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$L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators
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On nonexistence of solutions to some nonlinear parabolic inequalities
On weighted mixed-norm Sobolev estimates for some basic parabolic equations
1. | School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China |
2. | Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain |
3. | Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011, USA |
$ \partial_tu=\Delta u+f, $ |
$\partial_tu=\Delta u-|x|^2u+f, $ |
References:
[1] |
A. Bernardis, F. J. Martín-Reyes, P. R. Stinga and J. L. Torrea,
Maximum principles, extension problem and inversion for nonlocal one-sided equations, J. Differential Equations, 260 (2016), 6333-6362.
doi: 10.1016/j.jde.2015.12.042. |
[2] |
L. A. Caffarelli and P. R. Stinga,
Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincarė Anal. Non Linéaire, 33 (2016), 767-807.
doi: 10.1016/j.anihpc.2015.01.004. |
[3] |
A. P. Calderón and A. Zygmund,
On the existence of certain singular integrals, Acta Math., 88 (1952), 85-139.
doi: 10.1007/BF02392130. |
[4] |
A. P. Calderón and A. Zygmund,
Singular integral operators and differential equations, Amer. J. Math., 79 (1957), 901-921.
doi: 10.2307/2372441. |
[5] |
J. Duoandikoetxea, Fourier Analysis, translated and revised from the 1995 Spanish original by David Cruz-Uribe, Graduate Studies in Mathematics 29, Amer. Math. Soc., Providence, RI, 2001. |
[6] |
E. B. Fabes,
Singular integrals and partial differential equations of parabolic type, Studia Math., 28 (1966), 81-131.
|
[7] |
E. B. Fabes and C. Sadosky,
Pointwise convergence for parabolic singular integrals, Studia Math., 26 (1966), 225-232.
|
[8] |
J. E. Galé, P. J. Miana and P. R. Stinga,
Extension problem and fractional operators: semigroups and wave equations, J. Evol. Equ., 13 (2013), 343-368.
doi: 10.1007/s00028-013-0182-6. |
[9] |
R. Haller-Dintelmann, H. Heck and M. Hieber,
$L^p-L^q$ estimates for parabolic systems in non-divergence form with VMO coefficients, J. London Math. Soc., 74 (2006), 717-736.
|
[10] |
B. F.Jr Jones,
A class of singular integrals, Amer. J. Math., 86 (1964), 441-462.
doi: 10.2307/2373175. |
[11] |
M. Kemppainen, P. Sjögren and J. L. Torrea,
Wave extension problem for the fractional Laplacian, Discrete Contin. Dyn. Syst., 35 (2015), 4905-4929.
doi: 10.3934/dcds.2015.35.4905. |
[12] |
N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics 12, American Mathematical Society, Providence, R.I., 1996.
doi: 10.1090/gsm/012. |
[13] |
N. V. Krylov, The Calderón-Zygmund theorem and its applications to parabolic equations, (Russian) Algebra i Analiz, 13 (2001), 1-25; translation in St. Petersburg Math. J., 13 (2002), 509-526. |
[14] |
N. V. Krylov,
The Calderón-Zygmund theorem and parabolic equations in $L_p(\mathbb{R}.C.^{2+\alpha})$-spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci, I (2002), 799-820.
|
[15] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd., 1996.
doi: 10.1142/3302.![]() ![]() ![]() |
[16] |
N. N. Lebedev, Special Functions and Their Applications, Prentice-Hall, INC, Englewood Cliffs, N.J., 1965.
![]() ![]() |
[17] |
R. A. Macías, C. Segovia and J. L. Torrea,
Singular integral operators with non-necessarily bounded kernels on spaces of homogeneous type, Adv. Math., 93 (1992), 25-60.
doi: 10.1016/0001-8708(92)90024-F. |
[18] |
J.L. Rubio de Francia, F. J. Ruiz and J. L. Torrea,
Calderón-Zygmund theory for operator-valued kernels, Adv. in Math., 62 (1986), 7-48.
doi: 10.1016/0001-8708(86)90086-1. |
[19] |
F. J. Ruiz and J. L. Torrea,
Parabolic differential equations and vector-valued Fourier analysis, Colloq. Math., 58 (1989), 61-75.
|
[20] |
F. J. Ruiz and J. L. Torrea,
Vector-valued Calderón-Zygmund theory and Carleson measures on spaces of homogeneous nature, Studia Math., 88 (1988), 221-243.
|
[21] |
E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory.Annals of Mathematics Studies 63,, Princeton University Press, Princeton, N.J., 1970.
![]() ![]() |
[22] |
K. Stempak and J. L. Torrea,
Poisson integrals and Riesz transforms for Hermite function expansions with weights, J. Funct. Anal., 202 (2003), 443-472.
doi: 10.1016/S0022-1236(03)00083-1. |
[23] |
P. R. Stinga and J. L. Torrea,
Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.
doi: 10.1080/03605301003735680. |
[24] |
P. R. Stinga and J. L. Torrea, Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation, preprint, arXiv: 1511.01945. |
[25] |
S. Thangavelu, Lectures on Hermite and Laguerre expansions.Mathematical Notes 42,, Princeton University Press, Princeton, NJ, 1993.
![]() ![]() |
show all references
References:
[1] |
A. Bernardis, F. J. Martín-Reyes, P. R. Stinga and J. L. Torrea,
Maximum principles, extension problem and inversion for nonlocal one-sided equations, J. Differential Equations, 260 (2016), 6333-6362.
doi: 10.1016/j.jde.2015.12.042. |
[2] |
L. A. Caffarelli and P. R. Stinga,
Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincarė Anal. Non Linéaire, 33 (2016), 767-807.
doi: 10.1016/j.anihpc.2015.01.004. |
[3] |
A. P. Calderón and A. Zygmund,
On the existence of certain singular integrals, Acta Math., 88 (1952), 85-139.
doi: 10.1007/BF02392130. |
[4] |
A. P. Calderón and A. Zygmund,
Singular integral operators and differential equations, Amer. J. Math., 79 (1957), 901-921.
doi: 10.2307/2372441. |
[5] |
J. Duoandikoetxea, Fourier Analysis, translated and revised from the 1995 Spanish original by David Cruz-Uribe, Graduate Studies in Mathematics 29, Amer. Math. Soc., Providence, RI, 2001. |
[6] |
E. B. Fabes,
Singular integrals and partial differential equations of parabolic type, Studia Math., 28 (1966), 81-131.
|
[7] |
E. B. Fabes and C. Sadosky,
Pointwise convergence for parabolic singular integrals, Studia Math., 26 (1966), 225-232.
|
[8] |
J. E. Galé, P. J. Miana and P. R. Stinga,
Extension problem and fractional operators: semigroups and wave equations, J. Evol. Equ., 13 (2013), 343-368.
doi: 10.1007/s00028-013-0182-6. |
[9] |
R. Haller-Dintelmann, H. Heck and M. Hieber,
$L^p-L^q$ estimates for parabolic systems in non-divergence form with VMO coefficients, J. London Math. Soc., 74 (2006), 717-736.
|
[10] |
B. F.Jr Jones,
A class of singular integrals, Amer. J. Math., 86 (1964), 441-462.
doi: 10.2307/2373175. |
[11] |
M. Kemppainen, P. Sjögren and J. L. Torrea,
Wave extension problem for the fractional Laplacian, Discrete Contin. Dyn. Syst., 35 (2015), 4905-4929.
doi: 10.3934/dcds.2015.35.4905. |
[12] |
N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics 12, American Mathematical Society, Providence, R.I., 1996.
doi: 10.1090/gsm/012. |
[13] |
N. V. Krylov, The Calderón-Zygmund theorem and its applications to parabolic equations, (Russian) Algebra i Analiz, 13 (2001), 1-25; translation in St. Petersburg Math. J., 13 (2002), 509-526. |
[14] |
N. V. Krylov,
The Calderón-Zygmund theorem and parabolic equations in $L_p(\mathbb{R}.C.^{2+\alpha})$-spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci, I (2002), 799-820.
|
[15] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd., 1996.
doi: 10.1142/3302.![]() ![]() ![]() |
[16] |
N. N. Lebedev, Special Functions and Their Applications, Prentice-Hall, INC, Englewood Cliffs, N.J., 1965.
![]() ![]() |
[17] |
R. A. Macías, C. Segovia and J. L. Torrea,
Singular integral operators with non-necessarily bounded kernels on spaces of homogeneous type, Adv. Math., 93 (1992), 25-60.
doi: 10.1016/0001-8708(92)90024-F. |
[18] |
J.L. Rubio de Francia, F. J. Ruiz and J. L. Torrea,
Calderón-Zygmund theory for operator-valued kernels, Adv. in Math., 62 (1986), 7-48.
doi: 10.1016/0001-8708(86)90086-1. |
[19] |
F. J. Ruiz and J. L. Torrea,
Parabolic differential equations and vector-valued Fourier analysis, Colloq. Math., 58 (1989), 61-75.
|
[20] |
F. J. Ruiz and J. L. Torrea,
Vector-valued Calderón-Zygmund theory and Carleson measures on spaces of homogeneous nature, Studia Math., 88 (1988), 221-243.
|
[21] |
E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory.Annals of Mathematics Studies 63,, Princeton University Press, Princeton, N.J., 1970.
![]() ![]() |
[22] |
K. Stempak and J. L. Torrea,
Poisson integrals and Riesz transforms for Hermite function expansions with weights, J. Funct. Anal., 202 (2003), 443-472.
doi: 10.1016/S0022-1236(03)00083-1. |
[23] |
P. R. Stinga and J. L. Torrea,
Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.
doi: 10.1080/03605301003735680. |
[24] |
P. R. Stinga and J. L. Torrea, Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation, preprint, arXiv: 1511.01945. |
[25] |
S. Thangavelu, Lectures on Hermite and Laguerre expansions.Mathematical Notes 42,, Princeton University Press, Princeton, NJ, 1993.
![]() ![]() |
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