On 𝐿_{𝑝}-estimates for elliptic and parabolic equations with 𝐴_{𝑝} weights

Dong, Hongjie; Kim, Doyoon

doi:10.1090/tran/7161

On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights
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Trans. Amer. Math. Soc. 370 (2018), 5081-5130 Request permission

Abstract:

We prove generalized Fefferman-Stein type theorems on sharp functions with $A_p$ weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixed-norm weighted $L_p$-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains.

References

Hugo Aimar and Roberto A. Macías, Weighted norm inequalities for the Hardy-Littlewood maximal operator on spaces of homogeneous type, Proc. Amer. Math. Soc. 91 (1984), no. 2, 213–216. MR 740173, DOI 10.1090/S0002-9939-1984-0740173-5
Oleg V. Besov, Valentin P. Il′in, and Sergey M. Nikol′skiĭ, Integral representations of functions and imbedding theorems. Vol. II, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1979. Edited by Mitchell H. Taibleson. MR 521808
Anders Björn and Jana Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, vol. 17, European Mathematical Society (EMS), Zürich, 2011. MR 2867756, DOI 10.4171/099
Sun-Sig Byun, Jihoon Ok, Dian K. Palagachev, and Lubomira G. Softova, Parabolic systems with measurable coefficients in weighted Orlicz spaces, Commun. Contemp. Math. 18 (2016), no. 2, 1550018, 19. MR 3461425, DOI 10.1142/S0219199715500182
Sun-Sig Byun, Dian K. Palagachev, and Lubomira G. Softova, Global gradient estimates in weighted Lebesgue spaces for parabolic operators, Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 1, 67–83. MR 3467697, DOI 10.5186/aasfm.2016.4102
Sun-Sig Byun, Dian K. Palagachev, and Lihe Wang, Parabolic systems with measurable coefficients in Reifenberg domains, Int. Math. Res. Not. IMRN 13 (2013), 3053–3086. MR 3073000, DOI 10.1093/imrn/rns142
Sun-Sig Byun and Lihe Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math. 57 (2004), no. 10, 1283–1310. MR 2069724, DOI 10.1002/cpa.20037
Sun-Sig Byun and Lihe Wang, Elliptic equations with measurable coefficients in Reifenberg domains, Adv. Math. 225 (2010), no. 5, 2648–2673. MR 2680179, DOI 10.1016/j.aim.2010.05.014
L. A. Caffarelli and I. Peral, On $W^{1,p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), no. 1, 1–21. MR 1486629, DOI 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N
A.-P. Calderón, Inequalities for the maximal function relative to a metric, Studia Math. 57 (1976), no. 3, 297–306. MR 442579, DOI 10.4064/sm-57-3-297-306
Michael Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. MR 1096400, DOI 10.4064/cm-60-61-2-601-628
D. Cruz-Uribe, J. M. Martell, and C. Pérez, Extrapolation from $A_\infty$ weights and applications, J. Funct. Anal. 213 (2004), no. 2, 412–439. MR 2078632, DOI 10.1016/j.jfa.2003.09.002
David V. Cruz-Uribe, José Maria Martell, and Carlos Pérez, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2797562, DOI 10.1007/978-3-0348-0072-3
Hongjie Dong, Solvability of parabolic equations in divergence form with partially BMO coefficients, J. Funct. Anal. 258 (2010), no. 7, 2145–2172. MR 2584743, DOI 10.1016/j.jfa.2010.01.003
H. Dong, Parabolic equations with variably partially VMO coefficients, Algebra i Analiz 23 (2011), no. 3, 150–174; English transl., St. Petersburg Math. J. 23 (2012), no. 3, 521-539. MR 2896169, DOI 10.1090/S1061-0022-2012-01206-9
Hongjie Dong, Solvability of second-order equations with hierarchically partially BMO coefficients, Trans. Amer. Math. Soc. 364 (2012), no. 1, 493–517. MR 2833589, DOI 10.1090/S0002-9947-2011-05453-X
Hongjie Dong and Doyoon Kim, Elliptic equations in divergence form with partially BMO coefficients, Arch. Ration. Mech. Anal. 196 (2010), no. 1, 25–70. MR 2601069, DOI 10.1007/s00205-009-0228-7
Hongjie Dong and Doyoon Kim, Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains, J. Funct. Anal. 261 (2011), no. 11, 3279–3327. MR 2835999, DOI 10.1016/j.jfa.2011.08.001
Hongjie Dong and Doyoon Kim, $L_p$ solvability of divergence type parabolic and elliptic systems with partially BMO coefficients, Calc. Var. Partial Differential Equations 40 (2011), no. 3-4, 357–389. MR 2764911, DOI 10.1007/s00526-010-0344-0
Hongjie Dong and Doyoon Kim, On the $L_p$-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arch. Ration. Mech. Anal. 199 (2011), no. 3, 889–941. MR 2771670, DOI 10.1007/s00205-010-0345-3
Hongjie Dong and Doyoon Kim, Parabolic and elliptic systems in divergence form with variably partially BMO coefficients, SIAM J. Math. Anal. 43 (2011), no. 3, 1075–1098. MR 2800569, DOI 10.1137/100794614
Hongjie Dong and Doyoon Kim, Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces, Adv. Math. 274 (2015), 681–735. MR 3318165, DOI 10.1016/j.aim.2014.12.037
Doyoon Kim, Hongjie Dong, and Hong Zhang, Neumann problem for non-divergence elliptic and parabolic equations with $\textrm {BMO}_x$ coefficients in weighted Sobolev spaces, Discrete Contin. Dyn. Syst. 36 (2016), no. 9, 4895–4914. MR 3541508, DOI 10.3934/dcds.2016011
Hongjie Dong and Jingang Xiong, Boundary gradient estimates for parabolic and elliptic systems from linear laminates, Int. Math. Res. Not. IMRN 17 (2015), 7734–7756. MR 3403998, DOI 10.1093/imrn/rnu185
Hongjie Dong and Hong Zhang, Schauder estimates for higher-order parabolic systems with time irregular coefficients, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 47–74. MR 3385152, DOI 10.1007/s00526-014-0777-y
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
Nobuhiko Fujii, A proof of the Fefferman-Stein-Strömberg inequality for the sharp maximal functions, Proc. Amer. Math. Soc. 106 (1989), no. 2, 371–377. MR 946637, DOI 10.1090/S0002-9939-1989-0946637-8
Chiara Gallarati and Mark Veraar, Maximal regularity for non-autonomous equations with measurable dependence on time, Potential Anal. 46 (2017), no. 3, 527–567. MR 3630407, DOI 10.1007/s11118-016-9593-7
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
Loukas Grafakos, Classical Fourier analysis, 3rd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2014. MR 3243734, DOI 10.1007/978-1-4939-1194-3
Robert Haller-Dintelmann, Horst Heck, and Matthias Hieber, $L^p$-$L^q$ estimates for parabolic systems in non-divergence form with VMO coefficients, J. London Math. Soc. (2) 74 (2006), no. 3, 717–736. MR 2286441, DOI 10.1112/S0024610706023192
Tuomas Hytönen and Anna Kairema, Systems of dyadic cubes in a doubling metric space, Colloq. Math. 126 (2012), no. 1, 1–33. MR 2901199, DOI 10.4064/cm126-1-1
Doyoon Kim, Parabolic equations with measurable coefficients. II, J. Math. Anal. Appl. 334 (2007), no. 1, 534–548. MR 2332574, DOI 10.1016/j.jmaa.2006.12.077
Doyoon Kim, Parabolic equations with partially BMO coefficients and boundary value problems in Sobolev spaces with mixed norms, Potential Anal. 33 (2010), no. 1, 17–46. MR 2644213, DOI 10.1007/s11118-009-9158-0
Doyoon Kim and N. V. Krylov, Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others, SIAM J. Math. Anal. 39 (2007), no. 2, 489–506. MR 2338417, DOI 10.1137/050646913
Doyoon Kim and N. V. Krylov, Parabolic equations with measurable coefficients, Potential Anal. 26 (2007), no. 4, 345–361. MR 2300337, DOI 10.1007/s11118-007-9042-8
Vladimir Kozlov and Alexander Nazarov, The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients, Math. Nachr. 282 (2009), no. 9, 1220–1241. MR 2561181, DOI 10.1002/mana.200910796
Vladimir Kozlov and Alexander Nazarov, The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge, Math. Nachr. 287 (2014), no. 10, 1142–1165. MR 3231530, DOI 10.1002/mana.201100352
N. V. Krylov, Weighted Sobolev spaces and Laplace’s equation and the heat equations in a half space, Comm. Partial Differential Equations 24 (1999), no. 9-10, 1611–1653. MR 1708104, DOI 10.1080/03605309908821478
N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations 32 (2007), no. 1-3, 453–475. MR 2304157, DOI 10.1080/03605300600781626
N. V. Krylov, Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, J. Funct. Anal. 250 (2007), no. 2, 521–558. MR 2352490, DOI 10.1016/j.jfa.2007.04.003
N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Mathematics, vol. 96, American Mathematical Society, Providence, RI, 2008. MR 2435520, DOI 10.1090/gsm/096
N. V. Krylov, Second-order elliptic equations with variably partially VMO coefficients, J. Funct. Anal. 257 (2009), no. 6, 1695–1712. MR 2540989, DOI 10.1016/j.jfa.2009.06.014
Roberto A. Macías and Carlos Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 271–309. MR 546296, DOI 10.1016/0001-8708(79)90013-6
Roberto A. Macías and Carlos Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 257–270. MR 546295, DOI 10.1016/0001-8708(79)90012-4
Roberto A. Macías and Carlos A. Segovia, A well behaved quasi-distance for spaces of homogeneous type, volume 32 of Trabajos de Matemática, Inst. Argentino Mat., 1981.
José María Martell, Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications, Studia Math. 161 (2004), no. 2, 113–145. MR 2033231, DOI 10.4064/sm161-2-2
José L. Rubio de Francia, Factorization theory and $A_{p}$ weights, Amer. J. Math. 106 (1984), no. 3, 533–547. MR 745140, DOI 10.2307/2374284
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192