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Bourgain's analytic projection revisited

Author(s): S. V. Kislyakov
Journal: Proc. Amer. Math. Soc. 126 (1998), 3307-3314.
MSC (1991): Primary 30D55, 46E15
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Abstract: For a positive function $w$ on the unit circle with $\log w\in L^1$, the following two statements are equivalent: (a) $\log w\in BMO$; (b) there is an operator $Q$ projecting $L^p(w)$ onto $H^p(w)$ for all $1<p<\infty$ at once and having weak type (1,1) with respect to $w$.

References:

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J. Bourgain, Bilinear forms on $H^\infty$ and bounded bianalytic functions, Trans. Amer. Math. Soc. 286 (1986), no. 1, 313-337. MR 86c:46060

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J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North Holland, Amsterdam, 1985. MR 87d:42023

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N. J. Kalton, Complex interpolation of Hardy-type subspaces, Math. Nachr. 171 (1995), 227-258. MR 96a:46055

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S. V. Kislyakov, Absolutely summing operators on the disk algebra, St. Petersburg Math. J. 3 (1992), no. 3. MR 93b:47062

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S. V. Kislyakov, Interpolation of $H^p$-spaces: some recent developments, Function spaces, Interpolation spaces, and Related topics, Israel Mathematical Conferences Proceedings, Vol. 12, Amer. Math. Soc., Providence, RI, 1998.

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S. V Kislyakov and Quanhua Xu, Interpolation of weighted and vector-valued Hardy spaces, Trans. Amer. Math. Soc. 343 (1994), no. 3, 1-34. MR 95b:46038

[P]
G. Pisier, Interpolation between $H^p$ spaces and noncommutative generalizations, I, Pacific J. Math. 155 (1992), 341-368. MR 93f:46111

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Additional Information:

S. V. Kislyakov
Affiliation: Université Bordeaux I, Laboratoire de Mathématiques Pures, 351 cours de la Libération, F-33405 Talence Cedex 05, France - Steklov Mathematical Institute, St. Petersburg Branch, Fontanka 27, 191011 St. Petersburg, Russia
Email: skis@math.u-bordeaux.fr, skis@pdmi.ras.ru

DOI: 10.1090/S0002-9939-98-04502-X
PII: S 0002-9939(98)04502-X
Keywords: Weight, analytic decomposition of unity, analytic projection, $BMO$-regular space
Received by editor(s): March 30, 1997
Communicated by: Theodore W. Gamelin
Copyright of article: Copyright 1998, American Mathematical Society

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