The Banach envelope of Paley-Wiener type spaces

Hoffmann, Mark

doi:10.1090/S0002-9939-02-06581-4

The Banach envelope of Paley-Wiener type spaces
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Proc. Amer. Math. Soc. 131 (2003), 543-548 Request permission

Abstract:

We give an explicit computation of the Banach envelope for the Paley-Wiener type spaces $E^p, 0<p<1$. This answers a question by Joel Shapiro.

References

Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in $L^{p}$, Representation theorems for Hardy spaces, Astérisque, vol. 77, Soc. Math. France, Paris, 1980, pp. 11–66. MR 604369
Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
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), 32–60. MR 259579
Carolyn Eoff, The discrete nature of the Paley-Wiener spaces, Proc. Amer. Math. Soc. 123 (1995), no. 2, 505–512. MR 1219724, DOI 10.1090/S0002-9939-1995-1219724-X
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
N. J. Kalton, N. T. Peck, and James W. Roberts, An $F$-space sampler, London Mathematical Society Lecture Note Series, vol. 89, Cambridge University Press, Cambridge, 1984. MR 808777, DOI 10.1017/CBO9780511662447
N. J. Kalton and D. A. Trautman, Remarks on subspaces of $H_{p}$ when $0<p<1$, Michigan Math. J. 29 (1982), no. 2, 163–170. MR 654477
Paul Koosis, Introduction to $H_p$ spaces, 2nd ed., Cambridge Tracts in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998. With two appendices by V. P. Havin [Viktor Petrovich Khavin]. MR 1669574
Osvaldo Mendez and Marius Mitrea, The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations, J. Fourier Anal. Appl. 6 (2000), no. 5, 503–531. MR 1781091, DOI 10.1007/BF02511543
M. Mitrea, Banach envelopes of holomorphic Hardy spaces, preprint, 2000.
A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. MR 126145, DOI 10.4064/sm-19-2-209-228
M. Plancherel and G. Pólya, Fonctions entières et intègrales de Fourier multiples, Comment. Math. Helv. 10 (1937) 110-163.
Fulvio Ricci and Mitchell Taibleson, Boundary values of harmonic functions in mixed norm spaces and their atomic structure, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 1, 1–54. MR 713108
Hans Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992. MR 1163193, DOI 10.1007/978-3-0346-0419-2
Joel H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces, Duke Math. J. 43 (1976), no. 1, 187–202. MR 500100
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
W. J. Stiles, Some properties of $l_{p}$, $0<p<1$, Studia Math. 42 (1972), 109–119. MR 308726
P. Wojtaszczyk, $H_{p}$-spaces, $p\leq 1$, and spline systems, Studia Math. 77 (1984), no. 3, 289–320. MR 745285, DOI 10.4064/sm-77-3-289-320