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Thai Duong Do

Publications and source records attributed to Thai Duong Do.

13 recordsLinked to original sources
arXiv2026

Local-to-global maximality for plurisubharmonic functions with locally analytic singularities

We prove that a locally maximal plurisubharmonic function on a domain in $\bC^n$, $n\geq2$, is maximal if it has locally analytic singularities with a two-sided locally bounded remainder. Using McAdam's theorem, we show that local maximality forces the analytic spread of each defining ideal to be less than $n$. We use normalized blow-ups and Siu's analyticity theorem to compare singularities through generic divisorial Lelong numbers. We also obtain a criterion for maximality in terms of maximality outside the pole set and the analytic spread of the local defining ideals.

math.CV↗
arXiv2026

Comparison principles for local plurisubharmonic potentials on complex manifolds and applications

We prove a weighted comparison principle for local plurisubharmonic potentials in Cegrell's class on an arbitrary complex manifold under suitable boundary and gluing conditions. We obtain domination and uniqueness when the manifold admits a global psh function in $\mathcal E_{\mathrm{loc}}$ whose Monge-Ampère measure has an absolutely continuous part with positive density almost everywhere. We apply these results to plurisubharmonic functions on domains in $\mathbb C^2$ whose singularities satisfy local inequalities involving $α\log\|f\|$ and a Cegrell potential. For such functions, maximality is equivalent to the vanishing of the Monge-Ampère measure of the current remaining after subtraction of the Siu divisorial part. Consequently, maximality is a local property in this class.

math.CV↗
arXiv2026

Comparison principles for Monge-Ampère measures on pluripolar sets

In this paper, we introduce a notion of singularity comparison for plurisubharmonic functions based on the Bedford--Taylor capacity. We establish comparison principles for the complex Monge--Ampère operator on pluripolar sets in the Cegrell classes. As applications, we obtain a characterization of this relation via auxiliary functions in the energy class and prove a corresponding uniqueness result for the Monge--Ampère equation.

math.CV↗
arXiv2026

Sets of cardinality seven are not sum-dominant

We give a self-contained elementary proof of the known result that every set $A\subset\R$ of cardinality seven satisfies $|A+A|\le |A-A|$. The argument adapts Hegarty's method, using representation counts and the largest positive differences. An exact counting identity, two applications of the Cauchy--Schwarz inequality, and an analysis of a symmetric six-element set with one point added complete the proof, with no computer enumeration. This answers in the affirmative a question of Chu for the seven-element case.

math.CO↗
arXiv2026

Minimal m-subharmonic functions with nonmaximal Hessian measures

For every $2\le m\le n$, we construct Hölder continuous functions on the closed unit ball that are minimal in the Cegrell class $\F_m$, although their Hessian measures are not maximal in the $m$-subharmonic ordering. This answers a question posed by the second author. We also obtain a minimality criterion based on partial pluricomplex energy and an explicit family of Monge--Ampère examples on a product domain. For $m=1$, minimality and maximality are equivalent on a ball.

math.CV↗
arXiv2026

Non-collapsing volume estimate for local Kähler metrics in big cohomology classes

We prove a uniform local non-collapsing volume estimate for a large family of singular metrics in the big cohomology classes, which are Kähler on an open Euclidean subset of the manifold. The key ingredient is a generalization of a mixed energy estimate for functions in the complex Sobolev space to the setting of big cohomology classes.

math.DG↗
arXiv2025

Higher complex Sobolev spaces on complex manifolds

We study higher complex Sobolev spaces and their corresponding functional capacities. In particular, we prove the Moser-Trudinger inequality for these spaces and discuss some relationships between these spaces and the complex Monge-Ampère equation.

math.CV↗
arXiv2025

A Dirichlet type problem for non-pluripolar complex Monge-Ampère equations

In this paper, we study a Dirichlet type problem for the non-pluripolar complex Monge - Ampère equation with prescribed singularity on a bounded domain of $\mathbb{C}^n$. We provide a local version for an existence and uniqueness theorem proved by Darvas, Di Nezza and Lu. Our work also extends a result of Ahag, Cegrell, Czyz and Pham.

math.CV↗
arXiv2023

On the finite energy classes of quaternionic plurisubharmonic functions

We investigate the finite $p$-energy classes $E_p$ of quaternionic plurisubharmonic functions of Cegrell type. We also construct an example to show that the optimal constant in the energy estimate is strictly bigger than $1$ for $p>0$, $p\neq 1$. This leads us to the fact that we can not use the variational method to solve the quaternionic Monge-Ampère equation for the classes $E_p$ when $p>0$, $p\neq 1$.

math.CV↗
arXiv2022

On the weighted m-energy classes

In this article, we investigate the weighted $m-$subharmonic functions. We shall give some properties of this class and consider its relation to the $m-$Cegrell classes. We also prove an integration theorem and an almost everywhere convergence theorem for this class.

math.CV↗
arXiv2019

Some remarks on the Cegrell's class $\mathcal{F}$

In this paper, we study the near-boundary behavior of functions $u\in\mathcal{F}(Ω)$ in the case where $Ω$ is strictly pseudoconvex. We also introduce a sufficient condition for belonging to $\mathcal{F}$ in the case where $Ω$ is the unit ball.

math.CV↗
arXiv2018

Complex Monge-Ampère equation in strictly pseudoconvex domains

We study the complex Monge-Ampère equation $(dd^c u)^n=μ$ in a strictly pseudoconvex domain $Ω$ with the boundary condition $u=φ$, where $φ\in C(\partialΩ)$. We provide a non-trivial sufficient condition for continuity of the solution $u$ outside "small sets".

math.CV↗