Advertisement

Potential Analysis

, Volume 30, Issue 4, pp 301–313 | Cite as

Compactness of Commutators of Riesz Potential on Morrey Spaces

  • Yanping Chen
  • Yong Ding
  • Xinxia Wang
Article

Abstract

In this paper, the authors give a characterization of the (L p, λ , L q, λ )-compactness for the Riesz potential commutator [b,I α ]. More precisely, the authors prove that the commutator [b,I α ] is a compact operator from the Morrey space L p, λ (ℝ n ) to L q, λ (ℝ n ) if and only if b ∈ VMO(ℝ n ), the BMO-closure of \(C_c^\infty({\Bbb R}^n)\).

Keywords

Riesz potential Commutators Compactness VMO Morrey space 

Mathematics Subject Classifications (2000)

42B20 42B25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beatrous, F., Li, S.-Y.: Boundedness and compactness of operators of Hankel type. J. Funct. Anal. 111, 350C379 (1993)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Chanillo, S.: A note on commutators. Indiana Univ. Math. J. 31, 7–16 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Caffarelli, L.: Elliptic second order equations. Rend. Sem. Mat. Fis. Milano 58, 253–284 (1990)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Coifman, R., Lions, P., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247C286 (1993)MathSciNetGoogle Scholar
  5. 5.
    Coifman, R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103, 611–635 (1976)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ding, Y.: A characterization of BMO via commutators for some operators. Northeast. Math. J. 13, 422–432 (1997)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Di Fazio, G., Palagachev, D., Ragusa, M.: Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients. J. Funct. Anal. 166, 179–196 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Huang, Q.: Estimates on the generalized Morrey spaces \(L^{2,\lambda}_{\varphi}\) and BMO for linear elliptic systems. Indiana Univ. Math. J. 45, 397–439 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Iwaniec, T., Sboedone, C.: Riesz treansform and elliptic PDE’s with VMO-coefficients. Jour. D’Analyse Math. 74, 183–212 (1998)CrossRefzbMATHGoogle Scholar
  10. 10.
    Krantz1, S., Li, S.-Y.: Boundedness and compactness of integral operators on spaces of homogeneous type and applications, I. J. Math. Anal. Appl. 258, 629–641 (2001)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Krantz1, S., Li, S.-Y.: Boundedness and compactness of integral operators on spaces of homogeneous type and applications, II. J. Math. Anal. Appl. 258, 642–657 (2001)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Mazzucato, A.: Besov-Morrey spaces: functions space theory and applications to non-linear PDE. Trans. Amer. Math. Soc. 355, 1297–1364 (2002)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Morrey, C.: On the solutions of quasi linear elleptic partial diferential equations. Trans. Amer. Math. Soc. 43, 126–166 (1938)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Palagachev, D., Softova, L.: Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s. Potential Anal. 20, 237–263 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Ruiz, A., Vega, L.: On local regularity of Schrödinger equations. Int. Math. Res. Not. 1993(1), 13–27 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Shen, Z.: Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains. Am. J. Math. 125, 1079–1115 (2003)CrossRefzbMATHGoogle Scholar
  17. 17.
    Stein, E.M.: Singular Integral and Differentiability Properties of Functions. Princeton University Press, Princeton (1971)Google Scholar
  18. 18.
    Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  19. 19.
    Stein, E., Weiss, G.: Introdution to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)Google Scholar
  20. 20.
    Uchiyama, A.: On the compactness of operators of Hankel type. Tôhoku Math. 30, 163–171 (1976)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Wang, S.: The compactness of the commutator of fractional integral operator (in Chinese). Chin. Ann. Math. 8(A), 475–482 (1987)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Mathematics and Mechanics, Applied Science SchoolUniversity of Science and Technology BeijingBeijingChina
  2. 2.Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, School of Mathematical SciencesBeijing Normal UniversityBeijingChina
  3. 3.The College of Mathematics and System ScienceXinjiang UniversityUrumqiChina

Personalised recommendations