Related papers: Indicators for Plurisubharmonic Functions of Logar…

We consider growth conditions for (frequently) Birkhoff-universal functions of exponential type with respect to the different rays emanating from the origin. For that purpose, we investigate their (conjugate) indicator diagram or,…

We recall known and establish new properties of the Dieudonn\'e and Moore determinants of quaternionic matrices.Using these linear algebraic results we develop a basic theory of plurisubharmonic functions of quaternionic variables. Then we…

In a recent preprint published on arXiv (see arXiv:2308.02993v2, referred here as \cite{NXH}), N.X. Hong stated that every plurifinely open set $U\subset \mathbb{C}^n$, $n\geq 1$, is of the form $U=\bigcup \{\varphi_j>-1\}$, where each…

Let $f$ be a complex H\'enon map and $\mu$ its unique measure of maximal entropy. We prove that $\mu$ is exponentially mixing of all orders for all (not necessarily bounded) plurisubharmonic observables, and that all plurisubharmonic…

We show that a recent result of Demailly and Pham Hoang Hiep \cite{DH} implies a description of plurisubharmonic functions with given Monge-Amp\`ere mass and smallest possible log canonical threshold. We also study an equality case for the…

Let $\mu^z$ be the measure obtained by sweeping out the Monge-Amp\`ere measure of the pluricomplex Green function with pole at $z. $ We prove that $\mu^z$ vanish on Levi flat parts of the boundary for 1) every relatively compact analytic…

We study $L^p(\mu) \to L^q(\nu)$ mapping properties of the convolution operator $ T_{\lambda}f(x)=\lambda*(f\mu)(x)$ and of the corresponding maximal operator $ {\mathcal T}_{\lambda}f(x)=\sup_{t>0} |\lambda_t*(f\mu)(x)|$, where $\lambda$…

We show that unary log-analytic functions are polynomially bounded. In the higher dimensional case globally a log-analytic function can have exponential growth. We show that a log-analytic function is polynomially bounded on a definable set…

The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros is extended to subharmonic functions in $\mathbb{R}^n, n\geq 3$. We also show that the existence of the limit $\lim_{\rti} \frac{\log…

In this paper, we consider the nodal set of a bi-harmonic function $u$ on an $n$ dimensional $C^{\infty}$ Riemannian manifold $M$, that is, $u$ satisfies the equation $\triangle_M^2u=0$ on $M$, where $\triangle_M$ is the Laplacian operator…

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…

Let $u\not\equiv -\infty$ and $M\not\equiv -\infty$ are two subharmonic functions in the complex plane $\mathbb C$ with the Riesz measures $\nu_u$ and $\mu_M$ such that $u(z)\leq O(|z|)$ and $M(z)\leq O(|z|)$ as $z\to \infty$. If the growth…

For any proper polynomial map $f:C^k\longrightarrow C^k$ define the function \alpha as $$\alpha(z):=\limsup_{n\to\infty} \frac{\log^+\log^+|f^n(z)|}{n} where \log^+:=\max(\log, 0).$$ Let f=(P_1,...,P_k) be a proper polynomial map. We define…

Two properties of plurisubharmonic functions are proven. The first result is a Skoda type integrability theorem with respect to a Monge-Amp\`ere mass with H\"older continuous potential. The second one says that locally, a p.s.h. function is…

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 $f$ be a H\'enon-Sibony map (regular polynomial automorphism) of $\mathbb{C}^k$ and let $\mu$ be the equilibrium measure of $f$. In this paper we prove that $\mu$ is exponentially mixing for plurisubharmonic test functions.

Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. Then for any function $r\colon\mathbb C\to (0,1]$ satisfying the condition $$\inf_{z\in\mathbb C}\frac{\ln r(z)}{\ln(2+|z|)}>-\infty,$$ there is an entire…

Let $\varphi$ be a plurisubharmonic function on a pseudoconvex domain $D \subset \mathbb C^n$. We show that there exists a nonzero holomorphic function $f$ on $D$ such that some local mean value of $\varphi$ with logarithmic additional…

Consider a compact K\"ahler manifold $(X,\omega)$ and the space $\cal E(X,\omega)=\cal E$ of $\omega$--plurisubharmonic functions of full Monge--Amp\`ere mass on it. We introduce a quantity $\rho[u,v]$ to measure the distance between $u,…

We prove the existence of plurisubharmonic functions with prescribed logarithmic singularities on complex 3-folds equipped with a nef class of positive volume. We prove the same result for rational classes on Moishezon n-folds.