Carleson Measures for Spaces of Dirichlet Type |
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Authors: | Daniel Girela José Ángel Peláez |
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Institution: | (1) Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain |
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Abstract: | If 0 < p < ∞ and α > − 1, the space
consists of those functions f which are analytic in the unit disc
and have the property that f ′ belongs to the weighted Bergman space Aαp. In 1999, Z. Wu obtained a characterization of the Carleson measures for the spaces
for certain values of p and α. In particular, he proved that, for 0 < p ≤ 2, the Carleson measures for the space
are precisely the classical Carleson measures. Wu also conjectured that this result remains true for 2 < p < ∞. In this paper we prove that this conjecture is false. Indeed, we prove that if 2 < p < ∞, then there exists g analytic in
such that the measure μg,p on
defined by dμg,p (z) = (1 − |z|2)p - 1| g ′ (z)|p dx dy is not a Carleson measure for
but is a classical Carleson measure. We obtain also some sufficient conditions for multipliers of the spaces
![$$\mathcal{D}_{p - 1}^p .$$](/content/y32t0257j71g10uj/20_2005_Article_1391_TeX2GIFIEq8.gif) |
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Keywords: | Primary 30H05 Secondary 46J15 |
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