Embedding $L^{1}$ in $L^{1}/H^{1}$
HTML articles powered by AMS MathViewer
- by J. Bourgain PDF
- Trans. Amer. Math. Soc. 278 (1983), 689-702 Request permission
Abstract:
It is proved that ${L^1}$ is isomorphic to a subspace of ${L^1}/{H^1}$. More precisely, there exists a diffuse $\sigma$-algebra $\mathfrak {S}$ on the circle such that the corresponding expectation ${\mathbf {E}}:{H^\infty } \to {L^\infty }({\mathbf {C}})$ is onto. The method consists in studying certain martingales on the product ${\prod ^{\mathbf {N}}}$.References
- Aline Bonami, Étude des coefficients de Fourier des fonctions de $L^{p}(G)$, Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 2, 335–402 (1971) (French, with English summary). MR 283496, DOI 10.5802/aif.357
- J. Bourgain, On the embedding problem of $L^{1}$ in $L^{1}/H^{1}_{0}$, Bull. Soc. Math. Belg. Sér. B 34 (1982), no. 2, 187–194. MR 682996
- J. Bourgain and H. P. Rosenthal, Applications of the theory of semi-embeddings to Banach space theory, J. Funct. Anal. 52 (1983), no. 2, 149–188. MR 707202, DOI 10.1016/0022-1236(83)90080-0
- J. Bourgain, New Banach space properties of the disc algebra and $H^{\infty }$, Acta Math. 152 (1984), no. 1-2, 1–48. MR 736210, DOI 10.1007/BF02392189
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- John B. Garnett and Peter W. Jones, The distance in BMO to $L^{\infty }$, Ann. of Math. (2) 108 (1978), no. 2, 373–393. MR 506992, DOI 10.2307/1971171
- Adriano M. Garsia, Martingale inequalities: Seminar notes on recent progress, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. MR 0448538
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253 A. Pelczysnki, Banach spaces of analytic functions and absolutely summing operators, CBMS Regional Conf. Ser. in Math., no. 30, Amer. Math. Soc., Providence, R.I., 1976.
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Elias Stein, Classes $H^{p}$, multiplicateurs et fonctions de Littlewood-Paley, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A716–A719 (French). MR 225098
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 278 (1983), 689-702
- MSC: Primary 46E30; Secondary 42A99, 46B25, 60G46
- DOI: https://doi.org/10.1090/S0002-9947-1983-0701518-9
- MathSciNet review: 701518