The Bellman function, the two-weight Hilbert transform, and embeddings of the model spacesKθ



This paper is devoted to embedding theorems for the spaceKθ, where θ is an inner function in the unit disc D. It turns out that the question of embedding ofKθ into L2(Μ) is virtually equivalent to the boundedness of the two-weight Hilbert transform. This makes the embedding question quite difficult (general boundedness criteria of Hunt-Muckenhoupt-Wheeden type for the twoweight Hilbert transform have yet to be found). Here we are not interested in sufficient conditions for the embedding ofKg into L2(Μ) (equivalent to a certain two-weight problem for the Hilbert transform). Rather, we are interested in the fact that a certain natural set of conditions is not sufficient for the embedding ofKθ intoL2 (Μ) (equivalently, a certain set of conditions is not sufficient for the boundedness in a two-weight problem for the Hilbert transform). In particular, we answer (negatively) certain questions of W. Cohn about the embedding ofKθ into L2(Μ). Our technique leads naturally to the conclusion that there can be a uniform embedding of all the reproducing kernels ofKθ but the embedding of the wholeKθ intoL2(Μ) may fail. Moreover, it may happen that the embedding into a potentially larger spaceL2(μ) fails too.


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© Hebrew University of Jerusalem 2002

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Equipe d’AnalyseUniversité Paris VIParis CedexFrance

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