Geodesic
On the massiveness of exceptional sets of the maximum modulus principle
Izvestiya. Mathematics, Tome 74 (2010) no. 4, pp. 723-734

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We consider the sets $E_{\nu}(f)=\{z\colon |f(z)|\geqslant \nu\}$ for $\nu>\nu_0(f):=\limsup_{z\to\partial D}|f(z)|$ in the disc $D=\{z\colon |z|1\}$, 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. 
V. I. Danchenko. On the massiveness of exceptional sets of the maximum modulus principle. Izvestiya. Mathematics, Tome 74 (2010) no. 4, pp. 723-734. http://geodesic.mathdoc.fr/item/IM2_2010_74_4_a3/
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