Local inequalities for plurisubharmonic functions. (English) Zbl 0927.31001

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.


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
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