We extend the notion of quasibounded harmonic functions to the plurisubharmonic setting. As an application, using the theory of Jensen measures, we show that certain generalized Dirichlet problems with unbounded boundary data admit unique solutions, and that these solutions are continuous outside a pluripolar set.
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.
In this note, we study the approximation of singular plurifinely plurisubharmonic function $u$ defined on a plurifinely domain $Ω$. Under some conditions, we prove that $u$ can be approximated by an increasing sequence of plurisubharmonic functions defined on Euclidean neighborhoods of $Ω$.
Given a negative plurisubharmonic function $ϕ$ in a bounded pseudoconvex domain of ${\mathbb C}^n$, we introduce and study its residual function $g_ϕ$ determined by the asymptotic behavior of $ϕ$ near its singularity points, both inside the domain and on its boundary. For certain choices of $ϕ$, the function $g_ϕ$ coincides with different versions of pluricomplex Green functions. The considerations are motivated by a problem on when two given plurisubharmonic functions can be connected by a plurisubharmonic geodesic.
Let $Ω=\{r<0\}\subset\mathbb C^2$, with $r$ plurisubharmonic on $bΩ=\{r=0\}$. Let $ρ$ be another defining function for $Ω$. A formula for the determinant of the complex Hessian of $ρ$ in terms of $r$ is computed. This formula is used to give necessary and sufficient conditions that make $ρ$ (locally) plurisubharmonic.
Necessary conditions for a domain $Ω\subset \mathbb C^n$ 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 boundary when one exists. In some cases we show that the necessary conditions are also sufficient.
We give a geometric condition on a compact subset of a complex manifold which is necessary and sufficient for the existence of a smooth strictly plurisubharmonic function defined in a neighbourhood of this set.
We prove that for every $n \ge 2$, there exists a pseudoconvex domain $Ω\subset \mathbb{C}^n$ such that $\mathfrak{c}^0(Ω) \subsetneq \mathfrak{c}^1(Ω)$, where $\mathfrak{c}^k(Ω)$ denotes the core of $Ω$ with respect to $\mathcal{C}^k$-smooth plurisubharmonic functions on $Ω$. Moreover, we show that there exists a bounded continuous plurisubharmonic function on $Ω$ that is not the pointwise limit of a sequence of $\mathcal{C}^1$-smooth bounded plurisubharmonic functions on $Ω$.
We characterise in this work the $q$-plurisubharmonic functions in terms of the theory of viscosity solutions. We show that an upper semicontinuous function is $q$-plurisubharmonic if and only if its complex Hessian has at most $q$ strictly negative eigenvalues in the viscosity sense. This characterisation is then used to prove that the supremum convolution of a (strictly) $q$-plurisubharmonic function is again (strictly) $q$-plurisubharmonic on a maybe different set of definition. Finally, we use the supremum convolution to deduce a new characterisation for the $q$-pseudoconvex subsets in $\m…
Plurisubharmonic and holomorphic functions relative to the plurifine topology
引入了多重细多重次调和函数和多重细全纯函数的弱和强概念。强者意味着弱者。进一步研究了弱概念。一个函数f是弱多重细多重次调和的当且仅当f o h对所有复仿射线性映射h是细次调和的。因此，这类函数的点态上确界在plurifine拓扑中的正则化是弱plurifine plurisubharmonic的，并且它至多在一个多极集上不同于点态上确界。在弱多细全纯映射的复合下，弱多细次调和性和弱多细全纯性保持不变。
We show how the formalism of Levi currents on complex manifolds, as introduced by Sibony, can be used to study the analytic structure of singular sets associated to families of plurisubharmonic functions, in the sense of Slodkowski.
This paper studies the complex Monge-Ampère equations for $\mathcal F$-plurisubharmonic functions in bounded $\mathcal F$-hyperconvex domains. We give sufficient conditions for this equation to solve for measures with a singular part.
We prove an optimal Alexandrov-Bakelman-Pucci type estimate for plurisubharmonic functions without assuming their continuity. This generalizes a result of Y. Wang. As a corollary we generalize an estimate from \cite{DD18}. We also address a problem posed in \cite{Wan12}.
We give characterizations of (quasi-)plurisubharmonic functions in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic functions.
Let $u$ be a maximal plurisubharmonic function in a domain $Ω\subset\mathbb{C}^n$ ($n\geq 2$). It is classical that, for any $U\SubsetΩ$, there exists a sequence of bounded plurisubharmonic functions $PSH(U)\ni u_j\searrow u$ satisfying the property: $(dd^c u_j)^n$ is weakly convergent to $0$ as $j\rightarrow\infty$. In general, this property does not hold for arbitrary sequence. In this paper, we show that for any sequence of bounded plurisubharmonic functions $PSH(U)\ni u_j\searrow u$, $(|u_j|+1)^{-a} (dd^cu_j)^n$ is weakly convergent to $0$ as $j\rightarrow\infty$, where $a>n-1$. We also ge…
We study continuity properties of generalized Monge-Ampère operators for plurisubharmonic functions with analytic singularities. In particular, we prove continuity for a natural class of decreasing approximating sequences. We also prove a formula for the total mass of the Monge-Ampère measure of such a function on a compact Kähler manifold.
Let G/K be an irreducible Hermitian symmetric space and let D be a K-invariant domain in G/K. In this paper we characterize several classes of K-invariant plurisubharmonic functions on D in terms of their restrictions to a slice intersecting all K-orbits. As applications we show that K-invariant plurisubharmonic functions on D are necessarily continuous and we reproduce the classification of Stein K-invariant domains in G/K obtained by E. Bedford and J. Dadok.
Integration by parts for plurisubharmonic functions
Institute of Mathematics, Vietnam Academy of Science and Technology（越南科学院数学研究所）；Department of Mathematics, Hanoi National University of Education（河内国家教育大学数学系）
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 plurisubharmonic smooth local defining functions. Moreover, we give general criteria equivalent to the existence of plurisubharmonic smooth defining functions on or near the boundary of the domain. These equivalent characterizations are then explored for some classes of domains.
In this paper, we extend some recent results of Guedj-Lu-Zeriahi about psh envelopes of bounded functions on bounded domains in $\mathbb{C}^n$. We also present a result on the regularity of psh envelopes.
