Potential Analysis

, Volume 38, Issue 2, pp 653–681 | Cite as

The Local Trace Inequality for Potential Type Integral Operators

  • Hitoshi TanakaEmail author
  • Hendra Gunawan


The local trace inequality for potential type integral operator is shown and the trace inequality in the framework of Morrey spaces is obtained. A sufficient condition for the equivalence between the Kerman–Sawyer condition and the Adams condition is also presented.


Morrey space Potential type integral operator The Adams inequality Trace inequality The Olsen inequality 

Mathematics Subject Classifications (2010)

Primary 42B35 Secondary 42B25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, D.: Traces of potentials arising from translation invariant operators. Ann. Scuola Norm. Sup. Pisa 25(3), 203–217 (1971)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Adams, D.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Adams, D.: Weighted nonlinear potential theory. Trans. Am. Math. Soc. 297, 73–94 (1986)zbMATHCrossRefGoogle Scholar
  4. 4.
    Adams, D., Hedberg, L.: Function spaces and potential theory. In: Grundlehren der Mathematischen Wissenschaften, vol. 314. Springer, Berlin (1996). ISBN: 3-540-57060-8Google Scholar
  5. 5.
    Cascante, C., Ortega, J., Verbitsky, I.: Trace inequalities of Sobolev type in the upper triangle case. Proc. Lond. Math. Soc. 80(3), 391–414 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cascante, C., Ortega, J., Verbitsky, I.: Wolff’s inequality for radially nonincreasing kernels and applications to trace inequalities. Potential Anal. 16, 347–372 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cascante, C., Ortega, J., Verbitsky, I.: Nonlinear potentials and two weight trace inequalities for general dyadic and radial kernels. Indiana Univ. Math. J. 53, 845–882 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cascante, C., Ortega, J., Verbitsky, I.: On L p-L q trace inequalities. J. Lond. Math. Soc. 74(2), 497–511 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chiarenza, F., Frasca, M.: Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. Appl. 7, 273–279 (1987)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Cruz-Uribe, D.: New proofs of two-weight norm inequalities for the maximal operator. Georgian Math. J. 7, 33–42 (2000)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Eridani: On the boundedness of a generalized fractional integral on generalized Morrey spaces. Tamkang J. Math. 33, 335–340 (2002)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Eridani, Gunawan, H., Nakai, E.: On generalized fractional integral operators. Sci. Math. Jpn. 60, 539–550 (2004)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Gunawan, H., Sawano, Y., Sihwaningrum, I.: Fractional integral operators in nonhomogeneous spaces. Bull. Aust. Math. Soc. 80, 324–334 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hansson, K.: Imbedding theorems of Sobolev type in potential theory. Math. Scand. 45, 77–102 (1979)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Kerman, R., Sawyer, E.: The trace inequality and eigenvalue estimates for Schrödinger operators. Ann. Inst. Fourier (Grenoble) 36, 207–228 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Li, W.: Weighted inequalities for commutators of potential type operators. J. Korean Math. Soc. 44, 1233–1244 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Li, W., Yan, X., Yu, X.: Two-weight inequalities for commutators of potential operators on spaces of homogeneous type. Potential Anal. 31, 117–134 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Olsen, P.: Fractional integration, Morrey spaces and Schrödinger equation. Commun. Partial Differ. Equ. 20, 2005–2055 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Pérez, C.: Two weighted inequalities for potential and fractional type maximal operators. Indiana Univ. Math. J. 43, 663–684 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ragusa, A.: A new approach to some trace theorems. Matematiche (Catania) 48, 323–330 (1993)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Sawano, Y., Sugano, S., Tanaka, H.: Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces. Trans. Am. Math. Soc. 363, 6481–6503 (2011)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Sawano, Y., Sugano, S., Tanaka, H.: A note on generalized fractional integral operators on generalized morrey spaces. Bound. Value Probl. 2009, Art. ID 835865, 18 pp. (2009)Google Scholar
  23. 23.
    Sawano, Y., Sugano, S., Tanaka, H.: Orlicz–Morrey spaces and fractional operators. Potential Anal. 36, 517–556 (2012)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Sawyerm, E., Wheeden, T.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114, 813–874 (1992)CrossRefGoogle Scholar
  25. 25.
    Sihwaningrum, I., Suryawan, P., Gunawan, H.: Fractional integral operators and Olsen inequalities on non-homogeneous spaces. Aust. J. Math. Anal. Appl. 7(14), 6 (2010)MathSciNetGoogle Scholar
  26. 26.
    Tanaka, H.: Morrey spaces and fractional operators. J. Aust. Math. Soc. 88, 247–259 (2010)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan
  2. 2.Department of MathematicsBandung Institute of TechnologyBandungIndonesia

Personalised recommendations