## 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.

### 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
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