Related papers: Pluricomplex Green and Lempert functions for equal…

The classical Siciak-Zakharyuta theorem states that the Siciak-Zakharyuta function $V_{E}$ of a subset $E$ of $\mathbb C^n$, also called a pluricomplex Green function or global exremal function of $E$, equals the logarithm of the Siciak…

It has been recently established by the first and third author that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the…

For a given second-order linear elliptic operator $L$ which admits a positive minimal Green function, and a given positive weight function $W$, we introduce a family of weighted Lebesgue spaces $L^p(\phi_p)$ with their dual spaces, where…

To every labeled poset (P,\omega), one can associate a quasisymmetric generating function for its (P,\omega)-partitions. We ask: when do two labeled posets have the same generating function? Since the special case corresponding to skew…

We show that every bounded hyperconvex Reinhardt domain can be approximated by special polynomial polyhedra defined by homogeneous polynomial mappings. This is achieved by means of approximation of the pluricomplex Green function of the…

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or -- equivalently -- a…

We introduce a quantity which measures the singularity of a plurisubharmonic function f relative to another plurisubharmonic function g, at a point a. This quantity, which we denote by $\nu_{a,g}(f)$, can be seen as a generalization of the…

In this note, we consider meromorphic univalent functions $f(z)$ in the unit disc with a simple pole at $z=p\in(0,1)$ which have a $k$-quasiconformal extension to the extended complex plane $\hat{\mathbb C},$ where $0\leq k < 1$. We denote…

We study pluricomplex Green functions on algebraic sets. Let $f$ be a proper holomorphic mapping between two algebraic sets. Given a compact set $K$ in the range of $f$, we show how to estimate the pluricomplex Green functions of $K$ and of…

We study the problem of the product property for the Lempert function with many poles and consider some properties of this function mostly for plane domains.

We show existence, uniqueness and positivity for the Green's function of the operator $(\Delta_g + \alpha)^k$ in a closed Riemannian manifold $(M,g)$, of dimension $n>2k$, $k\in \mathbb{N}$, $k\geq 1$, with Laplace-Beltrami operator…

We construct the fundamental solution or Green function for a divergence form elliptic system in two dimensions with bounded and measurable coefficients. We consider the elliptic system in a Lipschitz domain with mixed boundary conditions.…

Let $X$ be a locally compact abelian group with countable base and let $\mathcal W$ be a convex cone of positive numerical functions on $X$ which is invariant under the group action and such that $(X,\mathcal W)$ is a balayage space or…

Let $u$ be a maximal plurisubharmonic function in a domain $\Omega\subset\mathbb{C}^n$ ($n\geq 2$). It is classical that, for any $U\Subset\Omega$, there exists a sequence of bounded plurisubharmonic functions $PSH(U)\ni u_j\searrow u$…

A simple model of noninteracting electrons with a separable one-body potential is used to discuss the possible pole structure of single particle Green's functions for fermions on unphysical sheets in the complex frequency plane as a…

For a chordal SLE$_\kappa$ ($\kappa\in(0,8)$) curve in a domain $D$, the $n$-point Green's function valued at distinct points $z_1,\dots,z_n\in D$ is defined to be $$G(z_1,\dots,z_n)=\lim_{r_1,\dots,r_n\downarrow 0} \prod_{k=1}^n r_k^{d-2}…

A seminal result of Agler characterizes the so-called Schur-Agler class of functions on the polydisk in terms of a unitary colligation transfer function representation. We generalize this to the unit ball of the algebra of multipliers for a…

We show an example of a sequence of elliptic operators in the unit ball with drifts that diverge as the inverse distance to the boundary, for which we do not get uniform upper estimates for the Green's function with the pole at the origin.…

We prove the the multipole Lempert function is monotone under inclusion of pole sets.

Let $L$ be a second-order, homogeneous, constant (complex) coefficient elliptic system in ${\mathbb{R}}^n$. The goal of this article is provide a qualitative and quantitative study of the nature of the Green function associated with the…