m-subharmonic and m-plurisubharmonic functions -- on two problems of Sadullaev

Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains

Let $D$ be a pseudoconvex domain in $\C^k_t\times\Cn_z$ and let $ϕ$ be a plurisubharmonic function in $D$. For each $t$ we consider the $n$-dimensional slice of $D$, $D_t=\{z; (t,z)\in D\}$, let $ϕ^t$ be the restriction of $ϕ$ to $D_t$ and denote by $K_t(z,ζ)$ the Bergman kernel of $D_t$ with the weight function $ϕ^t$. Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n=1$ and $ϕ=0$) we prove that $\log K_t(z,z)$ is a plurisubharmonic function in $D$. We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setti…

On a problem of CHirka

We observe that a slight adjustment of a method of Caffarelli, Li, and Nirenberg yields that plurisubharmonic functions extend across subharmonic singularities as long as the singularities form a closed set of measure zero. This solves a problem posed by Chirka.

Liouville-type theorems outside of small exceptional sets for functions of finite order

We prove that convex functions of finite order on the real line and subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some set of zero relative Lebesgue density, are bounded from above everywhere. It follows that subharmonic functions of finite order on the complex plane, entire and plurisubharmonic functions of finite order, and convex or harmonic functions of finite order bounded from above outside some set of zero relative Lebesgue density are constant.

Plurisubharmonic functions in calibrated geometry and q-convexity

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…

Liouville-type theorems with constraints outside of small sets on circles or spheres for functions of finite order

We prove that subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some asymptotically small sets on spheres, are bounded from above everywhere. It follows that subharmonic functions of finite order on the complex plane, entire and plurisubharmonic functions of finite order, and convex or harmonic functions of finite order bounded from above outside of such sets on spheres are constant. Our results and methods of proof are also new for functions of one complex variable.

Kahler流形中的核心集

Core Sets in Kahler Manifolds

本文在非紧Kahler流形上研究m-次调和函数的核心集，引入m-核心概念，并证明在维数至少2的Kahler流形中，m≥2的m-调和函数是多次调和函数。

Potential Theory on Almost Complex Manifolds

Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their "dual" objects, the plurisubharmonic functions. These functions are defined classically by requiring that the restriction to each pseudo-holomorphic curve is subharmonic. In this paper subharmonic functions are defined by applying the viscosity approach to a version of the complex hessian which exists intrinsically on any almost complex manifold. Three theorems are proved. The first is a restriction theorem which establishes the equivalence of our definition with the "classical" definition. I…

受限的佩龙包络与拟有界函数

Perron envelopes with globally bounded minorants

研究上半连续函数\(f\)的受限佩龙包络，证明经典位势理论中若正则化是次调和的则受限包络自动上半连续，通过构造反例表明多重位势理论中类似结论不成立，即不能去掉例外多重极集。

Plurisubharmonic functions and positive currents of type (1,1) over almost complex manifolds

If $(X,J)$ is an almost complex manifold, then a function $u$ is said to be plurisubharmonic on $X$ if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is conjectured that plurisubharmonicity is equivalent to the fact that the $(1,1)$-current $i\partial_{_J}\bar{\partial}_{_J}u$ is positive, (the $(1,1)$-current $i\partial_{_J}\bar{\partial}_{_J}u$ need not be closed here). The conjecture is trivial if $u$ is of class ${\cal C}^2$. The result is elementary in the complex integrable case because the operator $i\pa…

Hardy-Sobolev type inequalities and their applications

This paper is devoted to various applications of Hardy-Sobolev type inequalities. We derive a new $L^2$ estimate for the $\bar{\partial}-$equation on ${\mathbb C}^n$ which yields a quantitative generalization of the Hartogs extension theorem to the case when the singularity set is not necessary compact. We show that for any negative subharmonic function $ψ$ on ${\mathbb R}^n$, $n>2$, the BMO norm of $\log |ψ|$ is bounded above by $2\sqrt{n-2}$ and $|ψ|^γ$ satisfies a reverse Hölder inequality for every $0<γ<1$. We also show that every plurisubharmonic function is locally BMO. Several Liouville…