Global weighted estimates for second-order nondivergence elliptic and parabolic equations



In this paper, we obtain the global weighted \(L^p\) estimates for second-order nondivergence elliptic and parabolic equations with small BMO coefficients in the whole space. As a corollary, we obtain \(L^p\)-type regularity estimates for such equations.


Weighted \(L^p\) estimates second-order nondivergence small BMO elliptic parabolic the whole space 

2000 Mathematics Subject Classification

35K10 35J15 



The author wishes to thank the anonymous reviewer for valuable comments and suggestions that improved the expressions. This work is supported in part by the NSFC (11471207) and the Innovation Program of Shanghai Municipal Education Commission (14YZ027).


  1. 1.
    Acerbi E and Mingione G, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007) 285–320Google Scholar
  2. 2.
    Bramanti M and Cerutti M C, \(W^{1,2}_p\) solvability for the Cauchy Dirichlet problem for parabolic equations with VMO coefficients, Commun. Partial Differ. Equations, 18 (1993) 1735–1763CrossRefMATHGoogle Scholar
  3. 3.
    Byun S, Lee M and Palagachev Dian K, Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations, J. Differential Equations, 260(5) (2016) 4550–4571MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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(2) (2013) 291–313MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Byun S, Palagachev Dian K and Ryu S, Weighted \(W^{1,p}\) estimates for solutions of nonlinear parabolic equations over non-smooth domains, Bull. London Math. Soc. 45(4) (2013) 765–778MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Byun S and Wang L, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math. 57(10) (2004) 1283–1310MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Byun S and Wang L, Parabolic equations in Reifenberg domains, Arch. Ration. Mech. Anal. 176(2) (2005) 271–301MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Byun S and Wang L, Parabolic equations in time-dependent Reifenberg domains, Adv. Math. 212(2) (2007) 797–818MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Caffarelli L A and Peral I, On \(W^{1,p}\) estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1–21MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Calderón A P and Zygmund A, On the existence of certain singular integrals, Acta Math. 88 (1952) 85–139MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chiarenza F, Frasca M and Longo P, \(W^{2,p}\)-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336(2) (1993) 841–853MathSciNetMATHGoogle Scholar
  12. 12.
    Di Fazio G, \(L^p\) estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. A (7) 10(2) (1996) 409–420Google Scholar
  13. 13.
    Dong H, Solvability of parabolic equations in divergence form with partially BMO coefficients, J. Funct. Anal. 258 (2010) 2145–2172MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dong H and Kim D, Elliptic equations in divergence form with partially BMO coefficients, Arch. Rational Mech. Anal. 196 (2010) 25–70MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dong H and Kim D, Parabolic and elliptic systems in divergence form with variably partially BMO coefficients, SIAM J. Math. Anal. 43(3) (2011) 1075–1098MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dong H and Kim D, Parabolic equations in simple convex polytopes with time irregular coefficients, SIAM J. Math. Anal. 46(3) (2014) 1789–1819MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Giaquinta M, Multiple integrals in the calculus of variations and nonlinear elliptic systems (1983) (Princeton, NJ: Princeton University Press)MATHGoogle Scholar
  18. 18.
    Grafakos L, Classical Fourier analysis, Graduate Texts in Mathematics, 249 (2014) (New York: Springer) third editionGoogle Scholar
  19. 19.
    Jiménez Urrea J, The Cauchy problem associated to the Benjamin equation in weighted Sobolev spaces, J. Differential Equations, 254(4) (2013) 1863–1892MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kim D and Krylov N V, Parabolic Equations with Measurable Coefficients, Potential Anal. 26 (2007) 345–361MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Krylov N V, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations, 32(1–3) (2007) 453–475MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Krylov N V, Second-order elliptic equations with variably partially VMO coefficients, J. Funct. Anal. 257(6) (2009) 1695–1712MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kufner A, Weighted Sobolev spaces, translated from the Czech, a Wiley-Interscience Publication (1985) (New York: John Wiley & Sons Inc.)Google Scholar
  24. 24.
    Kuusi T and Mingione G, Universal potential estimates, J. Funct. Anal., 262 (2012) 4205–4269MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lieberman G M, Second order parabolic differential equations (1996) (River Edge, NJ: World Scientific Publishing Co. Inc.)Google Scholar
  26. 26.
    Li D and Wang L, A new proof for the estimates of Calderón–Zygmund type singular integrals, Arch. Math. (Basel), 87(5) (2006) 458–467MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Mengesha T and Phuc N, Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations, 250(5) (2011) 2485–2507MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Mengesha T and Phuc N, Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Ration. Mech. Anal. 203(1) (2012) 189–216MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Mingione G, The Calderón–Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6(2) (2007) 195–261.Google Scholar
  30. 30.
    Mingione G, Gradient estimates below the duality exponent, Math. Ann. 346 (2010) 571–627MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Muckenhoupt B, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972) 207–226MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Palagachev D K and Softova L G, A priori estimates and precise regularity for parabolic systems with discontinuous data, Discrete Contin. Dyn. Syst. 13(3) (2005) 721–742MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Stein E M, Harmonic Analysis (1993) (Princeton: Princeton University Press)Google Scholar
  34. 34.
    Torchinsky A, Real-Variable Methods in Harmonic Analysis, Pure Appl. Math., vol. 123, (1986) (Orlando, FL: Academic Press Inc.)Google Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina

Personalised recommendations