P. Åhag, U. Cegrell, S. Kołodziej, H. H. Pham, A. Zeriahi
We give a sufficient condition on the Monge-Ampère mass of a plurisubharmonic function $u$ for $\exp (- 2 u)$ to be locally integrable. This gives a pluripotential theoretic proof of a theorem by J-P. Demailly.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The pluricomplex Green function of the Monge-Ampère equation for $(n-1)$-plurisubharmonic functions and form type $k$-Hessian equations
本文引入$(n-1)$-多次调和函数的Monge-Ampère方程的多次复Green函数，通过求解Dirichlet问题并建立先验估计证明其$C^{1,\alpha}$正则性，且$k$-Hessian方程的奇异解光滑。
Let $u$ be a plurisubharmonic function, defined on a neighbourhood of a point $x,$ such that the complex Monge-Ampère operator is well-defined on $u.$ Suppose also that $u$ has a weak singularity, in the sense that the Lelong number of $u$ at $x$ vanish. Is it true that the residual mass of the measure $(dd^c{u})^n$ vanish at $x$? To our knowledge there is no known example that falsifies the posed question. In this paper some partial results are obtained. We find that for a significant subset of plurisubharmonic functions with well defined Monge-Ampère mass vanishing Lelong number does implies…
A variant of Siu's analyticity theorem is proved for relative types of plurisubharmonic functions. Some results on propagation of plurisubharmonic singularities and maximality of pluricomplex Green functions with analytic singularities are derived.
We introduce a new type of pluricomplex Green function which has a logarithmic pole along a complex subspace A of a complex manifold X. It is the largest negative plurisubharmonic function on X whose Lelong number is at least the Lelong number of log max{|f_1|,...,|f_m|}, where f_1,...,f_m are local generators for the ideal sheaf of A. The pluricomplex Green function with a single logarithmic pole or a finite number of weighted poles is a very special case of our construction. We give several equivalent definitions of this function and study its properties, including boundary behaviour, contin…
In this paper, we explore the existence of pluricomplex Green functions for Stein manifolds from a functional analysis point of view. For a Stein manifold $M$, we will denote by $O(M)$ the Fréchet space of analytic functions on $M$ equipped with the topology of uniform convergence on compact subsets. In the first section, we examine the relationship between the existence of pluricomplex Green functions and the diametral dimension of $O(M)$. This led us to consider negative plurisubharmonic functions on $M$ with a nontrivial relatively compact sublevel set (semi-proper). In section 2, we charac…
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.
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 singularity, as well as the residual Monge-Ampere mass of the function at this point. A pluricomplex Green function with respect to given indicators is constructed and applied to the Dirichlet problem for the Monge- Ampere operator in the class of plurisubharmonic functions with isolated singularities.
Let $D$ be a bounded strongly convex domain in the complex space of dimension $n$. Fixed a point $p\in \partial D$, we consider the solution of a homogeneous complex Monge-Ampere equation with simple pole at $p$. We prove that such a solution enjoys many properties of the classical Poisson kernel in the unit disc and thus deserves to be called the pluricomplex Poisson kernel of $D$ with pole at $p$. In particular we discuss extremality properties (such as a generalization of the classical Phragmen-Lindelof theorem), relations with the pluricomplex Green function of $D$, uniqueness in terms of …
In this paper we introduce, via a Phragmen-Lindelöf type theorem, a maximal plurisubharmonic function in a strongly pseudoconvex domain. We call such a function the {\sl pluricomplex Poisson kernel} because it shares many properties with the classical Poisson kernel of the unit disc. In particular, we show that such a function is continuous, it is zero on the boundary except at one boundary point where it has a non-tangential simple pole, and reproduces pluriharmonic functions. We also use such a function to obtain a new "intrinsic" version of the classical Julia's Lemma and Julia-Wolff-Carath…
The aim of this paper is to extend the domain of definition of $(dd^c\centerdot)^q\wedge T$ on some classes of plurisubharmonic (psh) functions, which are not necessary bounded, where $T$ is a positive closed current of bidimension $(q,q)$ on an open set $Ω$ of $\Bbb C^n$. We introduce two classes $\mathcal{F}_{p}^{T}(Ω)$ and $\mathcal{E}_p^T(Ω)$ and we show that they belong to the domain of definition of the operator $(dd^c\centerdot)^q\wedge T$. We also prove that all functions belong to these classes are $C_T$-quasicontinuous and that the comparison principle is valid in them.
By comparing Green functions of multi-circled plurisubharmonic singularities in the n-domensional complex space to their indicators, we obtain formulas for the higher Lelong numbers and integrability index for such singularities and extend Howald's result on multiplier ideals for monomial ideals to multi-circled singularities. This also leads to an elementary proof of the relations between the k-th Lelong numbers and the integrability index. For k=1 and arbitrary plurisubharmonic functions the inequality is due to Skoda, and for k=n and any plurisubharmonic function with isolated singularity t…
The aim of this paper is to study the Lelong number, the integrability index and the Monge-Ampère mass at the origin of an $S^1$-invariant plurisubharmonic function on a balanced domain in $\mathbb{C}^n$ under the Schwarz symmetrization. We prove that $n$ times the integrability index is exactly the Lelong number of the symmetrization, and if the function is further toric with a single pole at the origin, then the Monge-Ampère mass is always decreasing under the symmetrization
Using pluricomplex Green functions we introduce a compactification of a complex manifold $M$ invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form $V$ on $M$ such that all negative plurisubharmonic functions on $M$ are in $L^1(M,V)$. Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point $w\in M$ with the normalized pluricomplex Green function with pole at $w$ we get an imbedding of $M$ into a compact set and the closure of $M$ in this set is th…
$α$-Hölder Integrability Exponent
该研究针对多重次调和函数定义了α-Hölder可积性指数，得到了其以Lelong数和交数表示的改进下界，并给出了一类函数该指数的最优下界，推进了相关可积性问题的研究。
The pluricomplex Poisson kernel for convex finite type domains
研究有界凸有限型区域上齐次复蒙日 - 安培方程的解 \(\Omega_\xi\)，基于度量和缩放技术，证明其性质，包括与多复变格林函数关系等，它是单位圆盘泊松核推广，还可用于强凸域，过程中建立了小林距离新估计。
First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball $\B \sub \C^n$ with its relative logarithmic capacity in $\C^n$ with respect to the same ball $\B$. An analoguous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of $\C^n$ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of \psh lemniscates associated to the Lelong class of \psh functions of logarithmic singularities at in…
Two properties of plurisubharmonic functions are proven. The first result is a Skoda type integrability theorem with respect to a Monge-Ampère mass with Hölder continuous potential. The second one says that locally, a p.s.h. function is $k$-Lipschitz outside a set of Lebesgue measure smaller that $c/k^2$.
We give an estimate for the volume of an analytic variety (or more generally the mass of a positive closed current) close to a real submanifold $M$. Applications are given to the Hausdorff measure of the intersection of the variety with $M$ and the exponential integrability of plurisubharmonic functions on $M$.
In this article, we characterize plurisubharmonic functions of Lelong number one at the origin, such that the germ of the associated multiplier ideal sheaf is nontrivial: in arbitrary complex dimension, their singularity must be the sum of a germ of smooth divisor and of a plurisubharmonic function with zero Lelong number. We also present a new proof of the related well known integrability criterion due to Skoda.
Let $(M,ω)$ be a Kahler manifold. An integrable function on M is called $ω^q$-plurisubharmonic if it is subharmonic on all q-dimensional complex subvarieties. We prove that a smooth $ω^q$-plurisubharmonic function is q-convex. A continuous $ω^q$-plurisubharmonic function admits a local approximation by smooth, $ω^q$-plurisubharmonic functions. For any closed subvariety $Z\subset M$, $\dim Z < q$, there exists a strictly $ω^q$-plurisubharmonic function in a neighbourhood of $Z$ (this result is known for q-convex functions). This theorem is used to give a new proof of Sibony's lemma on integrabi…
We present a new approach to the study of multiplier ideals in a local, two-dimensional setting. Our method allows us to deal with ideals, graded systems of ideals and plurisubharmonic functions in a unified way. Among the applications are a formula for the complex integrability exponent of a plurisubharmonic function in terms of Kiselman numbers, and a proof of the openness conjecture by Demailly and Kollar. Our technique also yields new proofs of two recent results: one on the structure of the set of complex singularity exponents for holomorphic functions; the other by Lipman and Watanabe on…
We solve the Dirichlet problem for the quaternionic Monge-Ampère equation with a continuous boundary data and the right hand side in $L^p$ for $p>2$. This is the optimal bound on $p$. We prove also that the local integrability exponent of quaternionic plurisubharmonic functions is two which turns out to be less than an integrability exponent of the fundamental solution.
Characterization of Sobolev regularity of plurisubharmonic functions
本文刻画了由全纯芽构造的多次调和函数的局部Sobolev正则性，给出尖锐判据，并应用于Calderón-Zygmund理论反例与Monge-Ampère定义域刻画。
The goal of this short note is to relate the integrability property of the exponential $e^{-2ϕ}$ of a plurisubharmonic function $ϕ$ with isolated or compactly supported singularities, to a priori bounds for the Monge-Ampère mass of $(dd^cϕ)^n$. The inequality is valid locally or globally on an arbitrary open subset $Ω$ in $\bC^n$. We show that $\int_Ω(ddϕ)^n<n^n$ implies $\int_Ke^{-2ϕ}<+\infty$ for every compact subset $K$ in $Ω$, while functions of the form $ϕ(z)=n\log|z-z_0|$, $z_0\inΩ$, appear as limit cases. The result is derived from an inequality of pure local algebra, which turns out a …
Complex Monge-Amp{è}re equations with boundary values: a variational approach
针对复空间中有界超凸域上带边界值的复Monge-Ampère方程，引入具有限Φ-能量的多重次调和函数类，用变分方法刻画其Monge-Ampère测度的可积性条件，推广相关结果并通过对数变换得到实方程的类似结论。
We prove that if a smoothly bounded strongly pseudoconvex domain $D \subset \mathbb C^n$, $n \geq 2$, admits at least one Monge-Ampère exhaustion smooth up to the boundary (i.e. a plurisubharmonic exhaustion $τ: \overline D \to [0,1]$, which is $\mathcal C^\infty$ at all points except possibly at the unique minimum point $x$ and with $u := \log τ$ satisfying the homogeneous complex Monge-Ampère equation), then there exists a bounded open neighborhood $\mathcal U\subset D$ of the minimum point $x$, such that for each $y \in \mathcal U$ there exists a Monge-Ampère exhaustion with minimum at $y$.…
Some inequalities for the weighted log canonical thresholds
本文研究了加权对数canonical阈值的不等式，证明了其斜率受局部Lelong数控制，并导出了其增长下界。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。