Related papers: Newton Numbers and Residual Measures of Plurisubha…
A notion of indicator for a plurisubharmonic function u of logarithmic growth in C^n is introduced and studied. It is applied to evaluation of the total Monge-Amp\`ere measure (dd^cu)^n({C}^n). Upper bounds for the measure are obtained in…
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…
Let $u$ be a plurisubharmonic function, defined on a neighbourhood of a point $x,$ such that the complex Monge-Amp\`ere operator is well-defined on $u.$ Suppose also that $u$ has a weak singularity, in the sense that the Lelong number of…
Generalized Lelong numbers of plurisubharmonic functions with respect to plurisubharmonic weights (due to Demailly) are specified for weights with multicircled asymptotics. Explicit formulas for these values are obtained in terms of the…
We look for pointwise bounds on a plurisubharmonic function near its singularity point, given the value of its generalized Lelong number with respect to a plurisubharmonic weight. To this end, an extremal problem is considered. In certain…
Let $u$ be a pluriharmonic function on the unit ball in $\mathbb{C}^n$. I consider the relationship between the set of points $L_u$ on the boundary of the ball at which $u$ converges nontangentially and the set of points $\mathcal{L}_u$ at…
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…
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 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…
The paper considers a global version of the notion of log canonical threshold for plurisubharmonic functions $u$ of logarithmic growth in $\mathbb{C}^n$, aiming at description of the range of all $p>0$ such that $e^{-u}\in…
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 $M$…
Let $d\nu$ be a measure in $\mathbb{R}^d$ obtained from adding a set of mass points to another measure $d\mu$. Orthogonal polynomials in several variables associated with $d\nu$ can be explicitly expressed in terms of orthogonal polynomials…
We prove that a plurisubharmonic function on a domain in the complex Euclidean space is a locally VMO (Vanishing Mean Oscillation) function if and only if its Lelong number at each point vanishes. We also give a global version of this…
The main objective of this paper is to prove a new inequality for plurisubharmonic functions estimating their supremum over a ball by their supremum over a measurable subset of the ball. We apply this result to study local properties of…
We present a modification of Newton's method to restore quadratic convergence for isolated singular solutions of polynomial systems. Our method is symbolic-numeric: we produce a new polynomial system which has the original multiple solution…
A nonlocal mass operator can be consistently set in local form through the introduction of a set of additional fields with geometrical appropriated properties. A local and polynomial gauge invariant action is thus identified. Equations…
Under certain conditions, we give an estimate from above on the number of differential equations of order $r+1$ with prescribed regular singular points, prescribed exponents at singular points, and having a quasi-polynomial flag of…
We prove exponential estimates for plurisubharmonic functions with respect to Monge-Ampere measures with Holder continuous potential. As an application, we obtain several stochastic properties for the equilibrium measures associated to…
The main purpose of this paper is to introduce and study the notion of plurifinely-maximal plurifinely plurisubharmonic functions, which extends the notion of maximal plurisubharmonic functions on a Euclidean domain to a plurifine domain of…
We extend certain classical theorems in pluripotential theory to a class of functions defined on the support of a $(1,1)$-closed positive current $T$, analogous to plurisubharmonic functions, called $T$-plurisubharmonic functions. These…