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Proceedings of the American Mathematical Society

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Duality between $ H\sp{p}$ and $ H\sp{q}$ and associated projections

Author: Walter Pranger
Journal: Proc. Amer. Math. Soc. 49 (1975), 342-348
MSC: Primary 30A78
MathSciNet review: 0377064
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Abstract: If $ U/G$ represents a Riemann surface as the disk $ U$ modulo a discontinuous group $ G$ and if $ {L^p}/G$ denotes the $ {L^p}$ functions on the circle which are $ G$ invariant, then it is shown that $ {L^p}/G = {N_p} \oplus {K_p}$ if and only if $ {H^p}/G$ and $ {\bar H^q}/G$ are naturally dual. Here $ {K_p}$ is the subset of $ {L^p}/G$ consisting of those functions which are invariant and whose conjugates are invariant; $ {N_p}$ is $ E({H^p}) \cap E(\bar H_0^p)$ where $ E$ is the conditional expectation operator. $ {H^p}$ is the space of boundary values of holomorphic functions and $ 1 < p < \infty $.

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Article copyright: © Copyright 1975 American Mathematical Society