Mackey topologies, reproducing kernels,
and diagonal maps on the Hardy and Bergman spaces

 by Joel H. Shapiro


Duke Math. J. 43 (1976), 187--202.
 Abstract: The Mackey topology on an F-space is the strongest locally convex topology yielding the same dual. This paper presents a method for computing this topology which does not require determination of the dual space. The technique is applied to the Hardy spaces, Bergman spaces, and weighted Bergman spaces of holomorphic functions on the unit disk. For the Hardy spaces H^p, p < 1, the method gives a new proof of the result of P. L. Duren, B. W. Romberg and A. L. Shields [J. Reine Angew. Math. 238 (1969), 3260] which identifies the duals of H^p with the duals of certain weighted Bergman spaces. The duals of the weighted Bergman spaces with p < 1 are also identified. Connections between weighted Bergman spaces and the Hardy spaces of the polydisk are also discussed.


Download preprint (.pdf file, 524KB)