Parabolic equations involving Laguerre operators and weighted mixed-norm estimates

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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 December 2020 Early access 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 and 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.
[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.
[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.
[4]	A. Biswas, M. De León-Contreras and P. R. Stinga, Harnack inequalities and Hlöder estimates for master equations, arXiv: 1806.10072.
[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.
[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.
[7]	A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math., 88 (1952), 85-139. doi: 10.1007/BF02392130.
[8]	A. P. Calderón and A. Zygmund, Singular integral operators and differential equations, Am. J. Math., 79 (1957), 901-921. doi: 10.2307/2372441.
[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.
[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.
[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.
[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.
[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.
[14]	N. V. Krylov, The Calderón-Zygmund theorem and its applications to parabolic equations, Algebra i Anali, 13 (2001), 1-25.
[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.
[16]	N. N. Lebedev, Special Functions and Their Applications, Selected Russian Publications in the Mathematical Sciences. Prentice-Hall Inc., Englewood Cliffs (1965).
[17]	B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Am. Math. Soc., 139 (1969), 231-242. doi: 10.2307/1995316.
[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.
[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.
[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.
[21]	K. Stempak, Heat-diffusion and Poission integrals for Laguerre expansions, Tohoku Math. J., 46 (1994), 83-104. doi: 10.2748/tmj/1178225803.
[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, Commun. Partial Differ. Equ., 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, SIAM J. Math. Anal., 49 (2017), 3893-3924. doi: 10.1137/16M1104317.
[25]	G. Szegö, Orthogonal Polynomials, American Mathematical Society, Providence, RI, 1939.
[26]	S. Thangavelu, Lectures on Hermite and Laguerre expansions, Mathematical Notes 42, Princeton University Press, Princeton, NJ, 1993.

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.
[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.
[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.
[4]	A. Biswas, M. De León-Contreras and P. R. Stinga, Harnack inequalities and Hlöder estimates for master equations, arXiv: 1806.10072.
[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.
[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.
[7]	A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math., 88 (1952), 85-139. doi: 10.1007/BF02392130.
[8]	A. P. Calderón and A. Zygmund, Singular integral operators and differential equations, Am. J. Math., 79 (1957), 901-921. doi: 10.2307/2372441.
[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.
[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.
[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.
[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.
[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.
[14]	N. V. Krylov, The Calderón-Zygmund theorem and its applications to parabolic equations, Algebra i Anali, 13 (2001), 1-25.
[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.
[16]	N. N. Lebedev, Special Functions and Their Applications, Selected Russian Publications in the Mathematical Sciences. Prentice-Hall Inc., Englewood Cliffs (1965).
[17]	B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Am. Math. Soc., 139 (1969), 231-242. doi: 10.2307/1995316.
[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.
[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.
[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.
[21]	K. Stempak, Heat-diffusion and Poission integrals for Laguerre expansions, Tohoku Math. J., 46 (1994), 83-104. doi: 10.2748/tmj/1178225803.
[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, Commun. Partial Differ. Equ., 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, SIAM J. Math. Anal., 49 (2017), 3893-3924. doi: 10.1137/16M1104317.
[25]	G. Szegö, Orthogonal Polynomials, American Mathematical Society, Providence, RI, 1939.
[26]	S. Thangavelu, Lectures on Hermite and Laguerre expansions, Mathematical Notes 42, Princeton University Press, Princeton, NJ, 1993.

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