Related papers: Bergman subspaces and subkernels: Degenerate $L^p$…

Consider the Bergman kernel $K^B(z)$ of the domain $\ellip = \{z \in \Comp^n ; \sum_{j=1}^n |z_j|^{2m_j}<1 \}$, where $m=(m_1,\ldots,m_n) \in \Natl^n$ and $m_n \neq 1$. Let $z^0 \in \partial \ellip$ be any weakly pseudoconvex point, $k \in…

We give the parameter version of localization theorem for Bergman metric near the boundary points of strictly pseudoconvex domains. The approximation theorem for square integrable holomorphic functions on such domains in the spirit of…

We provide a rather complete description of the sharp regularity theory for a family of heterogeneous, two-phase variational free boundary problems, $\mathcal{J}_\gamma \to $ min, ruled by nonlinear, $p$-degenerate elliptic operators.…

We give a precise description of Bergman complete bounded pseudoconvex Reinhardt domains.

Let $\Omega \subset \mathbb{R}^{d}$ be a convex body with everywhere positive curvature except at the origin and with the boundary $\partial \Omega$ as the graph of the function $y=|x|^{\gamma}$ in a neighborhood of the origin with $\gamma…

We give a characterization of a variation of constants type estimate relating two positive semigroups on (possibly different) $L_p$-spaces to one another in terms of corresponding estimates for the respective generators and of estimates for…

We prove optimal estimates of the Bergman and Szeg\H{o} kernels on the diagonal, and the Bergman metric near the boundary of bounded smooth generalized decoupled pseudoconvex domains in $\mathbb{C}^n$. The generalized decoupled domains we…

We study integral functionals constrained to divergence-free vector fields in $L^p$ on a thin domain, under standard $p$-growth and coercivity assumptions, $1<p<\infty$. We prove that as the thickness of the domain goes to zero, the…

We study Bergman kernels $K_\Pi$ and projections $P_\Pi$ in unbounded planar domains $\Pi$, which are periodic in one dimension. In the case $\Pi$ is simply connected we write the kernel $K_\Pi$ in terms of a Riemann mapping $\varphi$…

We study the $\bar{\partial}_b$-Neumann problem for domains $\Omega$ contained in a strictly pseudoconvex manifold M^{2n+1} whose boundaries are noncharacteristic and have defining functions depending solely on the real and imaginary parts…

For a smooth $k$-dimensional submanifold $\Sigma$ of a $d$-dimensional compact Riemannian manifold $M$, we extend the $L^p(\Sigma)$ restriction bounds of Burq-G\'erard-Tzvetkov -- originally proved for individual Laplace--Beltrami…

We show that weighted Bergman projections, corresponding to weights of the form $M(z)(1-|z|^2)^{\alpha}$ where $\alpha>-1$ and $M(z)$ is a radially symmetric, strictly positive and at least $C^2$ function on the unit disc, are $L^p$…

We consider a thick p-brane embedded in an n-dimensional spacetime possessing radial symmetry in the directions orthogonal to the brane. We first consider a static brane, and find a general fine tuning relationship between the brane and…

The $p-$Bergman kernel $K_p(\cdot)$ is shown to be of $C^{1,1/2}$ for $1<p<\infty$. An unexpected relation between the off-diagonal $p-$Bergman kernel $K_p(\cdot,z)$ and certain weighted $L^2$ Bergman kernel is given for $1\le p\le 2$. As…

We propose a novel constructive framework for approaching the Hodge Conjecture via explicit degenerations. Building on limiting mixed Hodge structures (LMHS), we formulate a criterion under which a rational class of type (p, p) on a smooth…

Let $\gamma$ be a smooth, non-closed, simple curve whose image is symmetric with respect to the $y$-axis, and let $D$ be a planar domain consisting of the points on one side of $\gamma$, within a suitable distance $\delta$ of $\gamma$.…

The Bergman $p$-analytic content ($1\leq p<\infty $) of a planar domain $\Omega $ measures the $L^{p}(\Omega )$-distance between $\overline{z}$ and the Bergman space $A^{p}(\Omega )$ of holomorphic functions. It has a natural analogue in…

We introduce the PBW degeneration for basic classical Lie superalgebras and construct for all type I, $\mathfrak{osp}(1,2n)$ and exceptional Lie superalgebras new monomial bases. These bases are parametrized by lattice points in convex…

We analyze the main properties of the Bergman spaces of weak $L_p$- solutions for a biquaternionic Vekua equation of the form \[ \mathbf{D}w(x)-\mathbf{Q}_Aw(x)=0 \] on bounded domains of $\mathbb{R}^3$, where the operator $\mathbf{Q}_A$…

We investigate when the Bergman metric of a bounded domain is, up to a constant factor $\lambda$, induced by the Bergman metric of a finite-dimensional unit ball $\mathbb{B}^N$ via a holomorphic isometric immersion. For a strictly…