Carleson measures in Hardy and weighted Bergman spaces of polydiscs

Jafari, F.

doi:10.1090/S0002-9939-1991-1039533-0

Carleson measures in Hardy and weighted Bergman spaces of polydiscs

Author: F. Jafari
Journal: Proc. Amer. Math. Soc. 112 (1991), 771-781
MSC: Primary 47B38; Secondary 32A35, 46E15, 47B07
DOI: https://doi.org/10.1090/S0002-9939-1991-1039533-0
MathSciNet review: 1039533
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Abstract: The importance of theorems on Carleson measures has been well recognized [3]. In [1] Chang has given a characterization of the bounded measures on ${L^p}({T^n})$ using what one may characterize as the bounded identity operators from Hardy spaces of polydiscs in ${L^p}$ spaces. In [4] Hastings gives a similar result for (unweighted) Bergman spaces of polydiscs. In this paper we characterize the bounded identity operators from weighted Bergman spaces of polydiscs into ${L^p}$ spaces, and classify those operators which are compact on the Hardy and weighted Bergman spaces in terms of Carleson-type conditions. We give two immediate applications of these results here, and a much broader class of applications elsewhere [5].

[1] S.-Y. A. Chang, Carleson measure on the bi-disc, Ann. of Math. (2) 109 (1979), 613-620. MR 534766 (80j:32009)
[2] P. L. Duren, Theory of ${H^p}$ spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
[3] J. Garnett, Bounded analytic functions, Academic Press, New York, 1980. MR 628971 (83g:30037)
[4] W. W. Hastings, A Carleson measure theorem for Bergman spaces, Proc. Amer. Math. Soc. 52 (1975), 237-241. MR 0374886 (51:11082)
[5] F. Jafari, On bounded and compact composition operators in polydiscs, Canad. J. Math. (to appear). MR 1081000 (91k:47065)
[6] -, Composition operators in polydiscs, Dissertation, University of Wisconsin, Madison, 1989.
[7] B. D. MacCluer, Compact composition operators on ${H^p}({B_N})$ , Michigan Math. J. 32 (1985), 237-248. MR 783578 (86g:47037)
[8] B. D. MacCluer and J. H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), 878-906. MR 854144 (87h:47048)
[9] W. Rudin, Function theory in the unit ball of ${C^n}$ , Springer-Verlag, New York, 1980. MR 601594 (82i:32002)
[10] D. S. Stegenga, Multipliers of the Dirichlet spaces, Illinois J. Math. 24 (1980), 113-139. MR 550655 (81a:30027)