Related papers: Szeg\"o kernels for certain unbounded domains in $…

This paper provides a precise asymptotic expansion for the Bergman kernel on the non-smooth worm domains of Christer Kiselman in complex 2-space. Applications are given to the failure of Condition R, to deviant boundary behavior of the…

We describe the Szeg\"o kernel on a higher genus Riemann surface in terms of Szeg\"o kernel data coming from lower genus surfaces via two explicit sewing procedures where either two Riemann surfaces are sewn together or a handle is sewn to…

We construct a bounded plane domain which is Bergman complete but for which the Bergman kernel does not tend to infinity as the point approaches the boundary.

Let $G \subset \mathbb{C}^2$ be a smoothly bounded pseudoconvex domain and assume that the Bergman kernel of $G$ is algebraic of degree $d$. We show that the boundary $\partial G $ is of finite type and the type $r$ satisfies $r\leq 2d$.…

The bicomplex Bergman spaces are studied for any bounded bicomplex domain. Its Bergman kernel is computed in terms of the kernels of the complex projections of the domain. We also introduce two additional reproducing kernel Hilbert spaces…

We study diagonal estimates for the Bergman kernels of certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that…

In this article, we develop the theory of weighted $L^2$ Sobolev spaces on unbounded domains in $\mathbb R^n$. As an application, we establish the elliptic theory for elliptic operators and prove trace and extension results analogous to the…

We study the problem of the boundary behaviour of the Bergman kernel and the Bergman completeness in some classes of bounded pseudoconvex domains, which contain also non-hyperconvex domains. Among the classes for which we prove the Bergman…

In this paper we discuss some recent results concerning the regularity and irregularity of the Bergman and Szeg\H{o} projections on some weakly pseudoconvex domains that have the common feature to possess a nontrivial Nebenh\"ulle.

A characterization of non-hyperbolic pseudoconvex Reinhardt domains in $\mathbb C^2$ for which the answer to the Serre problem is positive is given.

We build a general theory of microlocal (homogeneous) Fourier Integral Operators in real-analytic regularity, following the general construction in the smooth case by H\"ormander and Duistermaat. In particular, we prove that the…

Bounded and unbounded weighted composition operators on $L^2$ spaces over $\sigma$-finite measure spaces are investigated. A variety of questions related to seminormality of such operators are discussed.

For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the…

We establish the full quasi-Banach range of $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \rightarrow L^p(\mathbb R)$ bounds for one-dimensional bilinear singular integral operators with homogeneous kernels whose restriction $\Omega$ to the…

We study several intrinsic properties of the Carath\'eodory and Szeg\"o metrics on finitely connected planar domains. Among them are the existence of closed geodesics and geodesic spirals, boundary behaviour of Gaussian curvatures, and…

We study the boundary behaviour of the Fefferman--Szeg\"o metric and several associated invariants in a $C^\infty$-smoothly bounded strictly pseudoconvex domain.

By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc, 28 (2000), 317-321], we study one-parameter semigroups generated by (the negative of) rather general Schroedinger operators, which may be unbounded from…

In this work, we consider the approximation capabilities of shallow neural networks in weighted Sobolev spaces for functions in the spectral Barron space. The existing literature already covers several cases, in which the spectral Barron…

Let $(X, T^{1,0}X)$ be a compact connected orientable CR manifold of dimension $2n+1$ with non-degenerate Levi curvature. Assume that $X$ admits a connected compact Lie group $G$ action. Under certain natural assumptions about the group $G$…

Using the machinery of unitary spherical harmonics due to Koornwinder, Folland and other authors, we~obtain expansions for the Szeg\"o and the weighted Bergman kernels of $M$-harmonic functions, i.e.~functions annihilated by the invariant…