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Regularity of Morrey commutators


Authors: David R. Adams and Jie Xiao
Journal: Trans. Amer. Math. Soc. 364 (2012), 4801-4818
MSC (2010): Primary 42B35, 46E35, 47G10
DOI: https://doi.org/10.1090/S0002-9947-2012-05595-4
Published electronically: April 6, 2012
MathSciNet review: 2922610
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Abstract: This paper is devoted to presenting a new proof of boundedness of the commutator $ bI_\alpha -I_\alpha b$ (in which $ I_\alpha $ and $ b$ are regarded as the Riesz and multiplication operators) acting on the Morrey space $ L^{p,\lambda }$ under $ b\in \operatorname {BMO}$, and naturally, developing a regularity theory of commutators for Morrey-Sobolev spaces $ I_\alpha (L^{p,\lambda })$ via a completely original iteration of $ I_\alpha $. Even in the special case of $ I_\alpha (L^p)$, this is a new theory.


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  • 1. D. R. Adams, Traces of potentials arising from translation invariant operators, Ann. Scuola Norm. Sup. Pisa (3) 25 (1971), 203-217. MR 0287301 (44:4508)
  • 2. D. R. Adams, A trace inequality for generalized potentials, Studia Math. 48 (1973), 99-105. MR 0336316 (49:1091)
  • 3. D. R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), 765-778. MR 0458158 (56:16361)
  • 4. D. R. Adams, On the existence of capacitary strong type estimates, Ark. Mat. 14 (1976), 125-140. MR 0417774 (54:5822)
  • 5. D. R. Adams, Lectures on $ L^p$-Potential Theory, Volume 2, Department of Mathematics, University of Umeå, 1981.
  • 6. D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. (2) 128 (1988), 385-398. MR 960950 (89i:46034)
  • 7. D. R. Adams and M. Frazier, Composition operators on potential spaces, Proc. Amer. Math. Soc. 114 (1992), 155-165. MR 1076570 (92e:46061)
  • 8. D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, Heidelberg, 1996. MR 1411441 (97j:46024)
  • 9. D. R. Adams and J. Xiao, Nonlinear analysis on Morrey spaces and their capacities, Indiana Univ. Math. J. 53 (2004), 1629-1663. MR 2106339 (2005h:31015)
  • 10. D. R. Adams and J. Xiao, Morrey spaces in harmonic analysis, Ark. Mat. DOI:10.1007/s11512-010-0134-0, 30 pp.
  • 11. D. R. Adams and J. Xiao, Morrey potentials and harmonic maps, Comm. Math. Phys. 308 (2011), 439-456. MR 2851148
  • 12. C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Inc., 1988. MR 928802 (89e:46001)
  • 13. Y. Ding, A characterization of BMO via commutators for some operators, Northeast. Math. J. 13(4) (1997), 422-432. MR 1612454 (2000d:42006)
  • 14. G. Di Fazio and M. A. Ragusa, Commutators and Morrey spaces, Bollettino U. M.I. 7(5-A) (1991), 321-332. MR 1138545 (93b:42028)
  • 15. G. Di Fazio and M. A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal. 112 (1993), 241-256. MR 1213138 (94e:35035)
  • 16. J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, 116. Notas de Matemática [Mathematical Notes], 104. North-Holland Publishing Co., Amsterdam, 1985. MR 807149 (87d:42023)
  • 17. M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Math. Studies 105, Princeton University Press, Princeton, N.J., 1983. MR 717034 (86b:49003)
  • 18. F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. MR 0131498 (24:A1348)
  • 19. Y. Komori and T. Mizuhara, Notes on commutators and Morrey spaces, Hokkaido Math. J. 32 (2003), 345-353. MR 1996283 (2004f:42038)
  • 20. J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs 51, Amer. Math. Soc., 1997. MR 1461542 (98h:35080)
  • 21. M. A. Ragusa, Regularity of solutions of divergence form elliptic equations, Proc. Amer. Math. Soc. 128 (1999), 533-540. MR 1641085 (2000c:35034)
  • 22. M. A. Ragusa, Commutators of fractional integral operators on vanishing-Morrey spaces, J. Glob. Optim. 40 (2008), 361-368. MR 2373563 (2008m:42026)
  • 23. C. Sadosky, Interpolation of Operators and Singular Integrals. Pure and Appl. Math., Marcel Dekker, Inc. New York and Basel, 1979. MR 551747 (81d:42001)
  • 24. J. Shatah and M. Struwe, Geometric Wave Equations, Courant Institute of Math. Sci., New York Univ., 1998. MR 1674843 (2000i:35135)
  • 25. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, New Jersey, 1993. MR 1232192 (95c:42002)
  • 26. A. Torchinsky, Real-variable Methods in Harmonic Analysis, Dover Publications, Inc., 2004. MR 2059284

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Additional Information

David R. Adams
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027
Email: dave@ms.uky.edu

Jie Xiao
Affiliation: Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Newfoundland & Labrador, Canada A1C 5S7
Email: jxiao@mun.ca

DOI: https://doi.org/10.1090/S0002-9947-2012-05595-4
Keywords: Morrey-Sobolev spaces, commutators, traces, weights, Choquet integrals, fractional Laplacians, Riesz integrals, maximal operators
Received by editor(s): November 4, 2010
Published electronically: April 6, 2012
Additional Notes: The second author was supported in part by NSERC of Canada.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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