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Carleson measures and operators on star-invariant subspaces. (English) Zbl 0615.47025

The author studies several questions related to the properties of Carleson measures for spaces of the form \(K=H^ 2\theta \phi H^ 2\), where \(\phi\) is an inner function. A (positive) measure \(\mu\) on the closed unit disk is a Carleson measure for K if it assigns no mass to the singular support of \(\phi\) and if there is a constant c such that \(\int | f|^ 2d\mu \leq c\| f\|^ 2_ 2\) for all f in K. A special case of these is obtained by \(\mu =wd\theta\), where w is a positive \(L^{\infty}\) function.
For an arbitrary inner function \(\phi\) and \(\mu =wd\theta\) as above, Volberg has established necessary and sufficient conditions for the inclusion operator \(I: K\to L^ 2(d\mu)\) to be bounded below and for it to be compact. Here, the author asks if these results can be generalized to hold for an arbitrary Carleson measure for K. In expanding the class of measures under consideration, the tools available force a restriction on the class of inner functions. Thus, the inner functions \(\phi\) in this paper satisfy the connected level set condition; that is, there exists \(0<r_ 0<1\) such that \(\{| \phi (z)| <r_ 0\}\) is connected. In this setting, the author produces necessary and sufficient conditions for the inclusion operator I to be compact and a sufficient condition for I to be bounded below. As well, he shows that, in this new setting, Volberg’s condition for I to be bounded below is no longer sufficient. The conditions obtained concern the behavior of the given Carleson measure on portions of the pre-image under \(\phi\) of the unit circle.
The author also presents a new characterization for Carleson measures living on the unit circle and provides, for \(\phi\) with the level set condition, an inequality which strengthens a previous inequality due to Axler, Chang, and Sarason (though theirs was valid for all inner functions). Specifically, for such \(\phi\), there is a constant c such that \[ \int_{\{| \phi | >r\}}| f'(z)|^ 2(1-| z|)dA\leq c(1-r)\| f\|^ 2_ 2, \] for \(r>r_ 0\) and f in K. As the author points out, it remains an interesting question whether the results herein can be extended to apply to all inner functions \(\phi\) and all Carleson measures for \(H^ 2\theta \phi H^ 2\).
Reviewer: T.Feeman

MSC:

47B38 Linear operators on function spaces (general)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
47A15 Invariant subspaces of linear operators
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