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Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry
Parabolic equations involving Laguerre operators and weighted mixed-norm estimates
1. | School of Mathematical Science, Zhejiang University, Hangzhou 310027, China |
2. | School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China |
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.
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. 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. |
[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. 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. |
[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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