Skip to main content
×
×
Home

Compactness of Commutators for Singular Integrals on Morrey Spaces

  • Yanping Chen (a1), Yong Ding (a2) and Xinxia Wang (a3)
Abstract

In this paper we characterize the compactness of the commutator [b, T] for the singular integral operator on the Morrey spaces . More precisely, we prove that if , the -closure of , then [b, T] is a compact operator on the Morrey spaces for ∞ < p < ∞ and 0 < ⋋ < n. Conversely, if and [b, T] is a compact operator on the for some p (1 < p < ∞), then . Moreover, the boundedness of a rough singular integral operator T and its commutator [b, T] on are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Compactness of Commutators for Singular Integrals on Morrey Spaces
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Compactness of Commutators for Singular Integrals on Morrey Spaces
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Compactness of Commutators for Singular Integrals on Morrey Spaces
      Available formats
      ×
Copyright
References
Hide All
[1] Adams, D. R., A note on Riesz potentials. Duke Math J. 42(1975), no. 4, 765778. http://dx.doi.org/10.1215/S0012-7094-75-04265-9
[2] Adams, D. R. and Xiao, J. Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53(2004), no. 6, 16291663.
[3] Adams, D. R. and Xiao, J., Morrey spaces in harmonic analysis. Ark. Mat. Published online March 4, 2011. http://dx.doi.org/10.1007/s11512-010-0134-0
[4] Adams, D. R. and Xiao, J., Morrey potentials and harmonic maps. Comm. Math. Phys., to appear.
[5] Adams, D. R. and Xiao, J., Regularity of Morrey commutators. Trans. Amer. Math. Soc., to appear.
[6] Beatrous, F. and Li, S.-Y., Boundedness and compactness of operators of Hankel type. J. Funct. Anal. 111(1993), no. 2, 350379. http://dx.doi.org/10.1006/jfan.1993.1017
[7] Caffarelli, L., Elliptic second order equations. Rend. Sem. Mat. Fis. Milano 58(1988), 253284. http://dx.doi.org/10.1007/BF02925245
[8] Calderón, A.-P., Commutators, singular integrals on Lipschitz curves and applications. In: Proceedings of the International Congress of Mathematicians. Acad, Sci. Fennica, Helsinki, 1980, pp. 8596.
[9] Calderón, A.-P. and Zygmund, A. On singular integrals. Amer. J. Math. 78(1956), 289309. http://dx.doi.org/10.2307/2372517
[10] Chen, Y. and Ding, Y. Compactness of the commutators of parabolic singular integrals. Sci. China Math. 53(2010), no. 10, 26332648. http://dx.doi.org/10.1007/s11425-010-4004-9
[11] Chen, Y., Ding, Y. and Wang, X., Compactness of commutators of Riesz potential on Morrey space. Potential Anal. 30(2009), no. 4, 301313. http://dx.doi.org/10.1007/s11118-008-9114-4
[12] Chiarenza, F., Frasca, M. and Longo, P. Interior W2, p estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche Mat. 40(1991), no. 1, 149168.
[13] Coifman, R., Lions, P. Meyer, Y. and Semmes, S. Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72(1993), no. 3, 247286.
[14] Coifman, R., Rochberg, R. and G.Weiss, Factorization theorems for Hardy spaces in several variables. Ann. of Math. 103(1976), no. 3, 611635. http://dx.doi.org/10.2307/1970954
[15] Deng, D., Duong, X. and Yan, L. A characterization of the Morrey-Campanato spaces. Math. Z. 250(2005), 641655. http://dx.doi.org/10.1007/s00209-005-0769-x
[16] Ding, Y., A characterization of BMO via commutators for some operators. Northeastern Math. J. 13(1997), no. 4, 422432.
[17] Ding, Y. and Lu, S. Homogeneous fractional integrals on Hardy spaces. Tôhoku Math. J. 52(2000), no. 1, 153162. http://dx.doi.org/10.2748/tmj/1178224663
[18] Di Fazio, G., Palagachev, D. and Ragusa, M. Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients. J. Funct. Anal. 166(1999), no. 2, 179196. http://dx.doi.org/10.1006/jfan.1999.3425
[19] Di Fazio, G. and Ragusa, M. Commutators and Morrey spaces. Boll. Un. Mat. Ital. A 5(1991), no. 3, 323332.
[20] Di Fazio, G. and Ragusa, M., Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 112(1993), no. 2, 241256. http://dx.doi.org/10.1006/jfan.1993.1032
[21] Duong, X., Xiao, J. and Yan, L. Old and new Morrey spaces via heat kernel bounds. J. Fourier Anal. Appl. 13(2007), no. 1, 87111. http://dx.doi.org/10.1007/s00041-006-6057-2
[22] Garćıa-Cuerva, J. and Rubio de, J. L. Francia, Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies 116. North-Holland, Amsterdam, 1985.
[23] Hu, G., Lp(Rn) boundedness for the commutator of a homogeneous singular integral operator. Studia Math. 154(2003), no. 1, 1327. http://dx.doi.org/10.4064/sm154-1-2
[24] Huang, Q., Estimates on the generalized Morrey spaces L2, _ ’ and BMO for linear elliptic systems. Indiana Univ. Math. J. 45(1996), no. 2, 397439.
[25] Iwaniec, T., Nonlinear commutators and Jacobians. J. Fourier Anal. Appl. 3(1997), Special Issue, 775796. http://dx.doi.org/10.1007/BF02656485
[26] Iwaniec, T. and Sboedone, C. Riesz treansform and elliptic PDE's with VMO-coefficients. J. Anal. Math. 74(1998), 183212. http://dx.doi.org/10.1007/BF02819450
[27] Janson, S., Mean oscillation and commutators of singular integral operators, Ark. Mat. 16(1978), no. 2, 263270. http://dx.doi.org/10.1007/BF02386000
[28] Kato, T., Strong solutions of the Navier-Stokes equation in Morrey spaces. Bol. Soc. Brasil. Mat. 22(1992), no. 2, 127155. http://dx.doi.org/10.1007/BF01232939
[29] Krantz, S. and Li, S.-Y., Boundedness and compactness of integral operators on spaces of homogeneous type and applications. I. II. J. Math. Anal. Appl. 258(2001), 629641, 642–657. http://dx.doi.org/10.1006/jmaa.2000.7402
[30] Lu, S., Ding, Y. and Yan, D. Singular Integral and Related Topics. World Scientific Publishing, Hackensack, NJ, 2007.
[31] Mazzucato, A., Besov-Morrey spaces: functions space theory and applications to non-linear PDE, Trans. Amer. Math. Soc. 355(2003), no. 4, 12971364. http://dx.doi.org/10.1090/S0002-9947-02-03214-2
[32] Morrey, C., On the solutions of quasi-linear elleptic partial diferential equations. Trans. Amer. Math. Soc. 43(1938), no. 1, 126166. http://dx.doi.org/10.1090/S0002-9947-1938-1501936-8
[33] Mizuhara, T., Boundedness of some classical operators on generalized Morrey spaces. In: Harmonic Analysis. Springer, Tokyo, 1991, pp. 183189.
[34] Nakai, E., Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166(1994), 95103. http://dx.doi.org/10.1002/mana.19941660108
[35] Palagachev, D. and Softova, L. Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s. Potential Anal. 20(2004), no. 3, 237263. http://dx.doi.org/10.1023/B:POTA.0000010664.71807.f6
[36] Pérez, C., Two weighted norm inequalities for Riesz potentials and uniform Lp-weighted Sobolev inequalities. Indiana Univ. Math. J. 39(1990), no. 1, 3144. http://dx.doi.org/10.1512/iumj.1990.39.39004
[37] Ruiz, A. and Vega, L. Unique continuation for Schrödinger operators with potential in Morrey spaces. Publ. Mat. 35(1991), no. 1, 291298.
[38] Sawano, Y., Generalized Morrey spaces for non-doubling measures. No DEA Nonlinear Differential Equations Appl. 15(2008), no. 4-5, 413425. http://dx.doi.org/10.1007/s00030-008-6032-5
[39] Sawano, Y. and Shirai, S. Compact commtators on Morrey spaces with non-doubling measures. Georgian Math. J. 15(2008), no. 2, 353376.
[40] Sawano, Y. and Tanaka, H. Morrey Spaces for non-doubling measures. Acta Math. Sin. (Engl. Ser.) 21(2005), no. 6, 15351544. http://dx.doi.org/10.1007/s10114-005-0660-z
[41] Sawano, Y. and Tanaka, H., Sharp maximal inequalities and commutators on Morrey spaces with non-doublin measueas. Taiwan. J. Math. 11(2007), no. 4, 10911112.
[42] Shen, Z., Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains. Amer. J. Math. 125(2003), no. 5, 10791115. http://dx.doi.org/10.1353/ajm.2003.0035
[43] Shen, Z., The periodic Schrödinger operators with potentials in the Morrey class. J. Funct. Anal. 193(2002), no. 2, 314345. http://dx.doi.org/10.1006/jfan.2001.3933
[44] Softova, L., Singular integrals and commutators in generalized Morrey spaces. Acta Math. Sin. (Engl. Ser.) 22(2006), no. 3, 757766. http://dx.doi.org/10.1007/s10114-005-0628-z
[45] Stein, E. M., Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43. Princeton University Press, Princeton, NJ, 1993.
[46] Stein, E. M. and G.Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series 32. Princeton University Press, Princeton, NJ, 1971.
[47] Taylor, M., Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Comm. Partial Differential Equations 17(1992). 14071456. http://dx.doi.org/10.1080/03605309208820892
[48] Uchiyama, A., On the compactness of operators of Hankel type. Tôhoku Math. J. 30(1978), no. 1, 163171. http://dx.doi.org/10.2748/tmj/1178230105
[49] Yan, L., Classes of Hardy spaces associated with operators, duality theorem and applications. Trans. Amer. Math. Soc. 360(2008), no. 8, 43834408. http://dx.doi.org/10.1090/S0002-9947-08-04476-0
[50] Yosida, K., Functional Analysis. Fifth edition. Grundlehren der Mathematischen Wissenschaften 123. Springer-Verlag, Berlin, 1978.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed