Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Interpolation of weighted and vector-valued Hardy spaces


Authors: Serguei V. Kisliakov and Quan Hua Xu
Journal: Trans. Amer. Math. Soc. 343 (1994), 1-34
MSC: Primary 46E15; Secondary 42B30, 46E40, 46M35
DOI: https://doi.org/10.1090/S0002-9947-1994-1236225-7
MathSciNet review: 1236225
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Real and complex interpolation methods, when applied to the couple $ ({H^{{p_0}}}({E_0};{w_0}),{H^{{p_1}}}({E_1};{w_1}))$, give what is expected if $ {E_0}$ and $ {E_1}$ are quasi-Banach lattices of measurable functions satisfying certain mild conditions and if $ \log (w_0^{1/{p_0}}w_1^{ - 1/{p_1}}) \in {\text{BMO}}\,({w_0},{w_1}$ being weights on the unit circle). The last condition is in fact necessary. (It is expected, of course, that the resulting spaces coincide with the subspaces of analytic functions in the corresponding interpolation spaces for the couple $ ({L^{{p_0}}}({E_0};{w_0}),{L^{{p_1}}}({E_1};{w_1})).)$


References [Enhancements On Off] (What's this?)

  • [AJ] K. F. Andersen and R. T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math. 69 (1980), 19-31. MR 604351 (82b:42015)
  • [BL] J. Bergh and J. Löfström, Interpolation spaces. An introduction, Springer-Verlag, 1976. MR 0482275 (58:2349)
  • [BX] O. Blasco and Q. Xu, Interpolation between vector-valued Hardy spaces, J. Funct. Anal. 102 (1991), 331-359. MR 1140630 (93e:46042)
  • [B1] J. Bourgain, New Banach space properties of the disk algebra and $ {H^\infty }$, Acta Math. 152 (1984), 1-46. MR 736210 (85j:46091)
  • [B2] -, Bilinear forms on $ {H^\infty }$ and bounded bianalytic functions, Trans. Amer. Math. Soc. 286 (1984), 313-338. MR 756042 (86c:46060)
  • [B3] -, Some remarks on Banach spaces in which martingale differences are unconditional, Ark. Mat. 21 (1983), 163-168. MR 727340 (85a:46011)
  • [Bu] D. L. Burkholder, Martingales and Fourier analysis in Banach spaces, Lecture Notes in Math., vol. 1206, Springer, 1983, pp. 61-108. MR 864712 (88c:42017)
  • [C] A. P. Calderón, Intermediate spaces and interpolation: The complex method, Studia Math. 24 (1964), 113-190. MR 0167830 (29:5097)
  • [CF] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. MR 0358205 (50:10670)
  • [CMW] M. Cwikel, J. E. McCarthy, and T. H. Wolff, Interpolation between weighted Hardy spaces, Preprint 1990. MR 1093595 (92m:46033)
  • [CMS] M. Cwikel, M. Milman, and Y. Sagher, Complex interpolation of some quasi-Banach spaces, J. Funct. Anal. 65 (1986), 339-347. MR 826431 (87h:46152)
  • [FRS] C. Fefferman, N. M. Riviere, and Y. Sagher, Interpolation between $ {H^p}$ spaces: The real method, Trans. Amer. Math. Soc. 191 (1974), 71-81. MR 0388072 (52:8909)
  • [GR] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland, Amsterdam, 1985. MR 807149 (87d:42023)
  • [G] J. Garnett, Bounded analytic functions, Pure and Appl. Math., vol. 96, Academic Press, 1981. MR 628971 (83g:30037)
  • [HP] U. Haagerup and G. Pisier, Factorization of analytic functions with values in non-commutative $ {L_1}$-spaces and applications, Canad. J. Math. 41 (1989), 882-906. MR 1015588 (91g:46068)
  • [JJ] S. Janson and P. W. Jones, Interpolation between $ {H^p}$-spaces: The complex method, J. Funct. Anal. 48 (1982), 58-80. MR 671315 (84c:46021)
  • [J] P. W. Jones, Interpolation between Hardy spaces, Proc. Conf. on Harmonic Analysis in Honor of Antoni Zygmund, (Chicago, 1981), Belmont Wadsworth International, 1983. MR 730083 (85a:42029)
  • [KA] L. V. Kantorovitsh and G. P. Akilov, Functional analysis, "Nauka", Moscow, 1977. (Russian) MR 0511615 (58:23465)
  • [K1] S. V. Kisliakov, Truncating fucntions in weighted $ {H^p}$ and two theorems of J. Bourgain, Preprint, Uppsala University, 1989.
  • [K2] -, Extension of (q, p)-summing operators defined on the disc-algebra with an appendix on Bourgain's analytic projection, Preprint 1990.
  • [K3] -, (q, p)-summing operators on the disc algebra and a weighted estimate for certain outer functions, LOMI Leningrad, 1989, preprint E-11-89.
  • [K4] -, Absolutely summing operators on the disc algebra, Algebra and Analysis 3 (1991), no. 4; English transl. in St. Petersburg Math. J. 3 (1992). MR 1152601 (93b:47062)
  • [KPS] S. G. Krein, Ju. I. Petunin, and E. M. Semenov, Interpolation of linear operators, Trans. Math. Monos, vol. 54, Amer. Math., Soc., Providence, R.I., 1980.
  • [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II, Springer-Verlag, Berlin, 1979. MR 540367 (81c:46001)
  • [P1] G. Pisier, A simple proof of a theorem of Jean Bourgain, Michigan J. Math. 39 (1992), 475-484. MR 1182502 (94g:46056)
  • [P2] -, Interpolation between $ {H^p}$ spaces and non-commutative generalizations. I, Pacific J. Math. 155 (1992), 341-368. MR 1178030 (93f:46111)
  • [RF] J. L. Rubio de Francia, Martingales and integral transforms of Banach space valued functions, Lecture Notes in Math., Springer, vol. 1221, 1985, pp. 195-222. MR 875011 (88g:42020)
  • [S] J. D. Stafney, The spectrum of an operator on an interpolation space, Trans. Amer. Math. Soc. 144 (1969), 333-349. MR 0253054 (40:6269)
  • [T] A. Torchinsky, Real-variable methods in harmonic anaysis, Pure and Appl. Math., vol. 123, Academic Press, 1986. MR 869816 (88e:42001)
  • [X1] Q. Xu, Real interpolation of some Banach lattices valued Hardy spaces, Bull. Sci. Math. 116 (1992), 227-246. MR 1168312 (93i:46057)
  • [X2] -, Notes on interpolation of Hardy spaces, Ann. Inst. Fourier (Grenoble) 42 (1992), 875-889; Erratum 43 (1993), 469. MR 1196097 (94e:46135a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46E15, 42B30, 46E40, 46M35

Retrieve articles in all journals with MSC: 46E15, 42B30, 46E40, 46M35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1236225-7
Keywords: Interpolation, Hardy space, Banach lattice, $ {A_p}$-weight, Hilbert transform, BMO
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society