Local inequalities for plurisubharmonic functions. (English) Zbl 0927.31001

Brudnyi, Alexander

Ann. Math. (2) 149, No. 2, 511-533 (1999).

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. A plurisubharmonic function \(f\): \(\mathbb{C}^n\rightarrow\mathbb{R}\) belongs to the class \(\mathcal{F}_r\) \((r>1)\) if it satisfies \(\sup_{B_c(0,r)}f=0;\quad \sup_{B_c(0,1)}f\geq -1 .\) Here and below \(B_c(x,\rho)\) and \(B(x,\rho)\) denote the Euclidean ball with center \(x\) and radius \(\rho\) in \(\mathbb{C}^n\) and \(\mathbb{R}^n\), respectively. Let the ball \(B(x,t)\) satisfy \(B(x,t)\subset B_c(x,at)\subset B_c(0,1)\), where \(a>1\) is a fixed constant.
The main result is the following: there are constants \(c=c(a,r)\) and \(d=d(n)\) such that the inequality \[ \sup_{B(x,t)}f\leq c\log\left(\frac{d| B(x,t)| }{| \omega| }\right)+\sup_{\omega}f \] holds for every \(f\in\mathcal{F}_r\) and every measurable subset \(\omega\subset B(x,t)\).
The author gives applications of the main theorem related to Yu. Brudnyǐ-Ganzburg type inequalities for polynomials, algebraic functions and entire functions of exponential type. He also gives applications to log-BMO properties of real analytic functions, which were known previously only for polynomials.

Reviewer: K.G.Malyutin (Sumy)

MSC:

31C10	Pluriharmonic and plurisubharmonic functions
32U05	Plurisubharmonic functions and generalizations
31B05	Harmonic, subharmonic, superharmonic functions in higher dimensions
46E15	Banach spaces of continuous, differentiable or analytic functions

Keywords:

plurisubharmonic function; BMO-function; Euclidean ball; Brudnyǐ-Ganzburg type inequality