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December  2020, 19(12): 5487-5508. doi: 10.3934/cpaa.2020249

Parabolic equations involving Laguerre operators and weighted mixed-norm estimates

Huiying Fan 1,, and  Tao Ma 2,

1. 

School of Mathematical Science, Zhejiang University, Hangzhou 310027, China
2. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

* Corresponding author

Received  March 2020 Revised  June 2020 Published  October 2020

Fund Project: The second author was supported by National Natural Science Foundation of China (Grant Nos. 11671308, 11971431)

In this paper, we study evolution equation $ \partial_t u = -L_\alpha u+f $ and the corresponding Cauchy problem, where $ L_\alpha $ represents the Laguerre operator $ L_\alpha = \frac 12(-\frac{d^2}{dx^2}+x^2+\frac 1{x^2}(\alpha^2-\frac 14)) $, for every $ \alpha\geq-\frac 12 $. We get explicit pointwise formulas for the classical solution and its derivatives by virtue of the parabolic heat-diffusion semigroup $ \{ e^{-\tau(\partial_t+L_\alpha)}\}_{\tau>0} $. In addition, we define the Poisson operator related to the fractional power $ (\partial_t+L_\alpha)^s $ and reveal weighted mixed-norm estimates for revelent maximal operators.

Keywords: Parabolic evolution equation, parabolic heat semigroups, Laguerre operator, mixed-norm estimates.
Mathematics Subject Classification: Primary: 35C15, 47D03; Secondary: 42B25.
Citation: Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249
References:
[1]

J. J. Betancor, A. J. Castro, J. C. Fari na and L. Rodríguez-Mesa, Conical square functions associated with Bessel, Laguerre and Schrödinger operators in UMD Banach spaces, J. Math. Anal. Appl., 447 (2017), 32-75.  doi: 10.1016/j.jmaa.2016.10.006.  Google Scholar

[2]

J. J. Betancor, R. Crescimbeni and J. L. Torrea, Oscillation and variation of the Laguerre heat and Poisson semigroups and Riesz transforms, Acta Math. Sci. Ser. B (Engl. Ed.), 32 (2012), 907-928.  doi: 10.1016/S0252-9602(12)60069-1.  Google Scholar

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J. J. Betancor and M. De León-Contreras, Parabolic equations involving Bessel operators and singular integrals, Integral Equ. Oper. Theory, 90 (2018), 18-58.  doi: 10.1007/s00020-018-2444-8.  Google Scholar

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A. Biswas, M. De León-Contreras and P. R. Stinga, Harnack inequalities and Hlöder estimates for master equations, arXiv: 1806.10072. Google Scholar

[5]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

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L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 767-807.  doi: 10.1016/j.anihpc.2015.01.004.  Google Scholar

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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

[8]

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

[9]

A. J. Castro, K. Nyström and O. Sande, Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients, Calc. Var. Partial Differ. Equ., 55 (2016), 49 pp. doi: 10.1007/s00526-016-1058-8.  Google Scholar

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R. R. Coifman and G. Weiss, Analyse Harmonique Non-commutative Sur Certains Espaces Homogènes, Lecture Notes in Math., Vol. 242 Springer-Verlag, Berlin, 1971.  Google Scholar

[11]

E. B. Fabes and C. Sadosky, Pointwise convergence for parabolic singular integrals, Studia Math., 26 (1966), 225-232.  doi: 10.4064/sm-26-3-225-232.  Google Scholar

[12]

C.E. Gutiérrez, A. Incognito and J. L. Torrea, Riesz transforms, $g$-functions and multipliers for the Laguerre semigroup, Houston J. Math., 27 (2001), 579-592.   Google Scholar

[13]

B. F. Jones, Singular integrals and parabolic equations, Bull. Am. Math. Soc., 69 (1963), 501-503.  doi: 10.1090/S0002-9904-1963-10977-5.  Google Scholar

[14]

N. V. Krylov, The Calderón-Zygmund theorem and its applications to parabolic equations, Algebra i Anali, 13 (2001), 1-25.   Google Scholar

[15]

N. V. Krylov, The Calderón-Zygmund theorem and parabolic equations in $L^p(\mathbb{R}, C^{2+d})$-spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., 1 (2002), 799-820.   Google Scholar

[16]

N. N. Lebedev, Special Functions and Their Applications, Selected Russian Publications in the Mathematical Sciences. Prentice-Hall Inc., Englewood Cliffs (1965).  Google Scholar

[17]

B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Am. Math. Soc., 139 (1969), 231-242.  doi: 10.2307/1995316.  Google Scholar

[18]

K. Nyström, $L^2$ Solvability of boundary value problems for divergence form parabolic equations with complex coefficients, J. Differ. Equ., 262 (2017), 2808-2939.  doi: 10.1016/j.jde.2016.11.011.  Google Scholar

[19]

L. Ping, P. R. Stinga and J. L. Torrea, On weighted mixed-norm Sobolev estimates for some basic parabolic equations, Commun. Pure Appl. Anal., 16 (2017), 855-882.  doi: 10.3934/cpaa.2017041.  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.  doi: 10.4064/sm-88-3-221-243.  Google Scholar

[21]

K. Stempak, Heat-diffusion and Poission integrals for Laguerre expansions, Tohoku Math. J., 46 (1994), 83-104.  doi: 10.2748/tmj/1178225803.  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, Commun. Partial Differ. Equ., 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, SIAM J. Math. Anal., 49 (2017), 3893-3924.  doi: 10.1137/16M1104317.  Google Scholar

[25]

G. Szegö, Orthogonal Polynomials, American Mathematical Society, Providence, RI, 1939.  Google Scholar

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

show all references

References:
[1]

J. J. Betancor, A. J. Castro, J. C. Fari na and L. Rodríguez-Mesa, Conical square functions associated with Bessel, Laguerre and Schrödinger operators in UMD Banach spaces, J. Math. Anal. Appl., 447 (2017), 32-75.  doi: 10.1016/j.jmaa.2016.10.006.  Google Scholar

[2]

J. J. Betancor, R. Crescimbeni and J. L. Torrea, Oscillation and variation of the Laguerre heat and Poisson semigroups and Riesz transforms, Acta Math. Sci. Ser. B (Engl. Ed.), 32 (2012), 907-928.  doi: 10.1016/S0252-9602(12)60069-1.  Google Scholar

[3]

J. J. Betancor and M. De León-Contreras, Parabolic equations involving Bessel operators and singular integrals, Integral Equ. Oper. Theory, 90 (2018), 18-58.  doi: 10.1007/s00020-018-2444-8.  Google Scholar

[4]

A. Biswas, M. De León-Contreras and P. R. Stinga, Harnack inequalities and Hlöder estimates for master equations, arXiv: 1806.10072. Google Scholar

[5]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[6]

L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 767-807.  doi: 10.1016/j.anihpc.2015.01.004.  Google Scholar

[7]

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

[8]

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

[9]

A. J. Castro, K. Nyström and O. Sande, Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients, Calc. Var. Partial Differ. Equ., 55 (2016), 49 pp. doi: 10.1007/s00526-016-1058-8.  Google Scholar

[10]

R. R. Coifman and G. Weiss, Analyse Harmonique Non-commutative Sur Certains Espaces Homogènes, Lecture Notes in Math., Vol. 242 Springer-Verlag, Berlin, 1971.  Google Scholar

[11]

E. B. Fabes and C. Sadosky, Pointwise convergence for parabolic singular integrals, Studia Math., 26 (1966), 225-232.  doi: 10.4064/sm-26-3-225-232.  Google Scholar

[12]

C.E. Gutiérrez, A. Incognito and J. L. Torrea, Riesz transforms, $g$-functions and multipliers for the Laguerre semigroup, Houston J. Math., 27 (2001), 579-592.   Google Scholar

[13]

B. F. Jones, Singular integrals and parabolic equations, Bull. Am. Math. Soc., 69 (1963), 501-503.  doi: 10.1090/S0002-9904-1963-10977-5.  Google Scholar

[14]

N. V. Krylov, The Calderón-Zygmund theorem and its applications to parabolic equations, Algebra i Anali, 13 (2001), 1-25.   Google Scholar

[15]

N. V. Krylov, The Calderón-Zygmund theorem and parabolic equations in $L^p(\mathbb{R}, C^{2+d})$-spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., 1 (2002), 799-820.   Google Scholar

[16]

N. N. Lebedev, Special Functions and Their Applications, Selected Russian Publications in the Mathematical Sciences. Prentice-Hall Inc., Englewood Cliffs (1965).  Google Scholar

[17]

B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Am. Math. Soc., 139 (1969), 231-242.  doi: 10.2307/1995316.  Google Scholar

[18]

K. Nyström, $L^2$ Solvability of boundary value problems for divergence form parabolic equations with complex coefficients, J. Differ. Equ., 262 (2017), 2808-2939.  doi: 10.1016/j.jde.2016.11.011.  Google Scholar

[19]

L. Ping, P. R. Stinga and J. L. Torrea, On weighted mixed-norm Sobolev estimates for some basic parabolic equations, Commun. Pure Appl. Anal., 16 (2017), 855-882.  doi: 10.3934/cpaa.2017041.  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.  doi: 10.4064/sm-88-3-221-243.  Google Scholar

[21]

K. Stempak, Heat-diffusion and Poission integrals for Laguerre expansions, Tohoku Math. J., 46 (1994), 83-104.  doi: 10.2748/tmj/1178225803.  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, Commun. Partial Differ. Equ., 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, SIAM J. Math. Anal., 49 (2017), 3893-3924.  doi: 10.1137/16M1104317.  Google Scholar

[25]

G. Szegö, Orthogonal Polynomials, American Mathematical Society, Providence, RI, 1939.  Google Scholar

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