Abstract:
The strong dual space of the Bergman space
B2(G)={f∈H(G):‖f‖2B2(G)=∫G|f(x)|2dv(z)<∞},
is described in terms of the Cauchy transformation, where v(z) is Lebesgue measure and G is a simply connected domain with boundary of class C1+0. As a normed space, B∗2(G) is isomorphic to the space
B12(C∖¯G)={γ(ζ)∈H(C∖¯G),γ(∞)=0:‖γ‖2B12(C∖¯G)=∫C∖¯G|γ′(ζ)|2dv(ζ)<∞}.
An example is given of a domain with nonsmooth boundary for which the spaces B∗2(G) and B12(C∖¯G) are not isomorphic.
Citation:
V. V. Napalkov, R. S. Yulmukhametov, “On the Cauchy transform of functionals on a Bergman space”, Russian Acad. Sci. Sb. Math., 82:2 (1995), 327–336
\Bibitem{NapYul94}
\by V.~V.~Napalkov, R.~S.~Yulmukhametov
\paper On the Cauchy transform of functionals on a~Bergman space
\jour Russian Acad. Sci. Sb. Math.
\yr 1995
\vol 82
\issue 2
\pages 327--336
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\crossref{https://doi.org/10.1070/SM1995v082n02ABEH003567}
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Linking options:
https://www.mathnet.ru/eng/sm912
https://doi.org/10.1070/SM1995v082n02ABEH003567
https://www.mathnet.ru/eng/sm/v185/i7/p77
This publication is cited in the following 4 articles:
Harti El, A. Elkachkouri, A. Ghanmi, “Solid Cauchy transform on the weighted poly-Bergman spaces”, Filomat, 37:3 (2023), 775
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
K. P. Isaev, R. S. Yulmukhametov, “Laplace transforms of functionals on Bergman spaces”, Izv. Math., 68:1 (2004), 3–41
R. S. Yulmukhametov, V. V. Napalkov, “On the Hilbert Transform in Bergman Space”, Math. Notes, 70:1 (2001), 61–70