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Morrey Potentials and Harmonic Maps

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An Erratum to this article was published on 11 July 2015

Abstract

This paper discusses trace estimates for Morrey potentials (i.e., Riesz potential integrals of Morrey functions), leading to a consideration of the C ∞ smoothness of a class of generalized harmonic maps.

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References

  1. Adams D.R.: Traces of potentials arising from translation invariant operators. Ann. Scuola Norm. Sup. Pisa 25(3), 203–217 (1971)

    MathSciNet  MATH  Google Scholar 

  2. Adams D.R.: Traces of potentials. II. Indiana Univ. Math. J. 22, 907–918 (1973)

    Article  MATH  Google Scholar 

  3. Adams D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Adams, D.R.: Lectures on L p-Potential Theory, Volume 2, Department of Mathematics, University of Umeå, 1981

  5. Adams D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128, 385–398 (1988)

    Article  MATH  Google Scholar 

  6. Adams D.R., Hedberg L.I.: Function Spaces and Potential Theory. Springer-Verlag, Berlin-Heidelberg (1996)

    Google Scholar 

  7. Adams D.R., Xiao J.: Nonlinear analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53, 1629–1663 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Adams, D.R., Xiao, J.: Morrey spaces in harmonic analysis. Ark. Mat. to appear, doi:10.1007/s11512-010-0134-0, 30 pp, March 2011

  9. Angelsberg G., Pumberger D.: A regularity result for polyharmonic maps with higher integrability. Ann. Glob. Anal. Geom. 35, 63–81 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bethuel F.: On the singular set of stationary harmonic maps. Manus. Math. 78, 417–443 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fuglede B.: The logarithmic potential in higher dimensions. Mat. Fys. Medd. Dan. Cid. Selsk. 33, 1–14 (1960)

    MathSciNet  Google Scholar 

  12. Garcia-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, 116. Notas de Matemática [Mathematical Notes], 104. Amsterdam: North-Holland Publishing Co., 1985

  13. Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Math. Studies 105, Princeton, NJ: Princeton University Press, 1983

  14. Hélein F.: Regularity of weakly harmonic maps between a surface and a Riemannian manifold. C. R. Acad. Sci. Paris Sr. I Math. 312, 591–596 (1991)

    MATH  Google Scholar 

  15. Hélein F.: Harmonic maps, conservation laws and moving frames. Cambridge Tracts in Mathematics 150. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  16. John F., Nirenberg L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14, 415–426 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mazýa V.G.: Sobolev Spaces. Springer, Berlin-New York (1985)

    Google Scholar 

  18. Mizuta Y., Shimomura T., Sobukawa T.: Sobolev’s inequality for Riesz potentials of functions in non-doubling Morrey spaces. Osaka J. Math. 46, 255–271 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Rivir̀e T., Struwe M.: Partial regularity for harmonic maps and related problems. Comm. Pure Appl. Math. 61, 451–463 (2008)

    Article  MathSciNet  Google Scholar 

  20. Rupflin, M.: What is a ... Morrey space? What is a ... Campanato space? Preprint, April 22, 2008

  21. Schoen, R., Yau, S.T.: Lectures on Harmonic Maps. Conference Proc. & Lecture Notes in Geometry & Topology, Vol. II, Somerville, MA: International Press, Inc., 1997

  22. Serrin J.: A remark on the Morrey potential. Contemp. Math. 426, 307–315 (2007)

    Article  MathSciNet  Google Scholar 

  23. Stein E.M.: Singular Integrals and Differentiability of Functions. Princeton Univ. Press, Princeton, NJ (1970)

    MATH  Google Scholar 

  24. Stein E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton, NJ (1993)

    MATH  Google Scholar 

  25. Torchinsky A.: Real-variable Methods in Harmonic Analysis. Dover Publications, Inc., New York (2004)

    MATH  Google Scholar 

  26. Xiao J.: Homothetic variant of fractional Sobolev space with application to Navier-Stokes system. Dyn. Part. Diff. Eq. 4, 227–245 (2007)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, Lexington, KY, 40506-0027, USA

    David R. Adams

  2. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada

    Jie Xiao

Authors
  1. David R. Adams
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  2. Jie Xiao
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Corresponding author

Correspondence to Jie Xiao.

Additional information

Communicated by P. Constantin

Jie Xiao was in part supported by NSERC of Canada.

An erratum to this article is available at http://dx.doi.org/10.1007/s00220-015-2409-6.

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Adams, D.R., Xiao, J. Morrey Potentials and Harmonic Maps. Commun. Math. Phys. 308, 439–456 (2011). https://doi.org/10.1007/s00220-011-1319-5

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  • DOI: https://doi.org/10.1007/s00220-011-1319-5

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