Characterization of Clifford-valued Hardy spaces and compensated compactness

Peng, Lizhong; Zhao, Jiman

doi:10.1090/S0002-9939-03-07034-5

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