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Surjective isometries of weighted Bergman spaces. (English) Zbl 0679.46023

In the reviewed paper the author investigates automorphisms \(\Phi\) of \(\Omega\) which generate surjective isometries f of the weighted Bergman space \(B^ p_ F(\Omega)\), where the weight function F is defined on \(\Omega\). The paper contains a characteristic of those automorphisms, where \(f\to g\cdot (f\circ \Phi)\) and the corresponding function g exists.
The author considers two cases of surjective isometries, for \(B^ p_ F(\Omega)\) and \(A^{p,r}(\Omega)\) in parts 3 and 4, respectively.
In the last part of the paper the author has presented a very important theorem, which says that if the group Aut(\(\Omega)\) acts transitively on \(\Omega\), then \(\Phi\in Aut(\Omega)\) generates a surjective isometry of \(A^{p,r}(\Omega)\) for \(p>0\), \(r>K(\Omega)\), and the corresponding function g is defined by the formula \(g=(J\Phi)^{(2r+2)/p}\), where \(J\Phi\) denotes the complex Jacobian of \(\Phi\).
Reviewer: A.Waszak

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
30H05 Spaces of bounded analytic functions of one complex variable
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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References:

[1] Charles Horowitz, Zeros of functions in the Bergman spaces, Duke Math. J. 41 (1974), 693 – 710. · Zbl 0293.30035
[2] Clinton J. Kolaski, Isometries of Bergman spaces over bounded Runge domains, Canad. J. Math. 33 (1981), no. 5, 1157 – 1164. · Zbl 0487.32002 · doi:10.4153/CJM-1981-087-1
[3] Clinton J. Kolaski, Isometries of weighted Bergman spaces, Canad. J. Math. 34 (1982), no. 4, 910 – 915. · Zbl 0459.32010 · doi:10.4153/CJM-1982-063-5
[4] Daniel H. Luecking, Representation and duality in weighted spaces of analytic functions, Indiana Univ. Math. J. 34 (1985), no. 2, 319 – 336. · Zbl 0538.32004 · doi:10.1512/iumj.1985.34.34019
[5] Walter Rudin, Function theory in the unit ball of \?\(^{n}\), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York-Berlin, 1980. · Zbl 0495.32001
[6] A. Selberg, Automorphic functions and integral operators, Seminars on Analytic Functions, II, Institute for Advanced Study, Princeton, N.J., 1957, pp. 152-161.
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