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$L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators
Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China |
We prove weighted Lorentz estimates of the Hessian of strong solution for nondivergence linear elliptic equations $a_{ij}(x)D_{ij}u(x)=f(x)$. The leading coefficients are assumed to be measurable with respect to one variable and have small BMO semi-norms with respect to the other variables. Here, an approximation method, Lorentz boundedness of the Hardy-Littlewood maximal operators and an equivalent representation of Lorentz norm are employed.
References:
[1] |
E. Acerbi and G. Mingione,
Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285-320.
doi: 10.1215/S0012-7094-07-13623-8. |
[2] |
S. Agmon, A. Douglisa and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math., 17 (1964), 35-92.
doi: 10.1002/cpa.3160170104. |
[3] |
P. Baroni,
Lorentz estimates for obstacle parabolic problems, Nonlinear Anal., 96 (2014), 167-188.
doi: 10.1016/j.na.2013.11.004. |
[4] |
P. Baroni,
Lorentz estimates for degenerate and singular evolutionary systems, J. Differ. Equ., 255 (2013), 2927-2951.
doi: 10.1016/j.jde.2013.07.024. |
[5] |
M. Bramanti and M. Cerutti,
Wp1,2 solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differ. Equ., 18 (1993), 1735-1763.
doi: 10.1080/03605309308820991. |
[6] |
S. S. Byun and Y. Kim,
Elliptic equations with measurable nonlinearities in nonsmooth domains, Adv. Math., 288 (2016), 152-200.
doi: 10.1016/j.aim.2015.10.015. |
[7] |
S. S. Byun and M. Lee,
On weighted W2,p estimates for elliptic equations with BMO coefficients in nondivergence form, International Journal of Mathematics, 26 (2015), 1550001.
doi: 10.1142/S0129167X15500019. |
[8] |
S. S. Byun and M. Lee,
Weighted estimates for nondivergence parabolic equations in Orlicz spaces, J. Funct. Anal., 269 (2015), 2530-2563.
doi: 10.1016/j.jfa.2015.07.009. |
[9] |
S. S. Byun and D. K. Palagachev,
Weighted Lp-estimates for elliptic equations with measurable coefficients in nonsmooth domains, Potential Anal., 41 (2014), 51-79.
doi: 10.1007/s11118-013-9363-8. |
[10] |
L. A. Caffarelli and I. Peral,
On W1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51 (1998), 1-21.
doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N. |
[11] |
F. Chiarenza, M. Frasca and P. Longo,
Interior W2,p estimates for nondivergence elliptic equations with discontinuous coeffcients, Ricerche Mat., 40 (1991), 149-168.
|
[12] |
F. Chiarenza, M. Frasca and P. Longo,
W2,p solvability of the Dirichlet problem for nonlinear elliptic equations with VMO coeffcients, Trans. Amer. Math. Soc., 336 (1993), 841-853.
doi: 10.2307/2154379. |
[13] |
H. Dong,
Solvability of second-order equations with hierarchically patially BMO coefficients, Trans. Amer. Math. Soc., 364 (2012), 493-517.
doi: 10.1090/S0002-9947-2011-05453-X. |
[14] |
H. Dong,
Parabolic equations with variably partially VMO coefficients, St. Petersburg Math. J., 23 (2012), 521-539.
doi: 10.1090/S1061-0022-2012-01206-9. |
[15] |
H. Dong and D. Kim, On Lp-estimates for elliptic and parabolic equations with Ap weights, preprint, arXiv: 1603.07844. |
[16] |
Q. Han and F. Lin, Elliptic Partial Differential Equation, American Mathematical Soc., 2011. |
[17] |
D. Kim and A. V. Krylov,
Elliptic differenrial equations with coefficients measurable with respact to one variable and VMO with respect to the others, SIAM J. Math. Anal., 32 (2007), 489-506.
doi: 10.1137/050646913. |
[18] |
D. Kim and N. V. Krylov,
Parabolic equations with measurable coefficients, Potential Anal., 26 (2007), 345-361.
doi: 10.1007/s11118-007-9042-8. |
[19] |
N. V. Krylov,
Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differ. Equ., 32 (2007), 453-475.
doi: 10.1080/03605300600781626. |
[20] |
N. V. Krylov,
Second order parabolic equations with variably partially VMO coeffcients, J. Funct. Anal., 257 (2009), 1695-1712.
doi: 10.1016/j.jfa.2009.06.014. |
[21] |
G. M. Lieberman,
A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients, J. Funct. Anal., 201 (2003), 457-479.
doi: 10.1016/S0022-1236(03)00125-3. |
[22] |
A. Maugeri, D. K. Palagachev and L. G. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, 1 edition, Wiley-vch Verlag, Berlin, 2000.
doi: 10.1002/3527600868.![]() ![]() ![]() |
[23] |
T. Mengesha and N. C. Phuc,
Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Rational Mech. Anal., 203 (2012), 189-216.
doi: 10.1007/s00205-011-0446-7. |
[24] |
N. G. Meyers,
An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17 (1963), 189-206.
|
[25] |
G. Mingione,
Gradient estimates below the duality exponent, Ann. Math., 346 (2010), 571-627.
doi: 10.1007/s00208-009-0411-z. |
[26] |
M. V. Safonov,
Harnack inequality for elliptic equations and the Hölder property of their solutions, J. Soviet Math., 21 (1983), 851-863.
|
[27] |
G. Talenti,
Elliptic Equations and Rearrangements, Ann Scuola Norm. Sup. Pisa Cl. Sci., 3 (1976), 697-718.
|
[28] |
B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, 1 edition, Springer-Verlag Berlin Heidelberg, New York, 2000.
doi: 10.1007/BFb0103908.![]() ![]() ![]() |
[29] |
L. Wang,
A geometric approach to the Calderon-Zygmmund estimates, Acta Math. Sin., 19 (2003), 381-396.
doi: 10.1007/s10114-003-0264-4. |
[30] |
C. Zhang and S. Zhou,
Global weighted estimates for quasilinear elliptic equations with nonstandard growth, J. Funct. Anal., 267 (2014), 605-642.
doi: 10.1016/j.jfa.2014.03.022. |
show all references
References:
[1] |
E. Acerbi and G. Mingione,
Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285-320.
doi: 10.1215/S0012-7094-07-13623-8. |
[2] |
S. Agmon, A. Douglisa and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math., 17 (1964), 35-92.
doi: 10.1002/cpa.3160170104. |
[3] |
P. Baroni,
Lorentz estimates for obstacle parabolic problems, Nonlinear Anal., 96 (2014), 167-188.
doi: 10.1016/j.na.2013.11.004. |
[4] |
P. Baroni,
Lorentz estimates for degenerate and singular evolutionary systems, J. Differ. Equ., 255 (2013), 2927-2951.
doi: 10.1016/j.jde.2013.07.024. |
[5] |
M. Bramanti and M. Cerutti,
Wp1,2 solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differ. Equ., 18 (1993), 1735-1763.
doi: 10.1080/03605309308820991. |
[6] |
S. S. Byun and Y. Kim,
Elliptic equations with measurable nonlinearities in nonsmooth domains, Adv. Math., 288 (2016), 152-200.
doi: 10.1016/j.aim.2015.10.015. |
[7] |
S. S. Byun and M. Lee,
On weighted W2,p estimates for elliptic equations with BMO coefficients in nondivergence form, International Journal of Mathematics, 26 (2015), 1550001.
doi: 10.1142/S0129167X15500019. |
[8] |
S. S. Byun and M. Lee,
Weighted estimates for nondivergence parabolic equations in Orlicz spaces, J. Funct. Anal., 269 (2015), 2530-2563.
doi: 10.1016/j.jfa.2015.07.009. |
[9] |
S. S. Byun and D. K. Palagachev,
Weighted Lp-estimates for elliptic equations with measurable coefficients in nonsmooth domains, Potential Anal., 41 (2014), 51-79.
doi: 10.1007/s11118-013-9363-8. |
[10] |
L. A. Caffarelli and I. Peral,
On W1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51 (1998), 1-21.
doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N. |
[11] |
F. Chiarenza, M. Frasca and P. Longo,
Interior W2,p estimates for nondivergence elliptic equations with discontinuous coeffcients, Ricerche Mat., 40 (1991), 149-168.
|
[12] |
F. Chiarenza, M. Frasca and P. Longo,
W2,p solvability of the Dirichlet problem for nonlinear elliptic equations with VMO coeffcients, Trans. Amer. Math. Soc., 336 (1993), 841-853.
doi: 10.2307/2154379. |
[13] |
H. Dong,
Solvability of second-order equations with hierarchically patially BMO coefficients, Trans. Amer. Math. Soc., 364 (2012), 493-517.
doi: 10.1090/S0002-9947-2011-05453-X. |
[14] |
H. Dong,
Parabolic equations with variably partially VMO coefficients, St. Petersburg Math. J., 23 (2012), 521-539.
doi: 10.1090/S1061-0022-2012-01206-9. |
[15] |
H. Dong and D. Kim, On Lp-estimates for elliptic and parabolic equations with Ap weights, preprint, arXiv: 1603.07844. |
[16] |
Q. Han and F. Lin, Elliptic Partial Differential Equation, American Mathematical Soc., 2011. |
[17] |
D. Kim and A. V. Krylov,
Elliptic differenrial equations with coefficients measurable with respact to one variable and VMO with respect to the others, SIAM J. Math. Anal., 32 (2007), 489-506.
doi: 10.1137/050646913. |
[18] |
D. Kim and N. V. Krylov,
Parabolic equations with measurable coefficients, Potential Anal., 26 (2007), 345-361.
doi: 10.1007/s11118-007-9042-8. |
[19] |
N. V. Krylov,
Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differ. Equ., 32 (2007), 453-475.
doi: 10.1080/03605300600781626. |
[20] |
N. V. Krylov,
Second order parabolic equations with variably partially VMO coeffcients, J. Funct. Anal., 257 (2009), 1695-1712.
doi: 10.1016/j.jfa.2009.06.014. |
[21] |
G. M. Lieberman,
A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients, J. Funct. Anal., 201 (2003), 457-479.
doi: 10.1016/S0022-1236(03)00125-3. |
[22] |
A. Maugeri, D. K. Palagachev and L. G. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, 1 edition, Wiley-vch Verlag, Berlin, 2000.
doi: 10.1002/3527600868.![]() ![]() ![]() |
[23] |
T. Mengesha and N. C. Phuc,
Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Rational Mech. Anal., 203 (2012), 189-216.
doi: 10.1007/s00205-011-0446-7. |
[24] |
N. G. Meyers,
An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17 (1963), 189-206.
|
[25] |
G. Mingione,
Gradient estimates below the duality exponent, Ann. Math., 346 (2010), 571-627.
doi: 10.1007/s00208-009-0411-z. |
[26] |
M. V. Safonov,
Harnack inequality for elliptic equations and the Hölder property of their solutions, J. Soviet Math., 21 (1983), 851-863.
|
[27] |
G. Talenti,
Elliptic Equations and Rearrangements, Ann Scuola Norm. Sup. Pisa Cl. Sci., 3 (1976), 697-718.
|
[28] |
B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, 1 edition, Springer-Verlag Berlin Heidelberg, New York, 2000.
doi: 10.1007/BFb0103908.![]() ![]() ![]() |
[29] |
L. Wang,
A geometric approach to the Calderon-Zygmmund estimates, Acta Math. Sin., 19 (2003), 381-396.
doi: 10.1007/s10114-003-0264-4. |
[30] |
C. Zhang and S. Zhou,
Global weighted estimates for quasilinear elliptic equations with nonstandard growth, J. Funct. Anal., 267 (2014), 605-642.
doi: 10.1016/j.jfa.2014.03.022. |
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