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Thai Duong Do

Publications and source records attributed to Thai Duong Do.

13 recordsLinked to original sources
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

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. McAdam's theorem gives a necessary bound on the analytic spread of the local defining ideals. Comparison on a normalized blow-up then controls the singularities of competing functions. As a consequence, maximality is characterized by the vanishing of the Bedford--Taylor Monge--Ampère measure off the pole set together with the analytic spread bound.

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↗