Local inequalities for plurisubharmonic functions

Alexander Brudnyi

Review from Zentralblatt MATH:

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 $f$: $\Bbb{C}^n\rightarrow\Bbb{R}$ belongs to the class $\cal{F}_r$ $(r>1)$ 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 $\Bbb{C}^n$ and $\Bbb{R}^n$, 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 $f\in\cal{F}_r$ and every measurable subset $\omega\subset B(x,t)$.

The author gives applications of the main theorem related to Yu. Brudnyi-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.

Reviewed by K.G.Malyutin

Keywords: plurisubharmonic function; BMO-function; Euclidean ball; Brudnyi-Ganzburg type inequality

Classification (MSC2000): 31C10 32U05 31B05 46E15

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