Rubio de Francia extrapolation theorem and related topics in the theory of elliptic and parabolic equations. A survey

Krylov, N.

doi:10.1090/spmj/1653

Rubio de Francia extrapolation theorem and related topics in the theory of elliptic and parabolic equations. A survey
HTML articles powered by AMS MathViewer

by N. V. Krylov

St. Petersburg Math. J. 32, 389-413

DOI: https://doi.org/10.1090/spmj/1653

Published electronically: May 11, 2021

PDF | Request permission

Abstract:

This is a brief historical overview of the Sobolev mixed norm theory of linear elliptic and parabolic equations and the recent development in this theory based on the Rubio de Francia extrapolation theorem. A self-contained proof of this theorem along with other relevant tools of Real Analysis are also presented as well as an application to mixed norm estimates for fully nonlinear equations.

References

H. Amann, Maximal regularity for nonautonomous evolution equations, Adv. Nonlinear Stud. 4 (2004), no. 4, 417–430. MR 2100906, DOI 10.1515/ans-2004-0404
Marco Bramanti and M. Cristina Cerutti, $W_p^{1,2}$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1735–1763. MR 1239929, DOI 10.1080/03605309308820991
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
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
Piermarco Cannarsa and Vincenzo Vespri, On maximal $L^p$ regularity for the abstract Cauchy problem, Boll. Un. Mat. Ital. B (6) 5 (1986), no. 1, 165–175 (English, with Italian summary). MR 841623
Filippo Chiarenza, Michele Frasca, and Placido Longo, Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), no. 1, 149–168. MR 1191890
Filippo Chiarenza, Michele Frasca, and Placido Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), no. 2, 841–853. MR 1088476, DOI 10.1090/S0002-9947-1993-1088476-1
Thierry Coulhon and Xuan Thinh Duong, Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss, Adv. Differential Equations 5 (2000), no. 1-3, 343–368. MR 1734546
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
Robert Denk, Matthias Hieber, and Jan Prüss, $\scr R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. 166 (2003), no. 788, viii+114. MR 2006641, DOI 10.1090/memo/0788
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
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, 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
Hongjie Dong and Doyoon Kim, On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights, Trans. Amer. Math. Soc. 370 (2018), no. 7, 5081–5130. MR 3812104, DOI 10.1090/tran/7161
Hongjie Dong and N. V. Krylov, Second-order elliptic and parabolic equations with $B(\Bbb R^2, \rm {VMO})$ coefficients, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6477–6494. MR 2678983, DOI 10.1090/S0002-9947-2010-05215-8
Hongjie Dong and N. V. Krylov, Fully nonlinear elliptic and parabolic equations in weighted and mixed-norm Sobolev spaces, Calc. Var. Partial Differential Equations 58 (2019), no. 4, Paper No. 145, 32. MR 3989949, DOI 10.1007/s00526-019-1591-3
Javier Duoandikoetxea, Extrapolation of weights revisited: new proofs and sharp bounds, J. Funct. Anal. 260 (2011), no. 6, 1886–1901. MR 2754896, DOI 10.1016/j.jfa.2010.12.015
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
A. Fröhlich, Stokes– und Navier–Stokes–Gleichungen in gewichteten Funktionenräumen, Dissertation, TU Darmstadt, 2001.
Chiara Gallarati, Emiel Lorist, and Mark Veraar, On the $\ell ^s$-boundedness of a family of integral operators, Rev. Mat. Iberoam. 32 (2016), no. 4, 1277–1294. MR 3593523, DOI 10.4171/RMI/916
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
Chiara Gallarati and Mark Veraar, Evolution families and maximal regularity for systems of parabolic equations, Adv. Differential Equations 22 (2017), no. 3-4, 169–190. MR 3611504
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
Mariko Giga, Yoshikazu Giga, and Hermann Sohr, $L^p$ estimate for abstract linear parabolic equations, Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), no. 6, 197–202. MR 1120516
Mariko Giga, Yoshikazu Giga, and Hermann Sohr, $L^p$ estimates for the Stokes system, Functional analysis and related topics, 1991 (Kyoto), Lecture Notes in Math., vol. 1540, Springer, Berlin, 1993, pp. 55–67. MR 1225811, DOI 10.1007/BFb0085474
Yoshikazu Giga and Hermann Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal. 102 (1991), no. 1, 72–94. MR 1138838, DOI 10.1016/0022-1236(91)90136-S
Davide Guidetti, General linear boundary value problems for elliptic operators with VMO coefficients, Math. Nachr. 237 (2002), 62–88. MR 1894353, DOI 10.1002/1522-2616(200204)237:1<62::AID-MANA62>3.3.CO;2-R
Robert Haller, Horst Heck, and Matthias Hieber, Muckenhoupt weights and maximal $L^p$-regularity, Arch. Math. (Basel) 81 (2003), no. 4, 422–430. MR 2057063, DOI 10.1007/s00013-003-0492-y
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
Horst Heck and Matthias Hieber, Maximal $L^p$-regularity for elliptic operators with VMO-coefficients, J. Evol. Equ. 3 (2003), no. 2, 332–359. MR 1980981, DOI 10.1007/978-3-0348-7924-8_{1}8
Matthias Hieber and Jan Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations 22 (1997), no. 9-10, 1647–1669. MR 1469585, DOI 10.1080/03605309708821314
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
N. V. Krylov, SPDEs in $L_Q((0,\tau ]\!],L_P)$ spaces, Electron. J. Probab. 5 (2000), Paper no. 13, 29. MR 1781025, DOI 10.1214/EJP.v5-69
N. V. Krylov, The heat equation in $L_q((0,T),L_p)$-spaces with weights, SIAM J. Math. Anal. 32 (2001), no. 5, 1117–1141. MR 1828321, DOI 10.1137/S0036141000372039
N. V. Krylov, The Calderón-Zygmund theorem and its applications to parabolic equations, Algebra i Analiz 13 (2001), no. 4, 1–25 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 4, 509–526. MR 1865493
N. V. Krylov, Parabolic equations in $L_p$-spaces with mixed norms, Algebra i Analiz 14 (2002), no. 4, 91–106 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 14 (2003), no. 4, 603–614. MR 1935919
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
Nicolai V. Krylov, On Bellman’s equations with VMO coefficients, Methods Appl. Anal. 17 (2010), no. 1, 105–121. MR 2735102, DOI 10.4310/MAA.2010.v17.n1.a4
N. V. Krylov, Sobolev and viscosity solutions for fully nonlinear elliptic and parabolic equations, Mathematical Surveys and Monographs, vol. 233, American Mathematical Society, Providence, RI, 2018. MR 3837125, DOI 10.1090/surv/233
Andrei K. Lerner, An elementary approach to several results on the Hardy-Littlewood maximal operator, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2829–2833. MR 2399047, DOI 10.1090/S0002-9939-08-09318-0
J. E. Lewis, Mixed estimates for singular integrals and an application to initial value problems in parabolic differential equations, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 218–231. MR 0234130
Antonino Maugeri, Dian K. Palagachev, and Lubomira G. Softova, Elliptic and parabolic equations with discontinuous coefficients, Mathematical Research, vol. 109, Wiley-VCH Verlag Berlin GmbH, Berlin, 2000. MR 2260015, DOI 10.1002/3527600868
P. Maremonti and V. A. Solonnikov, On estimates for the solutions of the nonstationary Stokes problem in S. L. Sobolev anisotropic spaces with a mixed norm, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222 (1995), no. Issled. po Lineĭn. Oper. i Teor. Funktsiĭ. 23, 124–150, 309 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 87 (1997), no. 5, 3859–3877. MR 1359996, DOI 10.1007/BF02355828
Dian Palagachev and Lubomira Softova, A priori estimates and precise regularity for parabolic systems with discontinuous data, Discrete Contin. Dyn. Syst. 13 (2005), no. 3, 721–742. MR 2153140, DOI 10.3934/dcds.2005.13.721
P. E. Sobolevskiĭ, Coerciveness inequalities for abstract parabolic equations, Dokl. Akad. Nauk SSSR 157 (1964), 52–55 (Russian). MR 0166487
Jonas Sauer, An extrapolation theorem in non-Euclidean geometries and its application to partial differential equations, J. Elliptic Parabol. Equ. 1 (2015), 403–418. MR 3487344, DOI 10.1007/BF03377388
Lubomira G. Softova and Peter Weidemaier, Quasilinear parabolic problem in spaces of maximal regularity, J. Nonlinear Convex Anal. 7 (2006), no. 3, 529–540. MR 2287547
Pablo Raúl Stinga and José L. Torrea, Hölder, Sobolev, weak-type and BMO estimates in mixed-norm with weights for parabolic equations, Sci. China Math. 64 (2021), no. 1, 129–154. MR 4200987, DOI 10.1007/s11425-018-1550-5
Peter Weidemaier, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm, Electron. Res. Announc. Amer. Math. Soc. 8 (2002), 47–51. MR 1945779, DOI 10.1090/S1079-6762-02-00104-X