We consider the $L^{p}$-regularity of the Szegö projection on the symmetrised polydisc $\mathbb {G}_{n}$. In the setting of the Hardy space corresponding to the distinguished boundary of the symmetrised polydisc, it is shown that this operator is $L^{p}$-bounded for $p\in (2-{1}/{n}, 2+{1}/{(n-1)})$.

In this paper we study boundedness and compactness characterizations of the commutators of Cauchy type integrals on bounded strongly pseudoconvex domains D in $\mathbb C^{n}$ with boundaries $bD$ satisfying the minimum regularity condition $C^{2}$ , based on the recent results of Lanzani–Stein and Duong et al. We point out that in this setting the Cauchy type integral is the sum of the essential part which is a Calderón–Zygmund operator and a remainder which is no longer a Calderón–Zygmund operator. We show that the commutator is bounded on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1<p<\infty $ ) if and only if b is in the BMO space on $bD$ . Moreover, the commutator is compact on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1<p<\infty $ ) if and only if b is in the VMO space on $bD$ .

Let $\mathcal {B}(\mathcal {H})$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathcal {H}$ . In this paper, we first establish several sharp improved and refined versions of the Bohr’s inequality for the functions in the class $H^{\infty }(\mathbb {D},\mathcal {B}(\mathcal {H}))$ of bounded analytic functions from the unit disk $\mathbb {D}:=\{z \in \mathbb {C}:|z|<1\}$ into $\mathcal {B}(\mathcal {H})$ . For the complete circular domain $Q \subset \mathbb {C}^{n}$ , we prove the multidimensional analogues of the operator valued Bohr-type inequality which can be viewed as a special case of the result by G. Popescu [Adv. Math. 347 (2019), 1002–1053] for free holomorphic functions on polyballs. Finally, we establish the multidimensional analogues of several improved Bohr’s inequalities for operator valued functions in Q.

In this paper, we give a generalization and improvement of the Pavlović result on the characterization of continuously differentiable functions in the Bloch space on the unit ball in $\mathbb {R}^{m}$. Then, we derive a Holland–Walsh type theorem for analytic normal mappings on the unit disk.

This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work by Goldberg and Keen [The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift. Invent. Math.101(2) (1990), 335–372] on degree two rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further is that, despite the presence of the essential singularity, our new functions exhibit a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted.

In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle ${\mathbb T}$ , there is a function f holomorphic in the unit disc ${{\mathbb D}}$ , having $\psi $ as radial limit a.e. on ${\mathbb T}$ . We consider an analogous boundary value problem, where the unit disc is replaced by a Stein domain on a complex manifold and radial approach to a boundary point p is replaced by (asymptotically) total approach to p.

Let $T = (T_1, \ldots , T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\mathcal{H}$. The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$. In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Let $n \geq 2$, and let $\mathcal{Q}_i$, $i = 1, \ldots , n$, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\mathbb {C}$. If $\mathcal{Q}_i^{\bot }$, $i = 1, \ldots , n$, is a zero-based shift invariant subspace, then the multiplicity of the joint $M_{\textbf {z}} = (M_{z_1}, \ldots , M_{z_n})$-invariant subspace $(\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{\perp }$ of the Dirichlet space or the Hardy space over the unit polydisc in $\mathbb {C}^{n}$ is given by

\[ \mbox{mult}_{M_{\textbf{{z}}}|_{ (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}}}} (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}} = \sum_{i=1}^{n} (\mbox{mult}_{M_z|_{\mathcal{Q}_i^{{\perp}}}} (\mathcal{Q}_i^{\bot})) = n. \]

We investigate integral means over spherical shell of holomorphic functions in the unit ball $\mathbb {B}_n$ of $\mathbb {C}^n$ with respect to the weighted volume measures and their relation with the weighted Hadamard product. The main result of this paper has many consequences which improve some well-known estimates related to the Hadamard product in Hardy spaces and weighted Bergman spaces.

Let $\Omega $ be a bounded Reinhardt domain in $\mathbb {C}^n$ and $\phi _1,\ldots ,\phi _m$ be finite sums of bounded quasi-homogeneous functions. We show that if the product of Toeplitz operators $T_{\phi _m}\cdots T_{\phi _1}=0$ on the Bergman space on $\Omega $ , then $\phi _j=0$ for some j.

In this paper, we construct a natural probability measure on the space of real branched coverings from a real projective algebraic curve $(X,c_X)$ to the projective line $(\mathbb{C} \mathbb {P}^1,\textit{conj} )$. We prove that the space of degree d real branched coverings having “many” real branched points (for example, more than $\sqrt {d}^{1+\alpha }$, for any $\alpha>0$) has exponentially small measure. In particular, maximal real branched coverings – that is, real branched coverings such that all the branched points are real – are exponentially rare.

We obtain sharp ranges of $L^p$ -boundedness for domains in a wide class of Reinhardt domains representable as sublevel sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating $L^p$ -boundedness on a domain and its quotient by a finite group. The range of p for which the Bergman projection is $L^p$ -bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases.

Spaces of power series solutions $y(\mathrm {t})$ in one variable $\mathrm {t}$ of systems of polynomial, algebraic, analytic or formal equations $f(\mathrm {t},\mathrm {y})=0$ can be viewed as ‘infinite-dimensional’ varieties over the ground field $\mathbf {k}$ as well as ‘finite-dimensional’ schemes over the power series ring $\mathbf {k}[[\mathrm {t}]]$ . We propose to call these solution spaces arquile varieties, as an enhancement of the concept of arc spaces. It will be proven that arquile varieties admit a natural stratification ${\mathcal Y}=\bigsqcup {\mathcal Y}_d$ , $d\in {\mathbb N}$ , such that each stratum ${\mathcal Y}_d$ is isomorphic to a Cartesian product ${\mathcal Z}_d\times \mathbb A^{\infty }_{\mathbf {k}}$ of a finite-dimensional, possibly singular variety ${\mathcal Z}_d$ over $\mathbf {k}$ with an affine space $\mathbb A^{\infty }_{\mathbf {k}}$ of infinite dimension. This shows that the singularities of the solution space of $f(\mathrm {t},\mathrm {y})=0$ are confined, up to the stratification, to the finite-dimensional part.

Our results are established simultaneously for algebraic, convergent and formal power series, as well as convergent power series with prescribed radius of convergence. The key technical tool is a linearisation theorem, already used implicitly by Greenberg and Artin, showing that analytic maps between power series spaces can be essentially linearised by automorphisms of the source space.

Instead of stratifying arquile varieties, one may alternatively consider formal neighbourhoods of their regular points and reprove with similar methods the Grinberg–Kazhdan–Drinfeld factorisation theorem for arc spaces in the classical setting and in the more general setting.

We construct a family of fibred threefolds $X_m \to (S , \Delta )$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,\Delta )$ of general type. Following Campana, the threefolds $X_m$ are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if m is big enough, the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalisations of known conjectures on exceptional loci that fit into Campana’s program and prove some cases over function fields.

Let $p:X\rightarrow Y$ be an algebraic fiber space, and let $L$ be a line bundle on $X$. In this article, we obtain a curvature formula for the higher direct images of $\unicode[STIX]{x1D6FA}_{X/Y}^{i}\otimes L$ restricted to a suitable Zariski open subset of $X$. Our results are particularly meaningful if $L$ is semi-negatively curved on $X$ and strictly negative or trivial on smooth fibers of $p$. Several applications are obtained, including a new proof of a result by Viehweg–Zuo in the context of a canonically polarized family of maximal variation and its version for Calabi–Yau families. The main feature of our approach is that the general curvature formulas we obtain allow us to bypass the use of ramified covers – and the complications that are induced by them.

The aim of this paper is twofold. The first aim is to describe the entire solutions of the partial differential equation (PDE) $u_{z_1}^2+2Bu_{z_1}u_{z_2}+u_{z_2}^2=e^g$, where B is a constant and g is a polynomial or an entire function in $\mathbb {C}^2$. The second aim is to consider the entire solutions of another PDE, which is a generalization of the well-known PDE of tubular surfaces.

The Fock–Bargmann–Hartogs domain $D_{n,m}(\,\unicode[STIX]{x1D707}):=\{(z,w)\in \mathbb{C}^{n}\times \mathbb{C}^{m}:\Vert w\Vert ^{2}<e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}\}$, where $\unicode[STIX]{x1D707}>0$, is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. We compute the weighted Bergman kernel of $D_{n,m}(\,\unicode[STIX]{x1D707})$ with respect to the weight $(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}$, where $\unicode[STIX]{x1D70C}(z,w):=\Vert w\Vert ^{2}-e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}$ and $\unicode[STIX]{x1D6FC}>-1$. Then, for $p\in [1,\infty ),$ we show that the corresponding weighted Bergman projection $P_{D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}}$ is unbounded on $L^{p}(D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}})$, except for the trivial case $p=2$. This gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is $L^{p}$ irregular when $p\in [1,\infty )\setminus \{2\}$, in contrast to the well-known positive $L^{p}$ regularity result on a bounded strongly pseudoconvex domain.

We characterize Fredholmness of Toeplitz operators acting on generalized Fock spaces of the n-dimensional complex space for symbols in the space of vanishing mean oscillation VMO. Our results extend the recent characterizations for Toeplitz operators on standard weighted Fock spaces to the setting of generalized weight functions and also allow for unbounded symbols in VMO for the first time. Another novelty is the treatment of small exponents 0 < p < 1, which to our knowledge has not been seen previously in the study of the Fredholm properties of Toeplitz operators on any function spaces. We accomplish this by describing the dual of the generalized Fock spaces with small exponents.

We strengthen, in various directions, the theorem of Garnett that every $\unicode[STIX]{x1D70E}$-compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some $\mathbb{C}^{N}$ so that $\widehat{K}\backslash K$ contains a Gleason part homeomorphic to $X$, and $\widehat{K}$ contains no analytic discs.

The relationship between interpolation and separation properties of hypersurfaces in Bargmann–Fock spaces over $\mathbb{C}^{n}$ is not well understood except for $n=1$. We present four examples of smooth affine algebraic hypersurfaces that are not uniformly flat, and show that exactly two of them are interpolating.

In the paper the correspondence between a formal multiple power series and a special type of branched continued fractions, the so-called ‘multidimensional regular C-fractions with independent variables’ is analysed providing with an algorithm based upon the classical algorithm and that enables us to compute from the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional regular C-fraction with independent variables. A few numerical experiments show, on the one hand, the efficiency of the proposed algorithm and, on the other, the power and feasibility of the method in order to numerically approximate certain multivariable functions from their formal multiple power series.