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)



Bilinear operators on Herz-type Hardy spaces

Authors: Loukas Grafakos, Xinwei Li and Dachun Yang
Journal: Trans. Amer. Math. Soc. 350 (1998), 1249-1275
MSC (1991): Primary 47H19, 42B20, 42B30
MathSciNet review: 1407489
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The authors prove that bilinear operators given by finite sums of products of Calderón-Zygmund operators on $\mathbb{R}^{n}$ are bounded from $ H\dot K_{q_{1}}^{\alpha _{1},p_{1}}\times H\dot K_{q_{2}}^{\alpha _{2},p_{2}}$ into $ H\dot K_{q}^{\alpha ,p}$ if and only if they have vanishing moments up to a certain order dictated by the target space. Here $H\dot K_{q}^{\alpha ,p}$ are homogeneous Herz-type Hardy spaces with $1/p=1/p_{1}+1/p_{2},$ $0<p_{i}\le \infty ,$ $1/q=1/q_{1}+1/q_{2},$ $1<q_{1},q_{2}<\infty ,$ $1\le q<\infty ,$ $\alpha =\alpha _{1}+\alpha _{2}$ and $-n/q_{i}<\alpha _{i}<\infty $. As an application they obtain that the commutator of a Calderón-Zygmund operator with a BMO function maps a Herz space into itself.

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

  • [1] A. Baernstein II and E. T. Sawyer, Embedding and Multiplier Theorems for $H^{p}(\mathbb R^n)$, Memoirs of the Amer. Math. Soc., vol. 59, no. 318, Amer. Math. Soc., Providence R. I., 1985. MR 86g:42036
  • [2] A. Beurling, Construction and analysis of some convolution algebras, Ann. Inst. Fourier Grenoble 14 (1964), 1-32. MR 31:321
  • [3] Y. Z. Chen and K. S. Lau, Some new classes of Hardy spaces, J. Funct. Anal. 84 (1989), 255-278. MR 90f:46059
  • [4] R. Coifman and L. Grafakos, Hardy space estimates for multilinear operators I, Rev. Mat. Iberoamericana 8 (1992), 45-62. MR 93j:42011
  • [5] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Appl. 72 (1993), 247-286. MR 95d:46033
  • [6] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. MR 54:843
  • [7] C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), 137-193. MR 56:6263
  • [8] H. G. Feichtinger, An elementary approach to Wiener's third Tauberian theorem for the Euclidean $n$-space, Proceedings of Conference at Cortona 1984, Symposia Mathematica, vol. 29, Academic Press, New York, 1987, pp. 267-301. MR 89i:42023
  • [9] T. M. Flett, Some elementary inequalities for integrals with applications to Fourier transforms, Proc. London Math. Soc. (3) 29 (1974), 538-556. MR 50:13406
  • [10] J. García-Cuerva, Hardy spaces and Beurling algebras, J. London Math. Soc. (2) 39 (1989), 499-513. MR 90i:42032
  • [11] J. García-Cuerva and M.-J. L. Herrero, A theory of Hardy spaces associated to the Herz spaces, Proc. London Math. Soc. (3) 69 (1994), 605-628. MR 96e:46037
  • [12] L. Grafakos, Hardy space estimates for multilinear operators II, Rev. Mat. Iberoamericana 8 (1992), 45-62. MR 93j:42012
  • [13] L. Grafakos and X. Li, Bilinear operators on homogeneous groups, submitted.
  • [14] E. Hernández and D. Yang, Interpolation of Herz-type spaces and applications, submitted.
  • [15] C. Herz, Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms, J. Math. Mech. 18 (1968), 283-324. MR 55:11128
  • [16] X. Li and D. Yang, Boundedness of some sublinear operators on Herz spaces, To appear in Illinois Journal of Math.
  • [17] S. Lu and F. Soria, On the Herz spaces with power weights, Fourier Analysis and Partial Differential Equations (J. Garc[??]i a-Cuerva, E. Hernández, F. Soria and J.-L. Torrea, eds.), Studies in advanced mathematics, CRC Press, Boca Raton, 1995, pp. 227-236. MR 96d:42030
  • [18] S. Lu and D. Yang, The Littlewood-Paley function and $\varphi -$transform characterizations of a new Hardy space $HK_{2}$ associated with the Herz space, Studia Math. 101 (1992), 285-298. MR 93b:42031
  • [19] S. Lu and D. Yang, Some new Hardy spaces associated with the Herz spaces and their applications (in Chinese), J. of Beijing Normal Univ. (Natur. Sci.) 29 (1993), 10-19. MR 94m:42078
  • [20] S. Lu and D. Yang, The local versions of $H^{p}(\mathbb R^n)$ spaces at the origin, Studia Math. 116 (1995), 103-131. MR 96h:42015
  • [21] S. Lu and D. Yang, The decomposition of weighted Herz space on $\mathbb{R}^n$ and its application, Science in China (Ser. A) 38 (1995), 147-158. MR 96c:46026
  • [22] S. Lu and D. Yang, The weighted Herz-type Hardy spaces and its applications, Science in China (Ser. A) 38 (1995), 662-673. MR 96i:42018
  • [23] S. Lu and D. Yang, Some characterizations of weighted Herz-type Hardy spaces and its applications, To appear in Acta Math. Sinica.
  • [24] S. Lu and D. Yang, Oscillatory singular integrals on Hardy spaces associated with Herz spaces, Proc. Amer. Math. Soc. 123 (1995), 1695-1709. MR 95g:42026
  • [25] S. Lu and D. Yang, Regularity of non-linear quantities in compensated compactness theory on Herz-type spaces, submitted.
  • [26] S. Lu and D. Yang, The molecular characterization of new Hardy $HA^{p}(\mathbb{R}^{n})$ spaces and some application, J. of Beijing Normal Univ. (Natur. Sci.) 27 (1991), 135-145. MR 93a:42008
  • [27] D. Yang, The real-variable characterizations of Hardy spaces $HK_{p}(\mathbb R^n)$ (in Chinese), Adv. in Math. (China) 24 (1995), 63-73. MR 96c:42043
  • [28] M. H. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Astérisque 77 (1980), 67-149. MR 81i:42013

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 47H19, 42B20, 42B30

Retrieve articles in all journals with MSC (1991): 47H19, 42B20, 42B30

Additional Information

Loukas Grafakos
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211-0001

Xinwei Li
Affiliation: Department of Mathematics, Washington University, Campus Box 1146, St. Louis, Missouri 63130-4899

Dachun Yang
Affiliation: Department of Mathematics, Beijing Normal University, 100875 Beijing, The People’s Republic of China

Keywords: Herz spaces, Beurling algebras, Hardy spaces, atoms, bilinear operators, Calder\'{o}n-Zygmund operators
Received by editor(s): January 15, 1996
Received by editor(s) in revised form: July 15, 1996
Additional Notes: The first author’s research was supported by the University of Missouri Research Board
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society