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Mat. Sb., 1994, Volume 185, Number 7, Pages 77–86 (Mi msb912)  

This article is cited in 3 scientific papers (total in 3 papers)

On the Cauchy transform of functionals on a Bergman space

V. V. Napalkov, R. S. Yulmukhametov

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences

Abstract: The strong dual space of the Bergman space
$$ B_2(G)=\{f\in H(G):\|f\|_{B_2(G)}^2=\int_G|f(x)|^2 d\mathrm{v}(z)<\infty\}, $$
is described in terms of the Cauchy transformation, where $\mathrm{v}(z)$ is Lebesgue measure and $G$ is a simply connected domain with boundary of class $C^{1+0}$. As a normed space, $B_2^*(G)$ is isomorphic to the space
$$ B_2^1(\mathbb{C}\setminus \overline G) =\{\gamma (\zeta )\in H(\mathbb{C}\setminus \overline G), \gamma (\infty )=0: \|\gamma \|_{B_2^1(\mathbb C\setminus\overline G)}^2 =\int_{{\mathbb C}\setminus {\overline G}} |\gamma'(\zeta )|^2 d\mathrm{v}(\zeta)<\infty\}. $$
An example is given of a domain with nonsmooth boundary for which the spaces $B_2^*(G)$ and $B_2^1(\mathbb C\setminus\overline G)$ are not isomorphic.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 82:2, 327–336

Bibliographic databases:

UDC: 517.5
MSC: Primary 47G10, 45P05, 46E20; Secondary 30D55, 32H10
Received: 24.06.1993

Citation: V. V. Napalkov, R. S. Yulmukhametov, “On the Cauchy transform of functionals on a Bergman space”, Mat. Sb., 185:7 (1994), 77–86; Russian Acad. Sci. Sb. Math., 82:2 (1995), 327–336

Citation in format AMSBIB
\Bibitem{NapYul94}
\by V.~V.~Napalkov, R.~S.~Yulmukhametov
\paper On the Cauchy transform of functionals on a~Bergman space
\jour Mat. Sb.
\yr 1994
\vol 185
\issue 7
\pages 77--86
\mathnet{http://mi.mathnet.ru/msb912}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1300133}
\zmath{https://zbmath.org/?q=an:0856.46031}
\transl
\jour Russian Acad. Sci. Sb. Math.
\yr 1995
\vol 82
\issue 2
\pages 327--336
\crossref{https://doi.org/10.1070/SM1995v082n02ABEH003567}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995RV83000005}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. R. S. Yulmukhametov, V. V. Napalkov, “On the Hilbert Transform in Bergman Space”, Math. Notes, 70:1 (2001), 61–70  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. K. P. Isaev, R. S. Yulmukhametov, “Laplace transforms of functionals on Bergman spaces”, Izv. Math., 68:1 (2004), 3–41  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. N. F. Abuzyarova, K. P. Isaev, R. S. Yulmukhametov, “Equivalence of norms of analytical functions on exterior of convex domain”, Ufa Math. J., 10:4 (2018), 3–11  mathnet  crossref  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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