, Volume 87, Issue 1, pp 385-414

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

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Abstract

This paper is devoted to embedding theorems for the spaceK θ , where θ is an inner function in the unit disc D. It turns out that the question of embedding ofK θ 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 ofK θ intoL 2 (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 ofK θ into L2(Μ). Our technique leads naturally to the conclusion that there can be a uniform embedding of all the reproducing kernels ofK θ but the embedding of the wholeK θ intoL 2(Μ) may fail. Moreover, it may happen that the embedding into a potentially larger spaceL 2(μ) fails too.

Both authors are supported by the NSF grant DMS 9970395 and joint American-Israeli grant BSF 00030.