Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-14T22:23:25.901Z Has data issue: false hasContentIssue false
  • English
  • Français

Hörmander's Carleson theorem for the ball

Published online by Cambridge University Press:  18 May 2009

S. C. Power
S. C. Power
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, U.S.A. Department of Mathematics, University of Lancaster, Bailrigg, Lancaster LA1 4YW, England
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let denote the unit ball in ℂ2 and let Sdenote its boundary, the unit sphere. For z ∈ B and δ>0, the following non isotropic balls are defined, where

A finite positive Borel measure μ, on B is called a Carleson measure if there exists a constant C for which

Here σ denotes normalized surface area measure on S. The following theorem was obtained by Hörmander [6] as a special case of more general variants for strictly pseudoconvex domains in ℂn. Recently Cima and Wogen [3] derived it from a Carleson measure theorem for Bergman spaces of the ball. A different direct approach to the Bergman context, and related settings, is given in Leucking [7].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 26 , Issue 1 , January 1985 , pp. 13 - 17
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

1.Carleson, L., Interpolation by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–599.CrossRefGoogle Scholar
2.Choi, M.-D., Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 90 (1983), 301–312.Google Scholar
3.Cima, J. A. and Wogen, W. R., A Carleson measure theorem for the Bergman space on the ball, J. Operator Theory 7 (1982), 157–165.Google Scholar
4.Coifman, R. R., Rochberg, R. and Weiss, G., Factorisation theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), 611–635.CrossRefGoogle Scholar
5.Duren, P. L., Theory of Hn spaces (Academic Press, 1970).Google Scholar
6.Hörmander, L., Lnestimates for (pluri-) subharmonic functions, Math. Scand. 20 (1967), 65–78.CrossRefGoogle Scholar
7.Leucking, D., A technique for characterising Carleson measures on Bergman spaces, Proc. Amer. Math. Soc. 87 (1983), 656–660.CrossRefGoogle Scholar
8.Magnus, W., On the spectrum of Hilbert's matrix, Amer. J. Math. 72 (1950), 699–704.Google Scholar
9.Power, S. C., Hankel operators on Hilbert space, Research Notes in Mathematics, No. 64 (Pitman, 1982).Google Scholar
10.Power, S. C., Finite rank multivariable Hankel forms, Linear Algebra Appl. 48 (1982), 237–244.Google Scholar
11.Rudin, W., Function theory in the unit ball of Cn (Springer-Verlag, 1980).Google Scholar
12.Sarason, D., Function theory on the unit circle, Lecture notes for a conference at Virginia Polytechnic and State University, Blacksburg, Virginia, 06 1978.Google Scholar
13.Shields, A. L., An analogue of the Fejéer-Riesz theorem for the Dirichlet space, in Beckner, W. et al. , eds. Conference on harmonic analysis in honour of Antoni Zygmund (Wadsworth, 1983), Vol. II, 810.Google Scholar