In this article, we study the $L^p$ Sobolev regularity of the Bergman projection on monomial polyhedra, which are a wide class of bounded singular Reinhardt domains defined as sublevel sets of holomorphic monomials. This work generalizes the previous results of Sobolev regularity of the Bergman projection on various special singular Reinhardt domains.

In this paper, we describe finite order entire solutions of nonlinear partial differential equation

\begin{equation*} \prod _{i=1}^n\left (a_{i 1} u_{z_1}+a_{i 2} u_{z_2}+\cdots +a_{i n} u_{z_n}\right )=1 \end{equation*}

Given a complex analytic family of complex manifolds, we consider canonical Aeppli deformations of $(p,q)$-forms and study its relations to the varying of dimension of the deformed Aeppli cohomology $\dim H^{\bullet ,\bullet }_{A\phi (t)}(X)$. In particular, we prove the jumping formula for the deformed Aeppli cohomology $H^{\bullet ,\bullet }_{A\phi (t)}(X)$. As a direct consequence, $\dim H^{p,q}_{A\phi (t)}(X)$ remains constant iff the Bott–Chern deformations of $(n-p,n-q)$-forms and the Aeppli deformations of $(n-p-1,n-q-1)$-forms are canonically unobstructed. Furthermore, the Bott–Chern/Aeppli deformations are shown to be unobstructed if some weak forms of ${ \partial }{ \bar {\partial } }$-lemma is satisfied.

We define the Schur–Agler class in infinite variables to consist of functions whose restrictions to finite-dimensional polydisks belong to the Schur–Agler class. We show that a natural generalization of an Agler decomposition holds and the functions possess transfer function realizations that allow us to extend the functions to the unit ball of $\ell ^\infty $. We also give a Pick interpolation type theorem which displays a subtle difference with finitely many variables. Finally, we make a brief connection to Dirichlet series derived from the Schur–Agler class in infinite variables via the Bohr correspondence.

We study the class of composition operators acting on the Fréchet space $\mathrm {Hol}(\mathbb {B}_N)$ of all holomorphic maps on the unit ball of $\mathbb {C}^N$. We describe the conditions to make these operators continuous, invertible and compact. We also do the spectral study of these operators, depending on the nature of its symbol.

In this article, we describe meromorphic solutions of certain partial differential equations, which are originated from the algebraic equation $P(f,g)=0$, where P is a polynomial on $\mathbb {C}^2$. As an application, with the theorem of Coman–Poletsky, we give a proof of the classic theorem: Every meromorphic solution $u(s)$ on $\mathbb {C}$ of $P(u,u')=0$ belongs to W, which is the class of meromorphic functions on $\mathbb {C}$ that consists of elliptic functions, rational functions and functions of the form $R(e^{a s})$, where R is rational and $a\in \mathbb {C}$. In addition, we consider the factorization of meromorphic solutions on $\mathbb {C}^n$ of some well-known PDEs, such as Inviscid Burgers’ equation, Riccati equation, Malmquist–Yosida equation, PDEs of Fermat type.

In this note, we give a new necessary condition for the boundedness of the composition operator on the Dirichlet-type space on the disc, via a two dimensional change of variables formula. With the same formula, we characterize the bounded composition operators on the anisotropic Dirichlet-type spaces $\mathfrak {D}_{\vec {a}}(\mathbb {D}^2)$ induced by holomorphic self maps of the bidisc $\mathbb {D}^2$ of the form $\Phi (z_1,z_2)=(\phi _1(z_1),\phi _2(z_2))$. We also consider the problem of boundedness of composition operators $C_{\Phi }:\mathfrak {D}(\mathbb {D}^2)\to A^2(\mathbb {D}^2)$ for general self maps of the bidisc, applying some recent results about Carleson measures on the Dirichlet space of the bidisc.

Motivated by recent papers [11] and [19] we prove a boundary Schwarz lemma (Burns-Krantz rigidity type theorem) for non-smooth boundary points of the polydisc and symmetrized bidisc. Basic tool in the proofs is the phenomenon of invariance of complex geodesics (and their left inverses) being somehow regular at the boundary point under the mapping satisfying the property as in the Burns-Krantz rigidity theorem that lets the problem reduce to one dimensional problem. Additionally, we make a discussion on bounded symmetric domains and suggest a way to prove the Burns-Krantz rigidity type theorem in these domains that however cannot be applied for all bounded symmetric domains.

Let $\varphi : B_d\to \mathbb {D}$, $d\ge 1$, be a holomorphic function, where $B_d$ denotes the open unit ball of $\mathbb {C}^d$ and $\mathbb {D}= B_1$. Let $\Theta : \mathbb {D} \to \mathbb {D}$ be an inner function, and let $K^p_\Theta $ denote the corresponding model space. For $p>1$, we characterize the compact composition operators $C_\varphi : K^p_\Theta \to H^p(B_d)$, where $H^p(B_d)$ denotes the Hardy space.

We provide two constructions of Gaussian random holomorphic sections of a Hermitian holomorphic line bundle $(L,h_{L})$ on a Hermitian complex manifold $(X,\Theta )$, that are particularly interesting in the case where the space of $\mathcal {L}^2$-holomorphic sections $H^{0}_{(2)}(X,L)$ is infinite dimensional. We first provide a general construction of Gaussian random holomorphic sections of L, which, if $H^{0}_{(2)}(X,L)$ is infinite dimensional, are almost never $\mathcal {L}^2$-integrable on X. The second construction combines the abstract Wiener space theory with the Berezin–Toeplitz quantization and yields a Gaussian ensemble of random $\mathcal {L}^2$-holomorphic sections. Furthermore, we study their random zeros in the context of semiclassical limits, including their distributions, large deviation estimates, local fluctuations and hole probabilities.

We give sufficient conditions for the essential spectrum of the Hermitian square of a class of Hankel operators on the Bergman space of the polydisc to contain intervals. We also compute the spectrum in case the symbol is a monomial.

In this paper, we establish a second main theorem for holomorphic maps with finite growth index on complex discs intersecting arbitrary families of hypersurfaces (fixed and moving) in projective varieties, which gives an above bound of the sum of truncated defects. Our result also generalizes and improves many previous second main theorems for holomorphic maps from ${\mathbb{C}}$ intersecting hypersurfaces (moving and fixed) in projective varieties.

We study the $L^p$ regularity of the Bergman projection P over the symmetrized polydisc in $\mathbb C^n$. We give a decomposition of the Bergman projection on the polydisc and obtain an operator equivalent to the Bergman projection over antisymmetric function spaces. Using it, we obtain the $L^p$ irregularity of P for $p=\frac {2n}{n-1}$ which also implies that P is $L^p$ bounded if and only if $p\in (\frac {2n}{n+1},\frac {2n}{n-1})$.

We study the $L^p$-boundedness of the Berezin transform on the generalised Hartogs triangles which are defined by

$$ \begin{align*}H_k:=\{(z, w)\in\mathbb C^n\times\mathbb C: |z_1|^2+\cdots+|z_n|^2<|w|^{2k}<1\},\end{align*} $$

where $z=(z_1, \ldots , z_n)$ and $k\in \mathbb N$. We prove that the Berezin transform is bounded on $L^p(H_k)$ if and only if $p>nk+1$.

This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions.

We introduce and study Dirichlet-type spaces $\mathcal D(\mu _1, \mu _2)$ of the unit bidisc $\mathbb D^2,$ where $\mu _1, \mu _2$ are finite positive Borel measures on the unit circle. We show that the coordinate functions $z_1$ and $z_2$ are multipliers for $\mathcal D(\mu _1, \mu _2)$ and the complex polynomials are dense in $\mathcal D(\mu _1, \mu _2).$ Further, we obtain the division property and solve Gleason’s problem for $\mathcal D(\mu _1, \mu _2)$ over a bidisc centered at the origin. In particular, we show that the commuting pair $\mathscr M_z$ of the multiplication operators $\mathscr M_{z_1}, \mathscr M_{z_2}$ on $\mathcal D(\mu _1, \mu _2)$ defines a cyclic toral $2$-isometry and $\mathscr M^*_z$ belongs to the Cowen–Douglas class $\mathbf {B}_1(\mathbb D^2_r)$ for some $r>0.$ Moreover, we formulate a notion of wandering subspace for commuting tuples and use it to obtain a bidisc analog of Richter’s representation theorem for cyclic analytic $2$-isometries. In particular, we show that a cyclic analytic toral $2$-isometric pair T with cyclic vector $f_0$ is unitarily equivalent to $\mathscr M_z$ on $\mathcal D(\mu _1, \mu _2)$ for some $\mu _1,\mu _2$ if and only if $\ker T^*,$ spanned by $f_0,$ is a wandering subspace for $T.$

We study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols $\varphi $ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.

We discuss the class of functions, which are well approximated on compacta by the geometric mean of the eigenvalues of a unital (completely) positive map into a matrix algebra or more generally a type $II_1$ factor, using the notion of a Fuglede–Kadison determinant. In two variables, the two classes are the same, but in three or more noncommuting variables, there are generally functions arising from type $II_1$ von Neumann algebras, due to the recently established failure of the Connes embedding conjecture. The question of whether or not approximability holds for scalar inputs is shown to be equivalent to a restricted form of the Connes embedding conjecture, the so-called shuffle-word-embedding conjecture.

Let $f(z)=\sum _{n=0}^{\infty }a_n z^n \in H(\mathbb {D})$ be an analytic function over the unit disk in the complex plane, and let $\mathcal {R} f$ be its randomization:

$$ \begin{align*}(\mathcal{R} f)(z)= \sum_{n=0}^{\infty} a_n X_n z^n \in H(\mathbb{D}),\end{align*} $$

where $(X_n)_{n\ge 0}$ is a standard sequence of independent Bernoulli, Steinhaus, or Gaussian random variables. In this note, we characterize those $f(z) \in H(\mathbb {D})$ such that the zero set of $\mathcal {R} f$ satisfies a Blaschke-type condition almost surely:

$$ \begin{align*}\sum_{n=1}^{\infty}(1-|z_n|)^t<\infty, \quad t>1.\end{align*} $$

The notion of indicator of an analytic function, that describes the function’s growth along rays, was introduced by Phragmen and Lindelöf. Trigonometric convexity is a defining property of the indicator. For multivariate cases, an analogous property of trigonometric convexity was not known so far. We prove the property of trigonometric convexity for the indicator of multivariate analytic functions, introduced by Ivanov. The results that we obtain are sharp. Derivation of a multidimensional analogue of the inverse Fourier transform in a sector and obtaining estimates on its decay is an important step of our proof.