Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces

Blasco, O.; Pełczyński, A.

doi:10.1090/S0002-9947-1991-0979957-5

Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces
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by O. Blasco and A. Pełczyński

Trans. Amer. Math. Soc. 323 (1991), 335-367

DOI: https://doi.org/10.1090/S0002-9947-1991-0979957-5

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Abstract:

We investigate the classes of Banach spaces where analogues of the classical Hardy inequality and the Paley gap theorem hold for vector-valued functions. We show that the vector-valued Paley theorem is valid for a large class of Banach spaces (necessarily of cotype $2$) which includes all Banach lattices of cotype $2$, all Banach spaces whose dual is of type $2$ and also the preduals of ${C^ * }$-algebras. For the trace class ${S_1}$ and the dual of the algebra of all bounded operators on a Hilbert space a stronger result holds; namely, the vector-valued analogue of the Fefferman theorem on multipliers from ${H^1}$ into ${l^1}$; in particular for the latter spaces the vector-valued Hardy inequality holds. This inequality is also true for every Banach space of type $> 1$ (Bourgain).

References

J. Bourgain, Vector valued singular integrals and the $H^1$-$\textrm {BMO}$ duality, Israel seminar on geometrical aspects of functional analysis (1983/84), Tel Aviv Univ., Tel Aviv, 1984, pp. XVI, 23. MR 827294
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
J. Bourgain, A Hausdorff-Young inequality for $B$-convex Banach spaces, Pacific J. Math. 101 (1982), no. 2, 255–262. MR 675400

Vector-valued Hausdorff-Young inequality and applications

J. Bourgain and W. J. Davis, Martingale transforms and complex uniform convexity, Trans. Amer. Math. Soc. 294 (1986), no. 2, 501–515. MR 825718, DOI 10.1090/S0002-9947-1986-0825718-5
Earl Berkson, T. A. Gillespie, and Paul S. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. (3) 53 (1986), no. 3, 489–517. MR 868456, DOI 10.1112/plms/s3-53.3.489
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
William J. Davis, D. J. H. Garling, and Nicole Tomczak-Jaegermann, The complex convexity of quasinormed linear spaces, J. Funct. Anal. 55 (1984), no. 1, 110–150. MR 733036, DOI 10.1016/0022-1236(84)90021-1
I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. Translated from the Russian by A. Feinstein. MR 0264447
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
Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
Uffe Haagerup and Gilles Pisier, Factorization of analytic functions with values in noncommutative $L_1$-spaces and applications, Canad. J. Math. 41 (1989), no. 5, 882–906. MR 1015588, DOI 10.4153/CJM-1989-041-6
Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
J. Lindenstrauss and H. P. Rosenthal, The ${\cal L}_{p}$ spaces, Israel J. Math. 7 (1969), 325–349. MR 270119, DOI 10.1007/BF02788865
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
Bernard Maurey and Gilles Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), no. 1, 45–90 (French). MR 443015, DOI 10.4064/sm-58-1-45-90
Mario Milman, Complex interpolation and geometry of Banach spaces, Ann. Mat. Pura Appl. (4) 136 (1984), 317–328. MR 765927, DOI 10.1007/BF01773388
Paul S. Muhly, Fefferman spaces and $C^*$-algebras, Banach space theory (Iowa City, IA, 1987) Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 371–385. MR 983395, DOI 10.1090/conm/085/983395
A. Pełczyński, Commensurate sequences of characters, Proc. Amer. Math. Soc. 104 (1988), no. 2, 525–531. MR 962823, DOI 10.1090/S0002-9939-1988-0962823-4
R. E. A. C. Paley, On the lacunary coefficients of power series, Ann. of Math. (2) 34 (1933), no. 3, 615–616. MR 1503129, DOI 10.2307/1968183
Jaak Peetre, Sur la transformation de Fourier des fonctions à valeurs vectorielles, Rend. Sem. Mat. Univ. Padova 42 (1969), 15–26 (French). MR 256153
G. Pisier, Les inégalités de Khintchine-Kahane, d’après C. Borell, Séminaire sur la Géométrie des Espaces de Banach (1977–1978), École Polytech., Palaiseau, 1978, pp. Exp. No. 7, 14 (French). MR 520209

Factorization of operator-valued analytic functions

Haskell P. Rosenthal, On subspaces of $L^{p}$, Ann. of Math. (2) 97 (1973), 344–373. MR 312222, DOI 10.2307/1970850
Donald Sarason, Generalized interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 127 (1967), 179–203. MR 208383, DOI 10.1090/S0002-9947-1967-0208383-8
Brent Smith, Two trigonometric designs: one-sided Riesz products and Littlewood products, General inequalities, 3 (Oberwolfach, 1981) Internat. Schriftenreihe Numer. Math., vol. 64, Birkhäuser, Basel, 1983, pp. 141–148. MR 785771

A proof of Fefferman’s theorem on multipliers

Nicole Tomczak-Jaegermann, The moduli of smoothness and convexity and the Rademacher averages of trace classes $S_{p}(1\leq p<\infty )$, Studia Math. 50 (1974), 163–182. MR 355667
Quan Hua Xu, Inégalités pour les martingales de Hardy et renormage des espaces quasi-normés, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 14, 601–604 (French, with English summary). MR 941635

Application du théorème de factorisation pour des fonctions à valeurs opérateurs