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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 4, Pages 63–74 (Mi izv4021)  

On the massiveness of exceptional sets of the maximum modulus principle

V. I. Danchenko

Vladimir State University

Abstract: We consider the sets $E_{\nu}(f)=ż\colon |f(z)|\ge \nu\}$ for $\nu>\nu_0(f):=\limsup_{z\to\partial D}|f(z)|$ in the disc $D=ż\colon |z|<1\}$, where $f(z)$, $z=x+iy$, are complex-valued functions defined on $D$ and having certain smoothness properties with respect to the real variables $x$ and $y$. We obtain estimates for some metric properties of the sets $E_{\nu}(f)$. For example, we prove that, if $\Delta f\in L_1(D)$, then the hyperbolic area of the set $E_\nu(f)$ cannot grow more rapidly than $\nu^{-1-o(1)}$ as $\nu\to 0$, where $o(1)$ is positive, and, if $f_{\bar{z}}\in L_2(D)$, then this area cannot grow more rapidly than $\nu^{-2-o(1)}$. The orders of these estimates with respect to $\nu$ are sharp.

Keywords: hyperbolic distance and area, capacity and potential, polyanalytic function, maximum modulus principle, Green's formulae.

DOI: https://doi.org/10.4213/im4021

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English version:
Izvestiya: Mathematics, 2010, 74:4, 723–734

Bibliographic databases:

UDC: 517.544.5+517.544.45
MSC: Primary 30C85; Secondary 31A15
Received: 18.09.2008

Citation: V. I. Danchenko, “On the massiveness of exceptional sets of the maximum modulus principle”, Izv. RAN. Ser. Mat., 74:4 (2010), 63–74; Izv. Math., 74:4 (2010), 723–734

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