Banach algebra of all bounded linear operators acting on complex Hilbert spaces $\mathcal{H}$. In this paper, we first establish several sharply refined versions of Bohr's inequality analogues with operator valued functions in the class $\mathcal{B}(\mathbb{D}, \mathcal{B}(\mathcal{H}))$ of bounded analytic functions from the unit disk $\mathbb{D}$ to $\mathcal{B}(\mathcal{H})$ with $\sup_{|z|<1}\Vert f(z)\leq 1$ by utilizing a certain power of the function's norm. Additionally, we establish several multidimensional analogues of refined Bohr's inequalities by using operator valued functions in the complete circular domain $\Omega\subset\mathbb{C}^n$. All of the results are sharp.

In this paper, we derive the sharp Bohr type inequality for the Ces\'aro operator, Bernardi integral operator, and discrete Fourier transform acting on the class of bounded analytic functions defined on shifted disks \beas \Omega_{\gamma}=\left\{z\in\mathbb{C}:\left|z+\frac{\gamma}{1-\gamma}\right|<\frac{1}{1-\gamma}\right\}\quad\text{for}\quad\gamma\in[0,1).\eeas

In this paper we generalize a result of Ye, Pang and Yang[12] on the normality of a family of holomorphic curves in $P^N(\mathbb{C})$. Further we obtain a normality criterion for family of meromorphic functions that partially share wandering holomorphic functions with their derivatives. We also devise a tractable representation of complex valued holomorphic functions from D as functions from D to $P^2(\mathbb{C})$ obtain a normality criterion that leads to a counterexample to the converse of Bloch's principle., Comment: 17 pages

In the theory of $g_\alpha$-potentials on a domain $D\subset\mathbb R^n$, $n\geqslant2$, $g_\alpha$ being the $\alpha$-Green kernel associated with the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$ of order $\alpha\in(0,n)$, $\alpha\leqslant2$, we establish the existence and uniqueness of the $g_\alpha$-balayage $\mu^F$ of a positive Radon measure $\mu$ onto a relatively closed set $F\subset D$, we analyze its alternative characterizations, and we provide necessary and/or sufficient conditions for $\mu^F(D)=\mu(D)$ to hold, given in terms of the $\alpha$-harmonic measure of suitable Borel subsets of $\overline{\mathbb R^n}$, the one-point compactification of $\mathbb R^n$. As a by-product, we find necessary and/or sufficient conditions for the existence of the $g_\alpha$-equilibrium measure $\gamma_F$, $\gamma_F$ being understood in an extended sense where $\gamma_F(D)$ might be infinite. We also discover quite a surprising version of Deny's principle of positivity of mass for $g_\alpha$-potentials, thereby significantly improving a previous result by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018). The results thus obtained are sharp, which is illustrated by means of a number of examples. Some open questions are also posed., Comment: 16 pages

We will cover the basics of several complex variables in 4 lectures: Basic properties of holomorphic functions in several variables, the notion of pseudoconvexity, CR functions and CR geometry, and the $\bar\partial$-problem. The main underlying idea is to connect various characterizations of domains of holomorphy, that is, the natural domains of definition for holomorphic functions. In the process we will connect the function theory on the domain to geometric properties of the boundary, and discuss the relationship between the boundary values and the functions themselves and extension of holomorphic functions from subspaces. These notes are based on a series of lectures held during the S-Matrix Marathon workshop at the Institute for Advanced Study on 11-22 March 2024., Comment: A chapter to be published by Springer Lecture Notes in Physics

The purpose of this paper is to provide a uniformization procedure for Gromov hyperbolic spaces, which need not be geodesic or proper. We prove that the conformal deformation of a Gromov hyperbolic space is a bounded uniform space. Further, we show that there is a natural quasi-isometry between the Gromov boundary and the metric boundary of the deformed space. Our main results are a generalization of the results of Bonk, Heninonen, and Koskela [Proposition 4.5, Proposition 4.13, Ast\'{e}risque 270 (2001)]., Comment: arXiv admin note: text overlap with arXiv:2408.01412

We consider rational maps $f$ on the Riemann sphere $\widehat {\mathbb{C}}$ with an $f$-invariant set $P\subset \widehat {\mathbb{C}}$ of four marked points containing the postcritical set of $f$. We show that the dynamics of the corresponding Thurston pullback map $\sigma_f$ on the completion $\overline{\mathcal{T}_P}$ of the associated Teichm\"uller space $\mathcal{T}_P$ with respect to the Weil-Petersson metric is easy to understand when $\overline{\mathcal{T}_P}$ admits a cover by sets with good combinatorial and dynamical properties. In particular, the map $f$ has a finite global curve attractor in this case. Using a result by Eremenko and Gabrielov, we also show that if $P$ contains all critical points of $f$ and each point in $P$ is periodic, then such a cover of $\overline{\mathcal{T}_P}$ can be obtained from a $\sigma_f$-invariant tessellation by ideal hyperbolic triangles., Comment: 12 pages