Weak-star convergence in multiparameter Hardy spaces

Pipher, Jill; Treil, Sergei

doi:10.1090/S0002-9939-2010-10567-1

Weak-star convergence in multiparameter Hardy spaces
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by Jill Pipher and Sergei Treil PDF

Proc. Amer. Math. Soc. 139 (2011), 1445-1454 Request permission

Abstract:

We prove a multiparameter version of a classical theorem of Jones and Journé on weak-star convergence in the Hardy space $H^1$.

References

Alain Bernard, Espaces $H^{1}$ de martingales à deux indices. Dualité avec les martingales de type “BMO”, Bull. Sci. Math. (2) 103 (1979), no. 3, 297–303 (French, with English summary). MR 539351
Óscar Blasco and Sandra Pott, Dyadic BMO on the bidisk, Rev. Mat. Iberoamericana 21 (2005), no. 2, 483–510. MR 2174914, DOI 10.4171/RMI/427
Sun-Yung A. Chang, Carleson measure on the bi-disc, Ann. of Math. (2) 109 (1979), no. 3, 613–620. MR 534766, DOI 10.2307/1971229
Sun-Yung A. Chang and Robert Fefferman, A continuous version of duality of $H^{1}$ with BMO on the bidisc, Ann. of Math. (2) 112 (1980), no. 1, 179–201. MR 584078, DOI 10.2307/1971324
R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), no. 3, 247–286 (English, with English and French summaries). MR 1225511
R. R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), no. 2, 249–254. MR 565349, DOI 10.1090/S0002-9939-1980-0565349-8
R. R. Coifman, R. Rochberg, and Guido Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611–635. MR 412721, DOI 10.2307/1970954
Burgess Davis, Hardy spaces and rearrangements, Trans. Amer. Math. Soc. 261 (1980), no. 1, 211–233. MR 576872, DOI 10.1090/S0002-9947-1980-0576872-9
R. Fefferman and J. Pipher, Multiparameter operators and sharp weighted inequalities, Amer. J. Math. 119 (1997), no. 2, 337–369. MR 1439553
Sarah H. Ferguson and Michael T. Lacey, A characterization of product BMO by commutators, Acta Math. 189 (2002), no. 2, 143–160. MR 1961195, DOI 10.1007/BF02392840
Sarah H. Ferguson and Cora Sadosky, Characterizations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures, J. Anal. Math. 81 (2000), 239–267. MR 1785283, DOI 10.1007/BF02788991
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, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2, 351–371. MR 658065
Jean-Lin Journé, Calderón-Zygmund operators on product spaces, Rev. Mat. Iberoamericana 1 (1985), no. 3, 55–91. MR 836284, DOI 10.4171/RMI/15
Svante Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), no. 2, 263–270. MR 524754, DOI 10.1007/BF02386000
Peter W. Jones and Jean-Lin Journé, On weak convergence in $H^1(\textbf {R}^d)$, Proc. Amer. Math. Soc. 120 (1994), no. 1, 137–138. MR 1159172, DOI 10.1090/S0002-9939-1994-1159172-3
Michael Lacey and Erin Terwilleger, Hankel operators in several complex variables and product BMO, Houston J. Math. 35 (2009), no. 1, 159–183. MR 2491875
Michael T. Lacey, Erin Terwilleger, and Brett D. Wick, Remarks on product VMO, Proc. Amer. Math. Soc. 134 (2006), no. 2, 465–474. MR 2176015, DOI 10.1090/S0002-9939-05-07974-8
Michael T. Lacey, Stefanie Petermichl, Jill C. Pipher, and Brett D. Wick, Multiparameter Riesz commutators, Amer. J. Math. 131 (2009), no. 3, 731–769. MR 2530853, DOI 10.1353/ajm.0.0059
Jill Pipher and Lesley A. Ward, BMO from dyadic BMO on the bidisc, J. Lond. Math. Soc. (2) 77 (2008), no. 2, 524–544. MR 2400405, DOI 10.1112/jlms/jdm114
Donald Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. MR 377518, DOI 10.1090/S0002-9947-1975-0377518-3
Alexei Alexandrov, Anton Baranov, and Sergey Kislyakov (eds.), Linear and complex analysis, American Mathematical Society Translations, Series 2, vol. 226, American Mathematical Society, Providence, RI, 2009. Dedicated to V. P. Havin on the occasion of his 75th birthday; Advances in the Mathematical Sciences, 63. MR 2517045, DOI 10.1090/trans2/226