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Regularity estimates in weighted Orlicz spaces for Calderón–Zygmund type singular integral operators

  • Fengping Yao EMAIL logo
Published/Copyright: June 11, 2016

Abstract

In this paper we obtain regularity estimates in weighted Orlicz spaces for the Calderón–Zygmund singular integral operators under certain optimal conditions.

MSC 2010: 32A55; 46E30

Communicated by Frank Duzaar


Award Identifier / Grant number: 11471207

Award Identifier / Grant number: 14YZ027

Funding statement: This work is supported in part by the NSFC (11471207) and the Innovation Program of Shanghai Municipal Education Commission (14YZ027).

Acknowledgements

The author wishes to thank the anonymous reviewer for the valuable comments and suggestions to improve the expressions.

References

[1] Acerbi E. and Mingione G., Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007), 285–320. 10.1215/S0012-7094-07-13623-8Search in Google Scholar

[2] Adams R. A. and Fournier J. J. F., Sobolev Spaces, 2nd ed., Academic Press, New York, 2003. Search in Google Scholar

[3] Benkirane A. and Elmahi A., An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces, Nonlinear Anal. 36 (1999), 11–24. 10.1016/S0362-546X(97)00612-3Search in Google Scholar

[4] Bögelein V. and Duzaar F., Higher integrability for parabolic systems with non-standard growth and degenerate diffusions, Publ. Mat. 55 (2011), no. 1, 201–250. 10.5565/PUBLMAT_55111_10Search in Google Scholar

[5] Bögelein V., Duzaar F. and Mingione G., The regularity of general parabolic systems with degenerate diffusion, Mem. Amer. Math. Soc. 1041 (2013), 1–143. 10.1090/S0065-9266-2012-00664-2Search in Google Scholar

[6] Byun S., Ok J., Palagachev D. K. and Softova L. G., Parabolic systems with measurable coefficients in weighted Orlicz spaces, Commun. Contemp. Math. 18 (2016), no. 2, Article ID 1550018. 10.1142/S0219199715500182Search in Google Scholar

[7] Byun S., Palagachev D. K. and Ryu S., Weighted W1,p estimates for solutions of nonlinear parabolic equations over non-smooth domains, Bull. Lond. Math. Soc. 45 (2013), no. 4, 765–778. 10.1112/blms/bdt011Search in Google Scholar

[8] Byun S. and Ryu S., Global weighted estimates for the gradient of solutions to nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), no. 2, 291–313. 10.1016/j.anihpc.2012.08.001Search in Google Scholar

[9] Caffarelli L. A. and Peral I., On W1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), 1–21. 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.0.CO;2-GSearch in Google Scholar

[10] Calderón A. P. and Zygmund A., On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. 10.1007/978-94-009-1045-4_3Search in Google Scholar

[11] Donaldson T., Nonlinear elliptic boundary-value problems in Orlicz–Sobolev spaces, J. Differential Equations 10 (1971), 507–528. 10.1016/0022-0396(71)90009-XSearch in Google Scholar

[12] Donaldson T., Inhomogeneous Orlicz–Sobolev spaces and nonlinear parabolic initial value problems, J. Differential Equations 16 (1974), 201–256. 10.1016/0022-0396(74)90012-6Search in Google Scholar

[13] Fiorenza A. and Krbec M., Indices of Orlicz spaces and some applications, Comment. Math. Univ. Carolin. 38 (1997), no. 3, 433–451. Search in Google Scholar

[14] Jiménez Urrea J., The Cauchy problem associated to the Benjamin equation in weighted Sobolev spaces, J. Differential Equations 254 (2013), no. 4, 1863–1892. 10.1016/j.jde.2012.11.016Search in Google Scholar

[15] Kerman R. A. and Torchinsky A., Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1982), 277–284. 10.4064/sm-71-3-277-284Search in Google Scholar

[16] Kokilashvili V. and Krbec M., Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, Hackensack, 1991. 10.1142/1367Search in Google Scholar

[17] Kufner A., Weighted Sobolev Spaces, John Wiley & Sons, New York, 1985. Search in Google Scholar

[18] Li D. and Wang L., A new proof for the estimates of Calderón–Zygmund type singular integrals, Arch. Math. (Basel) 87 (2006), no. 5, 458–467. 10.1007/s00013-006-1774-ySearch in Google Scholar

[19] Mengesha T. and Phuc N., Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations 250 (2011), no. 5, 2485–2507. 10.1016/j.jde.2010.11.009Search in Google Scholar

[20] Mengesha T. and Phuc N., Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Ration. Mech. Anal. 203 (2012), no. 1, 189–216. 10.1007/s00205-011-0446-7Search in Google Scholar

[21] Mingione G., The Calderón–Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), no. 2, 195–261. 10.2422/2036-2145.2007.2.01Search in Google Scholar

[22] Orlicz W., Über eine gewisse Klasse von Räumen vom Typus B, Bull. Int. Acad. Polon. Sci. A 1932 (1932), no. 8–9, 207–220. Search in Google Scholar

[23] Rao M. and Ren Z., Applications of Orlicz Spaces, Marcel Dekker, New York, 2000. Search in Google Scholar

[24] Stein E. M., Harmonic Analysis, Princeton University Press, Princeton, 1993. Search in Google Scholar

[25] Torchinsky A., Real-Variable Methods in Harmonic Analysis, Pure Appl. Math. 123, Academic Press, Orlando, 1986. Search in Google Scholar

[26] Wang L., Yao F., Zhou S. and Jia H., Optimal regularity for the Poisson equation, Proc. Amer. Math. Soc. 137 (2009), 2037–2047. 10.1090/S0002-9939-09-09805-0Search in Google Scholar

Received: 2015-6-28
Revised: 2016-1-14
Published Online: 2016-6-11
Published in Print: 2017-1-1

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