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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Bonded projections, duality, and multipliers in spaces of analytic functions
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by A. L. Shields and D. L. Williams PDF
Trans. Amer. Math. Soc. 162 (1971), 287-302 Request permission

Abstract:

Let $\varphi$ and $\psi$ be positive continuous functions on $[0,1)$ with $\varphi (r) \to 0$ as $r \to 1$ and $\smallint _0^1\psi (r)\;dr < \infty$. Denote by ${A_0}(\varphi )$ and ${A_\infty }(\varphi )$ the Banach spaces of functions f analytic in the open unit disc D with $|f(z)|\varphi (|z|) = o(1)$ and $|f(z)|\varphi (|z|) = O(1),|z| \to 1$, respectively. In both spaces $\left \|f\right \|_\varphi = {\sup _D}|f(z)|\varphi (|z|)$. Let ${A^1}(\psi )$ denote the space of functions analytic in D with $\left \|f\right \|_\psi = \smallint {\smallint _D}|f(z)|\psi (|z|)\;dx\;dy < \infty$. The spaces ${A_0}(\varphi ),{A^1}(\psi )$, and ${A_\infty }(\varphi )$ are identified in the obvious way with closed subspaces of ${C_0}(D),{L^1}(D)$, and ${L^\infty }(D)$, respectively. For a large class of weight functions $\varphi ,\psi$ which go to zero at least as fast as some power of $(1 - r)$ but no faster than some other power of $(1 - r)$, we exhibit bounded projections from ${C_0}(D)$ onto ${A_0}(\varphi )$, from ${L^1}(D)$ onto ${A^1}(\psi )$, and from ${L^\infty }(D)$ onto ${A_\infty }(\varphi )$. Using these projections, we show that the dual of ${A_0}(\varphi )$ is topologically isomorphic to ${A^1}(\psi )$ for an appropriate, but not unique choice of $\psi$. In addition, ${A_\infty }(\varphi )$ is topologically isomorphic to the dual of ${A^1}(\psi )$. As an application of the above, the coefficient multipliers of ${A_0}(\varphi ),{A^1}(\psi )$, and ${A_\infty }(\varphi )$ are characterized. Finally, we give an example of a weight function pair $\varphi ,\psi$ for which some of the above results fail.
References
  • Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
  • P. L. Duren, B. W. Romberg, and A. L. Shields, Linear functionals on $H^{p}$ spaces with $0<p<1$, J. Reine Angew. Math. 238 (1969), 32โ€“60. MR 259579
  • P. L. Duren and A. L. Shields, Coefficient multipliers of $H^{p}$ and $B^{p}$ spaces, Pacific J. Math. 32 (1970), 69โ€“78. MR 255825
  • G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z. 34 (1932), no.ย 1, 403โ€“439. MR 1545260, DOI 10.1007/BF01180596
  • E. Landau, Darstellung und Begrรผndung einiger neuerer Ergebnisse der Funktionentheorie, Springer-Verlag, Berlin, 1929. J. Lindenstrauss and A. Peล‚czyรบski, Contributions to the theory of the classical Banach spaces (preprint).
  • L. A. Rubel and A. L. Shields, The second duals of certain spaces of analytic functions, J. Austral. Math. Soc. 11 (1970), 276โ€“280. MR 0276744
  • J. H. Shapiro, A. L. Shields and G. D. Taylor, The second duals of some function spaces (preprint).
  • A. Zygmund, On the preservation of classes of functions, J. Math. Mech. 8 (1959), 889-895; erratum 9 (1959), 663. MR 0117498, DOI 10.1512/iumj.1960.9.59040
  • โ€”, Trigonometric series, Vols. 1, 2, 2nd ed., Cambridge Univ. Press, London, 1968. MR 38 #4882.
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 162 (1971), 287-302
  • MSC: Primary 46.30; Secondary 30.00
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0283559-3
  • MathSciNet review: 0283559