## Characterization of Clifford-valued Hardy spaces and compensated compactness

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- by Lizhong Peng and Jiman Zhao PDF
- Proc. Amer. Math. Soc.
**132**(2004), 47-58 Request permission

## 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

- S. Bernstein,
*Operator calculus for elliptic boundary value problems in unbounded domains*, Z. Anal. Anwendungen**10**(1991), no. 4, 447–460. MR**1155823**, DOI 10.4171/ZAA/467 - F. Brackx, Richard Delanghe, and F. Sommen,
*Clifford analysis*, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR**697564** - R. Delanghe and F. Brackx,
*Hypercomplex function theory and Hilbert modules with reproducing kernel*, Proc. London Math. Soc. (3)**37**(1978), no. 3, 545–576. MR**512025**, DOI 10.1112/plms/s3-37.3.545 - Klaus Gürlebeck and Wolfgang Sprössig,
*Quaternionic analysis and elliptic boundary value problems*, Mathematical Research, vol. 56, Akademie-Verlag, Berlin, 1989. MR**1056478** - Marius Mitrea,
*Clifford wavelets, singular integrals, and Hardy spaces*, Lecture Notes in Mathematics, vol. 1575, Springer-Verlag, Berlin, 1994. MR**1295843**, DOI 10.1007/BFb0073556 - Svante Janson and Jaak Peetre,
*Paracommutators—boundedness and Schatten-von Neumann properties*, Trans. Amer. Math. Soc.**305**(1988), no. 2, 467–504. MR**924766**, DOI 10.1090/S0002-9947-1988-0924766-6 - Li Zhong Peng,
*On compactness of paracommutators*, Ark. Mat.**26**(1988), no. 2, 315–325. MR**1050111**, DOI 10.1007/BF02386126 - Lizhong Peng and M. W. Wong,
*Compensated compactness and paracommutators*, J. London Math. Soc. (2)**62**(2000), no. 2, 505–520. MR**1783641**, DOI 10.1112/S0024610700001290 - Tao Qian and John Ryan,
*Conformal transformations and Hardy spaces arising in Clifford analysis*, J. Operator Theory**35**(1996), no. 2, 349–372. MR**1401694** - F. Sommen,
*A product and an exponential function in hypercomplex function theory*, Applicable Anal.**12**(1981), no. 1, 13–26. MR**618518**, DOI 10.1080/00036818108839345 - Franciscus Sommen,
*Microfunctions with values in a Clifford algebra. II*, Sci. Papers College Arts Sci. Univ. Tokyo**36**(1986), no. 1, 15–37. MR**868638** - F. Sommen,
*Hypercomplex Fourier and Laplace transforms. II*, Complex Variables Theory Appl.**1**(1982/83), no. 2-3, 209–238. MR**690496**, DOI 10.1080/17476938308814016 - Zhijian Wu,
*Clifford algebras, Hardy spaces, and compensated compactness*, Clifford algebras in analysis and related topics (Fayetteville, AR, 1993) Stud. Adv. Math., CRC, Boca Raton, FL, 1996, pp. 217–238. MR**1383107** - Zhijian Wu,
*Commutators and related operators on harmonic Bergman space of $\textbf {R}^{n+1}_+$*, J. Funct. Anal.**144**(1997), no. 2, 448–474. MR**1432593**, DOI 10.1006/jfan.1996.3000

## Additional Information

**Lizhong Peng**- Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China
- Email: lzpeng@pku.edu.cn
**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
- Email: jmz70@sina.com
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**132**(2004), 47-58 - MSC (2000): Primary 15A66, 42B30, 46J15, 47B35
- DOI: https://doi.org/10.1090/S0002-9939-03-07034-5
- MathSciNet review: 2021247