Related papers: A disc formula for plurisubharmonic subextensions …

Let $D_j\subset\Bbb C^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluripolar set, $j=1,...,N$. Put$$X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N\subset\Bbb…

We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Am\`ere equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and H\"older…

In this paper, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Amp\`ere equation and has a convex level set. To prove our main theorem, we show a minimum principle of a maximal…

On any metric space, I provide an intrinsic characterization of those complex-valued functions which are uniform limits of Lipschitz functions. There are applications to function theory on complete Riemannian manifolds and, in particular,…

Let K be a subset of a smooth manifold M. In some cases functor calculus methods lead to a homotopical formula for M minus K in terms of the subspaces M minus S, where S runs through the finite subsets of K.

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…

We investigate the spectrum and the essential spectra of weighted automorphisms of the polydisc algebra $\mathbb{A}^n$. In the case $n=2$ we provide a detailed and, in most cases, complete description of these spectra.

We recall the complex structure on the generalised loop spaces $W^{k,2}(S,X)$, where $S$ is a compact real manifold with boundary and $X$ is a complex manifold, and prove a Hartogs-type extension theorem for holomorphic maps from certain…

We prove the following theorem: Let X be a Stein manifold of dimension at least 2 and Y a closed complete pluripolar subset of X. Given a point p in the complement of Y there is a proper holomorphic map f from the unit disc to X such that…

In this paper, we combine tools from pluripotential theory and commutative algebra to study singularity invariants of plurisubharmonic functions. We establish several relationships between the singularity invariants of plurisubharmonic…

We re-consider the procedure of ``taking a square root of the Dirac equation'' on the superspace and show that it leads to the well known superfield W_\alpha and to the proper equations of motion for the components, i.e. the Maxwell…

The general purpose of this paper is to investigate the notion of "pluriharmonics" for the general potential theory associated to a convex cone $F\subset {\rm Sym}^2({\bf R}^n)$. For such $F$ there exists a maximal linear subspace $E\subset…

We give the extension formulae on almost complex manifolds and give decompositions of the extension formulae. As applications, we study $(n,0)$-forms, the $(n,0)$-Dolbeault cohomology group and $(n,q)$-forms on almost complex manifolds.

We formulate a multi-valued version of the Tietze-Urysohn extension theorem. Precisely, we prove that any upper semicontinuous multi-valued map with nonempty closed convex values defined on a closed subset (resp. closed perfectly normal…

In this article, a novel method to compute all discrete polyharmonic functions in the quarter plane for models with small steps, zero drift and a finite group is proposed. A similar method is then introduced for continuous polyharmonic…

The celebrated Ohsawa--Takegoshi extension theorem for $L^2$ holomorphic functions on bounded pseudoconvex domains in $\mathbb C^n$ is a fundamental result in several complex variables and complex geometry. Ohsawa conjectured in 1995 that…

For a (not necessarily locally convex) topological vector space $\mathcal{X}$ of holomorphic functions in one complex variable, we show that the shift invariant subspace generated by a set of polynomials is $\mathcal{X}$ if and only if…

In this paper, we introduce the notion of strong locally irreducible complex spaces $\widetilde{X}$. Based on this notion we prove the equality $\bar{\nu}_{\varphi}(x)=$ mult$(\widetilde{X},x). \nu_{\varphi}(x)$ for all $x\in…

In this paper, we establish the log-plurisubharmonicity of fiberwise $\xi$-Bergman kernels for a family of variant functionals, thereby addressing a question posed by Bo Berndtsson to the authors. As an application, we prove that for a…

We prove a representation formula for superharmonic functions on the half-space $\mathbb{R}^N_+ := \mathbb{R}^{N-1}\times]0,+\infty[$. As a consequence, we derive some comparison principles and various positivity results.