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May  2017, 16(3): 855-882. doi: 10.3934/cpaa.2017041

On weighted mixed-norm Sobolev estimates for some basic parabolic equations

Ping Li 1,2, , Pablo Raúl Stinga 3,, and  José L. Torrea 2,

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

* Corresponding author

Received  July 2016 Revised  January 2017 Published  February 2017

Fund Project: The first author was supported by grants 11471251 and 11271293 from National Natural Science Foundation of China. The second and third authors were supported by grant MTM2015-66157-C2-1-P from Government of Spain

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation
$ \partial_tu=\Delta u+f, $
the harmonic oscillator evolution equation
$\partial_tu=\Delta u-|x|^2u+f, $
and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-\Delta)^su=f$ and $(\partial_t-\Delta+|x|^2)^su=f$, for $0 < s < 1$.
Keywords: Heat equation, harmonic oscillator evolution equation, degenerate parabolic extension problem, weighted Sobolev estimate, mixed-norm estimate, language of semigroups.
Mathematics Subject Classification: Primary: 35K10, 35B45, 42B37; Secondary: 58J35, 42B20.
Citation: Ping Li, Pablo Raúl Stinga, José L. Torrea. On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 855-882. doi: 10.3934/cpaa.2017041
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.  Google Scholar

[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.  Google Scholar

[3]

A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math., 88 (1952), 85-139.  doi: 10.1007/BF02392130.  Google Scholar

[4]

A. P. Calderón and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math., 79 (1957), 901-921.  doi: 10.2307/2372441.  Google Scholar

[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.  Google Scholar

[6]

E. B. Fabes, Singular integrals and partial differential equations of parabolic type, Studia Math., 28 (1966), 81-131.   Google Scholar

[7]

E. B. Fabes and C. Sadosky, Pointwise convergence for parabolic singular integrals, Studia Math., 26 (1966), 225-232.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

B. F.Jr Jones, A class of singular integrals, Amer. J. Math., 86 (1964), 441-462.  doi: 10.2307/2373175.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd., 1996.  doi: 10.1142/3302.  Google Scholar
[16] N. N. Lebedev, Special Functions and Their Applications, Prentice-Hall, INC, Englewood Cliffs, N.J., 1965.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[19]

F. J. Ruiz and J. L. Torrea, Parabolic differential equations and vector-valued Fourier analysis, Colloq. Math., 58 (1989), 61-75.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[25] S. Thangavelu, Lectures on Hermite and Laguerre expansions.Mathematical Notes 42,, Princeton University Press, Princeton, NJ, 1993.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math., 88 (1952), 85-139.  doi: 10.1007/BF02392130.  Google Scholar

[4]

A. P. Calderón and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math., 79 (1957), 901-921.  doi: 10.2307/2372441.  Google Scholar

[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.  Google Scholar

[6]

E. B. Fabes, Singular integrals and partial differential equations of parabolic type, Studia Math., 28 (1966), 81-131.   Google Scholar

[7]

E. B. Fabes and C. Sadosky, Pointwise convergence for parabolic singular integrals, Studia Math., 26 (1966), 225-232.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

B. F.Jr Jones, A class of singular integrals, Amer. J. Math., 86 (1964), 441-462.  doi: 10.2307/2373175.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd., 1996.  doi: 10.1142/3302.  Google Scholar
[16] N. N. Lebedev, Special Functions and Their Applications, Prentice-Hall, INC, Englewood Cliffs, N.J., 1965.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[19]

F. J. Ruiz and J. L. Torrea, Parabolic differential equations and vector-valued Fourier analysis, Colloq. Math., 58 (1989), 61-75.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[25] S. Thangavelu, Lectures on Hermite and Laguerre expansions.Mathematical Notes 42,, Princeton University Press, Princeton, NJ, 1993.   Google Scholar
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