Vol. 110, No. 1, 1984

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ISSN: 0030-8730
Closed ranged restriction operators on weighted Bergman spaces

Daniel Henry Luecking

Vol. 110 (1984), No. 1, 145–160
Abstract

We extend the results in [5] to several variables and to larger classes of domains. In particular it is shown that if B is the unit ball in Cn and G B is measurable then for any p < 0

∫               ∫
|f|pdV ≤ const.  |f|pdV,
B               G

for all analytic functions f in Lp(B) if and only if there exist δ > 0 and 0 < r < 1 such that |G Q(B(0,r))|≥ δ|Q(B(0,r))| for all automorphisms Q : B B. This is actually done for weighted integrals in more general domains. This is easily seen to be a criterion for the operation f f|G to have closed range. In addition, some partial results on the closed range of f →{f(zn)} in some weighted lp spaces are obtained for sequences {zn} in certain domains.

Mathematical Subject Classification 2000
Primary: 46E15
Secondary: 32H10
Milestones
Received: 9 March 1982
Published: 1 January 1984
Authors
Daniel Henry Luecking