RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find

 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

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

Full text: PDF file (544 kB)
References: PDF file   HTML file

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

Bibliographic databases:

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

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

Citation in format AMSBIB
\Bibitem{Dan10} \by V.~I.~Danchenko \paper On the massiveness of exceptional sets of the maximum modulus principle \jour Izv. RAN. Ser. Mat. \yr 2010 \vol 74 \issue 4 \pages 63--74 \mathnet{http://mi.mathnet.ru/izv4021} \crossref{https://doi.org/10.4213/im4021} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2730010} \zmath{https://zbmath.org/?q=an:1202.30041} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2010IzMat..74..723D} \elib{https://elibrary.ru/item.asp?id=20358754} \transl \jour Izv. Math. \yr 2010 \vol 74 \issue 4 \pages 723--734 \crossref{https://doi.org/10.1070/IM2010v074n04ABEH002504} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000281623100003} \elib{https://elibrary.ru/item.asp?id=16975300} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-78049345283}