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)

 
 

 

Hardy-Hodge decomposition of vector fields in $ \mathbb{R}^n$


Authors: Laurent Baratchart, Pei Dang and Tao Qian
Journal: Trans. Amer. Math. Soc. 370 (2018), 2005-2022
MSC (2010): Primary 42B30
DOI: https://doi.org/10.1090/tran/7202
Published electronically: September 15, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that an $ \mathbb{R}^{n+1}$-valued vector field on $ \mathbb{R}^n$ is the sum of the traces of two harmonic gradients, one in each component of $ \mathbb{R}^{n+1}\setminus \mathbb{R}^n$, and of an $ \mathbb{R}^n$-valued divergence free vector field. We apply this to the description of vanishing potentials in divergence form. The results are stated in terms of Clifford Hardy spaces, the structure of which is important for our study.


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

  • [1] L. Baratchart, D. P. Hardin, E. A. Lima, E. B. Saff, and B. P. Weiss, Characterizing kernels of operators related to thin-plate magnetizations via generalizations of Hodge decompositions, Inverse Problems 29 (2013), no. 1, 015004, 29. MR 3003011, https://doi.org/10.1088/0266-5611/29/1/015004
  • [2] Steven R. Bell, The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1228442
  • [3] L. Cohen, Time-frequency analysis, Prentice-Hall, Englewood Cliffs, NJ, 1995.
  • [4] P. A. Deift, Orthogonal polynomials and random matrices: A Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, vol. 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1677884
  • [5] A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 453 (1997), no. 1962, 1411-1443. MR 1469927, https://doi.org/10.1098/rspa.1997.0077
  • [6] 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
  • [7] Bruno Harris, Iterated integrals and cycles on algebraic manifolds, Nankai Tracts in Mathematics, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 2004. MR 2063961
  • [8] John E. Gilbert and Margaret A. M. Murray, Clifford algebras and Dirac operators in harmonic analysis, Cambridge Studies in Advanced Mathematics, vol. 26, Cambridge University Press, Cambridge, 1991. MR 1130821
  • [9] Tadeusz Iwaniec and Gaven Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2001. MR 1859913
  • [10] J. D. Jackson, Classical electrodynamics, Third Edition, John Wiley & Sons Inc., Hoboken, New Jersey, 1999.
  • [11] Kit-Ian Kou and Tao Qian, The Paley-Wiener theorem in $ \mathbf{R}^n$ with the Clifford analysis setting, J. Funct. Anal. 189 (2002), no. 1, 227-241. MR 1887633, https://doi.org/10.1006/jfan.2001.3848
  • [12] Chun Li, Alan McIntosh, and Tao Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Rev. Mat. Iberoamericana 10 (1994), no. 3, 665-721. MR 1308706, https://doi.org/10.4171/RMI/164
  • [13] Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Vol. 1: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002; Vol. 2: Model operators and systems, vol. 93, 2002. MR 1864396, MR 1892647
  • [14] O. G. Parfënov, Estimates for singular numbers of the Carleson embedding operator, Mat. Sb. (N.S.) 131(173) (1986), no. 4, 501-518 (Russian); English transl., Math. USSR-Sb. 59 (1988), no. 2, 497-514. MR 881910
  • [15] R. L. Parker, Geophysical inverse theory, Princeton Univ. Press, Princeton, N. J., 1994.
  • [16] Vladimir V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1949210
  • [17] Tao Qian, Fourier analysis on starlike Lipschitz surfaces, J. Funct. Anal. 183 (2001), no. 2, 370-412. MR 1844212, https://doi.org/10.1006/jfan.2001.3750
  • [18] Tao Qian and Yan-Bo Wang, Adaptive Fourier series--a variation of greedy algorithm, Adv. Comput. Math. 34 (2011), no. 3, 279-293. MR 2776445, https://doi.org/10.1007/s10444-010-9153-4
  • [19] N. I. Muskhelishvili, Singular integral equations: Boundary problems of function theory and their application to mathematical physics, Dover Publications, Inc., New York, 1992. Translated from the second (1946) Russian edition and with a preface by J. R. M. Radok. Corrected reprint of the 1953 English translation. MR 1215485
  • [20] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [21] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
  • [22] Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR 869816

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 42B30

Retrieve articles in all journals with MSC (2010): 42B30


Additional Information

Laurent Baratchart
Affiliation: INRIA, 2004 route de Lucioles, 06902 Sophia-Antipolis Cedex, France
Email: Laurent.Baratchart@inria.fr

Pei Dang
Affiliation: Faculty of Information Technology, Macau University of Science and Technology, Macao, China
Email: pdang@must.edu.mo

Tao Qian
Affiliation: Department of Mathematics, University of Macau, Macao, China
Email: fsttq@umac.mo

DOI: https://doi.org/10.1090/tran/7202
Received by editor(s): July 9, 2016
Published electronically: September 15, 2017
Additional Notes: This work was supported by the Macao Science and Technology Development Fund: FDCT 045/2015/A2 and FDCT 098/2012, the Chinese National Natural Science Funds for Young Scholars: 11701597, and the University of Macau Multi-Year Research Grant MYRG116 (Y1-L3)-FST13-QT. The third author is the corresponding author.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society