Related papers: Plurisubharmonic Defining Functions in $\mathbb C^…

Necessary conditions for a domain $\Omega\subset \mathbb C^n$ admitting a local plurisubharmonic defining function on the boundary are given. In tandem, we give an algorithm to construct a local plurisubharmonic defining function on the…

We investigate the question of existence of plurisubharmonic defining functions for smoothly bounded, pseudoconvex domains in $\mathbb{C}^2$. In particular, we construct a family of simple counterexamples to the existence of…

In this paper, we use Takeuchi's Theorem to show that for every Lipschitz pseudoconvex domain $\Omega$ in $\mathbb{CP}^n$ there exists a Lipschitz defining function $\rho$ and an exponent $0<\eta<1$ such that $-(-\rho)^\eta$ is strictly…

We extend a recent result of Avelin, Hed, and Persson about approximation of functions $u$ that are plurisubharmonic on a domain $\Omega$ and continuous on $\bar\Omega$, with functions that are plurisubharmonic on (shrinking) neighborhoods…

We study the problem of approximating plurisubharmonic functions on a bounded domain $\Omega$ by continuous plurisubharmonic functions defined on neighborhoods of $\bar\Omega$. It turns out that this problem can be linked to the problem of…

Let $H$ be a complex Hilbert space and let $\Omega\subset H$ be a domain. In infinite dimensions, there is no canonical complex Monge--Amp\`ere operator and no basis-free determinant of the Levi form. Hence, a determinant-type…

A noncommutative (nc) function in $x_1,\dots,x_g,x_1^*,\dots,x_g$ is called plurisubharmonic (plush) if its nc complex Hessian takes only positive semidefinite values on an nc neighborhood of 0. The main result of this paper shows that an…

In this paper we prove a criterion for plurisubharmonic functions in terms of integral mean by complex ellipsoids. Moreover, by using the criterion we prove an analogue of Blaschke-Privalov theorem for plurisubharmonic functions.

For a continuous function, we prove that the function is pluriharmonic if and only if the equality part of the optimal Ohsawa--Takegoshi $L^2$-extension theorem is satisfied with respect to the metric having the function as a weight. This…

In this note, we study the approximation of singular plurifinely plurisubharmonic function $u$ defined on a plurifinely domain $\Omega$. Under some conditions, we prove that $u$ can be approximated by an increasing sequence of…

Let $\varphi $ be a negative plurisubharmonic function in a pseudoconvex domain $\Omega$ in $\mathbb{C}^{n}$ and $f$ be a bounded holomorphic function belonging to $L^{2}(\Omega, \varphi)$. For all negative plurisubharmonic functions $\psi$…

Two properties of plurisubharmonic functions are proven. The first result is a Skoda type integrability theorem with respect to a Monge-Amp\`ere mass with H\"older continuous potential. The second one says that locally, a p.s.h. function is…

A notion of local indicator for a plurisubharmonic function is introduced. The indicator is a certain plurisubharmonic function in the unit polydisc, which controls the behavior of the considered function near a fixed point of its…

We prove that for every $n \ge 2$, there exists a pseudoconvex domain $\Omega \subset \mathbb{C}^n$ such that $\mathfrak{c}^0(\Omega) \subsetneq \mathfrak{c}^1(\Omega)$, where $\mathfrak{c}^k(\Omega)$ denotes the core of $\Omega$ with…

In this paper, we investigate the geometric properties of complex-valued pluriharmonic mappings defined over convex Reinhardt domains in $\mathbb{C}^n$. We first establish a multidimensional analogue of the Noshiro-Warschawski Theorem,…

First, we give the definition for quasi-nearly subharmonic functions, now for general, not necessarily nonnegative functions, unlike previously. We point out that our function class incudes, among others, quasisubharmonic functions, nearly…

We will give a concise formula for the Hessian determinant of a smooth function $y:\mathbb R^n\supseteq\Omega\to\mathbb R$ such that its graph is contained in a quadratic hypersurface. The proof will make heavy use of matrix algebra.

Let $n \geq 4$ and let $\Omega$ be a bounded hyperconvex domain in $\mathbb{C}^{n}$. Let $\varphi$ be a negative exhaustive smooth plurisubharmonic function on $\Omega$. We show that any holomorphic function defined on a connected open…

In this note we study the plurifinely locally maximal plurifinely plurisubharmonic functions and improve some known results on these functions. We prove in particular that any locally bounded plurifinely locally maximal plurifinely…

We show that every bounded domain $D$ in $\mathbb R^n$ with smooth $p$-convex boundary for $2\le p < n$ admits a smooth defining function $\rho$ which is $p$-plurisubharmonic on $\overline D$; if in addition $bD$ has no $p$-flat points then…