DOI QR코드

DOI QR Code

CHARACTERIZATION OF FUNCTIONS VIA COMMUTATORS OF BILINEAR FRACTIONAL INTEGRALS ON MORREY SPACES

  • Mao, Suzhen (School of Mathematical Sciences Xiamen University) ;
  • Wu, Huoxiong (School of Mathematical Sciences Xiamen University)
  • Received : 2015.07.01
  • Published : 2016.07.31

Abstract

For $b{\in}L^1_{loc}({\mathbb{R}}^n)$, let ${\mathcal{I}}_{\alpha}$ be the bilinear fractional integral operator, and $[b,{\mathcal{I}}_{\alpha}]_i$ be the commutator of ${\mathcal{I}}_{\alpha}$ with pointwise multiplication b (i = 1, 2). This paper shows that if the commutator $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 is bounded from the product Morrey spaces $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to the Morrey space $L^{q,{\lambda}}({\mathbb{R}}^n)$ for some suitable indexes ${\lambda}$, ${\lambda}_1$, ${\lambda}_2$ and $p_1$, $p_2$, q, then $b{\in}BMO({\mathbb{R}}^n)$, as well as that the compactness of $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 from $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to $L^{q,{\lambda}}({\mathbb{R}}^n)$ implies that $b{\in}CMO({\mathbb{R}}^n)$ (the closure in $BMO({\mathbb{R}}^n)$of the space of $C^{\infty}({\mathbb{R}}^n)$ functions with compact support). These results together with some previous ones give a new characterization of $BMO({\mathbb{R}}^n)$ functions or $CMO({\mathbb{R}}^n)$ functions in essential ways.

Keywords

bilinear fractional integrals;commutators;Morrey spaces;$BMO({\mathbb{R}}^n)$;$CMO({\mathbb{R}}^n)$;boundeness;compactness

Acknowledgement

Supported by : NNSF of China, NSF of Fujian Province of China

References

  1. A. Benyi, W. Damian, K. Moen, and R. Torres, Compactness properties of commutators of bilinear fractional integrals, Math. Z. 280 (2015), no. 1-2, 569-582. https://doi.org/10.1007/s00209-015-1437-4
  2. A. Benyi, W. Damian, K. Moen, and R. Torres, Compact bilinear commutators: the weighted case, Michigan Math. J. 64 (2015), no. 1, 39-51. https://doi.org/10.1307/mmj/1427203284
  3. A. Benyi and R. Torres, Compact bilinear operators and commutators, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3609-3621. https://doi.org/10.1090/S0002-9939-2013-11689-8
  4. L. Caffarelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano 58 (1988), 253-284. https://doi.org/10.1007/BF02925245
  5. L. Chaffee, Characterizations of BMO through commutators of bilinear singular integral operator, arXiv:1410.4587v3.
  6. L. Chaffee and R. Torres, Characterizations of compactness of the commutators of bilinear fractional integral operators, Potential Anal. 43 (2015), no. 3, 481-494. https://doi.org/10.1007/s11118-015-9481-6
  7. S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), no. 1, 7-16. https://doi.org/10.1512/iumj.1982.31.31002
  8. Y. Chen and Y. Ding, Compactness of the commutators of parabolic singular integrals, Sci. China Math. 53 (2010), no. 10, 2633-2648. https://doi.org/10.1007/s11425-010-4004-9
  9. Y. Chen, Y. Ding, and X. Wang, Compactness of commutators of Riesz potential on Morrey spaces, Potential Anal. 30 (2009), no. 4, 301-313. https://doi.org/10.1007/s11118-008-9114-4
  10. Y. Chen, Y. Ding, and X. Wang, Compactness for commutators of Marcinkiewicz integral in Morrey spaces, Taiwanese J. Math. 15 (2011), no. 2, 633-658. https://doi.org/10.11650/twjm/1500406226
  11. Y. Chen, Y. Ding, and X. Wang, Compactness of commutators for singular integrals on Morrey spaces, Canad. J. Math. 64 (2012), no. 2, 257-281. https://doi.org/10.4153/CJM-2011-043-1
  12. S. Chen and H. Wu, Multiple weighted estimates for commutators of multilinear fractional integral operators, Sci. China Math. 56 (2013), no. 9, 1879-1894. https://doi.org/10.1007/s11425-013-4607-z
  13. X. Chen and Q. Xue, Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl. 362 (2010), no. 2, 355-373. https://doi.org/10.1016/j.jmaa.2009.08.022
  14. F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Math. Appl. 7 (1987), no. 3-4, 273-279.
  15. F. Chiarenza, M. Frasca, and P. Longo, Interior $W^{2,p}$-estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche. Mat. 40 (1991), no. 1, 149-168.
  16. Y. Ding, A characterization of BMO via commutators for some operators, Northeast. Math. J. 13 (1997), no. 4, 422-432.
  17. Y. Ding and T. Mei, Boundedness and compactness for the commutators of bilinear operators on Morrey spaces, Potential Anal. 42 (2015), no. 3, 717-748. https://doi.org/10.1007/s11118-014-9455-0
  18. J. Lian, B. Ma, and H. Wu, Commutators of multilinear fractional integrals with weighted Lipschitz functions, Acta Math. Sci. 33A (2013), no. 3, 494-509.
  19. J. Lian and H. Wu, A class of commutators for multilinear fractional integrals in nonhomogeneous spaces, J. Inequal. Appl. 2008 (2008), Article ID 373050, 17 pages.
  20. C. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126-166. https://doi.org/10.1090/S0002-9947-1938-1501936-8
  21. D. Palagachev and L. Softova, Singular integral operators, Morrey spaces and fine regularity of solutions to PDE's, Potential Anal. 20 (2004), no. 3, 237-263. https://doi.org/10.1023/B:POTA.0000010664.71807.f6
  22. A. Ruiz and L. Vega, Unique continuation for Schrodinger operators with potential in Morrey spaces, Publ. Mat. 35 (1991), no. 1, 291-298. https://doi.org/10.5565/PUBLMAT_35191_15
  23. Z. Shen, The periodic Schrodinger operators with potentials in the Morrey class, J. Funct. Anal. 193 (2002), no. 2, 314-345. https://doi.org/10.1006/jfan.2001.3933
  24. M. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), no. 9-10, 1407-1456. https://doi.org/10.1080/03605309208820892
  25. A. Uchiyama, On the compactness of operators of Hankel type, Tohoku Math. J. 30 (1978), no. 1, 163-171. https://doi.org/10.2748/tmj/1178230105
  26. H. Wang and W. Yi, Multilinear singular and fractional integral operators on weighted Morrey spaces, J. Funct. Spaces Appl. 2013 (2013), Art. ID 735795, 11 pages.
  27. S. Wang, The compactness of the commutator of fractional integral operator, China Ann. Math. Ser. A 4 (1987), no. 3, 475-482.
  28. J. Zhang and H. Wu, Oscillation and variation inequalities for singular integrals and commmutators on weighted Morrey spaces, Front. Math. China, DOI 10.1007/s11464-014-0186-5 (in press).