Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Characterization of Clifford-valued Hardy spaces and compensated compactness

Authors: Lizhong Peng and Jiman Zhao
Journal: Proc. Amer. Math. Soc. 132 (2004), 47-58
MSC (2000): Primary 15A66, 42B30, 46J15, 47B35
Published electronically: May 22, 2003
MathSciNet review: 2021247
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, the general Clifford $R_{n,s}$-valued Hardy spaces and conjugate Hardy spaces are characterized. In particular, each function in $R_n$-valued Hardy space can be determined by half of its function components through Riesz transform, and the explicit determining formulas are given. The products of two functions in the Hardy space give six kinds of compensated quantities, which correspond to six paracommutators, and their boundedness, compactness and Schatten-von Neumann properties are given.

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

  • 1. S. Bernstein, Operator Calculus for Elliptic Boundary Value Problems in Unbounded Domains, Zeitschrift für analysis und ihre Anwendungen, 10, (1991), 447-460. MR 93e:35015
  • 2. F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis, Pitman Advanced Publishing Program, Boston, MA, 1982. MR 85j:30103
  • 3. R. Delanghe and F. Brackx, Hypercomplex function theory and Hilbert modules with reproducing kernel, Proc. London Math. Soc.(3) 37, (1978), 545-576. MR 81j:46032
  • 4. K. Gürlebeck and W. Sprößig, Quaternionic Analysis and Elliptic Boundary Value Problems, Birkhäuser-Verlag, Basel, Berlin, 1990. MR 91k:35002b
  • 5. M. Mitrea, Clifford wavelets, singular integrals and Hardy space, Springer-Verlag, Berlin, Heidelberg, 1994. MR 96e:31005
  • 6. S. Janson and J. Peetre, Paracommutators--boundedness and Schatten-von Neumann properties, Trans. Amer. Math. Soc. 305, (1988), 467-504. MR 89g:47034
  • 7. L. Z. Peng, On the compactness of paracommutators, Ark. Mat. 26, (1988), 315-325. MR 91g:47020
  • 8. L. Z. Peng and M. W. Wong, Compensated compactness and paracommutators, J. London Math. Soc., 62:2, (2000), 505-520. MR 2001g:42035
  • 9. T. Qian and J. Ryan, Conformal transformations and Hardy spaces arising in Clifford analysis, J. Operator Theory, 35, (1996), 349-372. MR 97d:30064
  • 10. F. Sommen, A product and an exponential function in hypercomplex function theory, Appl. Anal. 12, (1981), 13-26. MR 84a:30087
  • 11. F. Sommen, Microfunctions with values in a Clifford algebra, 2, Scientific Papers in the College of Arts and Sciences, University of Tokyo, 36, (1986), 15-37. MR 88f:30083
  • 12. F. Sommen, Hypercomplex Fourier and Laplace Transforms 2, Complex Variables Theory Appl., 1, (1983), 209-238. MR 84g:46061
  • 13. Z. J. Wu, Clifford algebras, Hardy spaces and compensated compactness, Clifford algebras in analysis and related topics (John Ryan, ed.), Studies in Adv. Math., CRC Press, Boca Raton, FL, 1996, 217-238. MR 96m:30067
  • 14. Z. J. Wu, Commutators and related operators on Harmonic Bergman Space of $R^{n+1}_+$, J. Funct. Anal. 144, (1997), 448-474. MR 98a:47029

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 15A66, 42B30, 46J15, 47B35

Retrieve articles in all journals with MSC (2000): 15A66, 42B30, 46J15, 47B35

Additional Information

Lizhong Peng
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

Jiman Zhao
Affiliation: Department of Mathematics, Beijing Normal University, Beijing, 100875, People’s Republic of China – and – Academy of Mathematics and System Sciences, Chinese Academy of Sciences, 100080, People’s Republic of China

Keywords: Clifford algebra, Cauchy-Riemann operator, Hardy space, compensated compactness, paracommutator
Received by editor(s): November 15, 2001
Received by editor(s) in revised form: September 4, 2002
Published electronically: May 22, 2003
Additional Notes: Research supported by NNSF of China Nos. 90104004 and 69735020 and 973 project of China G1999075105
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society