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Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem
Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials
1. | S.M. Nikolskii Institute of Mathematics at RUDN University, 117198 Moscow, Russia |
2. | Department of Mathematics, Dumlupinar University, 43100 Kutahya, Turkey |
3. | Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ1141 Baku, Azerbaijan |
4. | Baku State University, AZ1141 Baku, Azerbaijan |
5. | Dipartimento di Matematica e Informatica, Universita di Catania, Catania, Italy |
We establish the boundedness of some Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.
References:
[1] |
A. Akbulut, V. S. Guliyev and M. N. Omarova, Marcinkiewicz integrals associated with Schrödinger operators and their commutators on vanishing generalized Morrey spaces, Bound. Value Probl., (2017), Paper No. 121, 16 pp.
doi: 10.1186/s13661-017-0851-4. |
[2] |
J. Alvarez, J. Lakey and M. Guzman-Partida,
Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures, Collect. Math., 51 (2000), 1-47.
|
[3] |
A. S. Berdyshev, B. J. Kadirkulov and J. J. Nieto,
Solvability of an elliptic partial differential equation with boundary condition involving fractional derivatives, Complex Var. Elliptic Equ., 59 (2014), 680-692.
doi: 10.1080/17476933.2013.777711. |
[4] |
B. Bongioanni, E. Harboure and O. Salinas,
Commutators of Riesz transforms related to Schödinger operators, J. Fourier Anal. Appl., 17 (2011), 115-134.
doi: 10.1007/s00041-010-9133-6. |
[5] |
T. Bui,
Weighted estimates for commutators of some singular integrals related to Schrödinger operators, Bull. Sci. Math., 138 (2014), 270-292.
doi: 10.1016/j.bulsci.2013.06.007. |
[6] |
D. Chen and L. Song,
The boundedness of the commutator for Riesz potential associated with Schrödinger operator on Morrey spaces, Anal. Theory Appl., 30 (2014), 363-368.
doi: 10.4208/ata.2014.v30.n4.3. |
[7] |
F. Chiarenza and M. Frasca,
Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl., 7 (1987), 273-279.
|
[8] |
D. Cruz-Uribe and A. Fiorenza,
Endpoint estimates and weighted norm inequalities for commutators of fractional integrals, Publ. Mat., 47 (2003), 103-131.
doi: 10.5565/PUBLMAT_47103_05. |
[9] |
G. Di Fazio and M. A. Ragusa,
Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal., 112 (1993), 241-256.
doi: 10.1006/jfan.1993.1032. |
[10] |
Y. Ding, S. Z. Lu and P. Zhang,
Weak estimates for commutators of fractional integral operators, Sci. China Ser. A, 44 (2001), 877-887.
doi: 10.1007/BF02880137. |
[11] |
D. Fan, S. Lu and D. Yang,
Boundedness of operators in Morrey spaces on homogeneous spaces and its applications, Acta Math. Sinica (N.S.), 14 (1998), 625-634.
|
[12] |
V. S. Guliyev, Integral Operators on Function Spaces on the Homogeneous Groups and on Domains in $\mathbb{R}^n$, Doctoral dissertation, Mat. Inst. Steklov, Moscow, 1994. Google Scholar |
[13] |
V. S. Guliyev, Function Spaces, Integral Operators and two Weighted Inequalities on Homogeneous Groups. Some Applications, Baku, Elm, 1999. Google Scholar |
[14] |
V. S. Guliyev, Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl., (2009), Art. ID 503948, 20 pp. |
[15] |
V. S. Guliyev,
Local generalized Morrey spaces and singular integrals with rough kernel, Azerb. J. Math., 3 (2013), 79-94.
|
[16] |
V. S. Guliyev,
Generalized local Morrey spaces and fractional integral operators with rough kernel, J. Math. Sci. (N.Y.), 193 (2013), 211-227.
doi: 10.1007/s10958-013-1448-9. |
[17] |
V. S. Guliyev,
Function spaces and integral operators associated with Schrödinger operators: An overview, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 40 (2014), 178-202.
|
[18] |
V. S. Guliyev, M. N. Omarova, M. A. Ragusa and A. Scapellato,
Commutators and generalized local Morrey spaces, J. Math. Anal. Appl., 457 (2018), 1388-1402.
doi: 10.1016/j.jmaa.2016.09.070. |
[19] |
V. S. Guliyev and L. Softova,
Global regularity in generalized Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients, Potential Anal., 38 (2013), 843-862.
doi: 10.1007/s11118-012-9299-4. |
[20] |
V. S. Guliyev and L. Softova,
Generalized Morrey estimates for the gradient of divergence form parabolic operators with discontinuous coefficients, J. Differential Equations, 259 (2015), 2368-2387.
doi: 10.1016/j.jde.2015.03.032. |
[21] |
T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (Sendai, 1990), ICM 90 Satellite Proceedings, Springer-Verlag, Tokyo, 1991,183–189. |
[22] |
C. Morrey,
On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166.
doi: 10.1090/S0002-9947-1938-1501936-8. |
[23] |
E. Nakai,
Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr., 166 (1994), 95-103.
doi: 10.1002/mana.19941660108. |
[24] |
J. J. Nieto and C. C. Tisdell,
Existence and uniqueness of solutions to first-order systems of nonlinear impulsive boundary-value problems with sub-, super-linear or linear growth, Electron. J. Differential Equations, 2007 (2007), 1-14.
|
[25] |
M. Otelbaev,
Imbedding theorems for spaces with weight and their use in investigating the spectrum of the Schrödinger operator, Trudy Mat. Inst. Steklov., 150 (1979), 265-305.
|
[26] |
M. Otelbaev, Spectrum Estimates of the Sturm-Liouville Operator, Alma Ata, Gylym, 1990. Google Scholar |
[27] |
C. Perez,
Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., 128 (1995), 163-185.
doi: 10.1006/jfan.1995.1027. |
[28] |
S. Polidoro and M. A. Ragusa,
Hölder regularity for solutions of ultraparabolic equations in divergence form, Potential Anal., 14 (2001), 341-350.
doi: 10.1023/A:1011261019736. |
[29] |
M. A. Ragusa,
Local Hölder regularity for solutions of elliptic systems, Duke Math. J., 113 (2002), 385-397.
doi: 10.1215/S0012-7094-02-11327-1. |
[30] |
M. A. Ragusa and A. Tachikawa,
On continuity of minimizers for certain quadratic growth functionals, J. Math. Soc. Japan, 57 (2005), 691-700.
doi: 10.2969/jmsj/1158241929. |
[31] |
Z. Shen,
$L_p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble), 45 (1995), 513-546.
doi: 10.5802/aif.1463. |
[32] |
L. Softova,
Singular integrals and commutators in generalized Morrey spaces, Acta Math. Sin. (Engl. Ser.), 22 (2006), 757-766.
doi: 10.1007/s10114-005-0628-z. |
[33] |
E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993. |
[34] |
L. Tang and J. Dong,
Boundedness for some Schrödinger type operator on Morrey spaces related to certain nonnegative potentials, J. Math. Anal. Appl., 355 (2009), 101-109.
doi: 10.1016/j.jmaa.2009.01.043. |
[35] |
R. Wheeden and A. Zygmund, Measure and Integral, An Introduction to Real Analysis, Pure and Applied Mathematics, 43, Marcel Dekker, Inc., New York-Basel, 1977. |
[36] |
H. K. Xu and J. J. Nieto,
Extremal solutions of a class of nonlinear integro-differential equations in Banach spaces, Proc. Amer. Math. Soc., 125 (1997), 2605-2614.
doi: 10.1090/S0002-9939-97-04149-X. |
[37] |
J. Zhong, Harmonic Analysis for some Schrödinger Type Operators, Ph.D. thesis, Princeton University, 1993. |
show all references
References:
[1] |
A. Akbulut, V. S. Guliyev and M. N. Omarova, Marcinkiewicz integrals associated with Schrödinger operators and their commutators on vanishing generalized Morrey spaces, Bound. Value Probl., (2017), Paper No. 121, 16 pp.
doi: 10.1186/s13661-017-0851-4. |
[2] |
J. Alvarez, J. Lakey and M. Guzman-Partida,
Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures, Collect. Math., 51 (2000), 1-47.
|
[3] |
A. S. Berdyshev, B. J. Kadirkulov and J. J. Nieto,
Solvability of an elliptic partial differential equation with boundary condition involving fractional derivatives, Complex Var. Elliptic Equ., 59 (2014), 680-692.
doi: 10.1080/17476933.2013.777711. |
[4] |
B. Bongioanni, E. Harboure and O. Salinas,
Commutators of Riesz transforms related to Schödinger operators, J. Fourier Anal. Appl., 17 (2011), 115-134.
doi: 10.1007/s00041-010-9133-6. |
[5] |
T. Bui,
Weighted estimates for commutators of some singular integrals related to Schrödinger operators, Bull. Sci. Math., 138 (2014), 270-292.
doi: 10.1016/j.bulsci.2013.06.007. |
[6] |
D. Chen and L. Song,
The boundedness of the commutator for Riesz potential associated with Schrödinger operator on Morrey spaces, Anal. Theory Appl., 30 (2014), 363-368.
doi: 10.4208/ata.2014.v30.n4.3. |
[7] |
F. Chiarenza and M. Frasca,
Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl., 7 (1987), 273-279.
|
[8] |
D. Cruz-Uribe and A. Fiorenza,
Endpoint estimates and weighted norm inequalities for commutators of fractional integrals, Publ. Mat., 47 (2003), 103-131.
doi: 10.5565/PUBLMAT_47103_05. |
[9] |
G. Di Fazio and M. A. Ragusa,
Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal., 112 (1993), 241-256.
doi: 10.1006/jfan.1993.1032. |
[10] |
Y. Ding, S. Z. Lu and P. Zhang,
Weak estimates for commutators of fractional integral operators, Sci. China Ser. A, 44 (2001), 877-887.
doi: 10.1007/BF02880137. |
[11] |
D. Fan, S. Lu and D. Yang,
Boundedness of operators in Morrey spaces on homogeneous spaces and its applications, Acta Math. Sinica (N.S.), 14 (1998), 625-634.
|
[12] |
V. S. Guliyev, Integral Operators on Function Spaces on the Homogeneous Groups and on Domains in $\mathbb{R}^n$, Doctoral dissertation, Mat. Inst. Steklov, Moscow, 1994. Google Scholar |
[13] |
V. S. Guliyev, Function Spaces, Integral Operators and two Weighted Inequalities on Homogeneous Groups. Some Applications, Baku, Elm, 1999. Google Scholar |
[14] |
V. S. Guliyev, Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl., (2009), Art. ID 503948, 20 pp. |
[15] |
V. S. Guliyev,
Local generalized Morrey spaces and singular integrals with rough kernel, Azerb. J. Math., 3 (2013), 79-94.
|
[16] |
V. S. Guliyev,
Generalized local Morrey spaces and fractional integral operators with rough kernel, J. Math. Sci. (N.Y.), 193 (2013), 211-227.
doi: 10.1007/s10958-013-1448-9. |
[17] |
V. S. Guliyev,
Function spaces and integral operators associated with Schrödinger operators: An overview, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 40 (2014), 178-202.
|
[18] |
V. S. Guliyev, M. N. Omarova, M. A. Ragusa and A. Scapellato,
Commutators and generalized local Morrey spaces, J. Math. Anal. Appl., 457 (2018), 1388-1402.
doi: 10.1016/j.jmaa.2016.09.070. |
[19] |
V. S. Guliyev and L. Softova,
Global regularity in generalized Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients, Potential Anal., 38 (2013), 843-862.
doi: 10.1007/s11118-012-9299-4. |
[20] |
V. S. Guliyev and L. Softova,
Generalized Morrey estimates for the gradient of divergence form parabolic operators with discontinuous coefficients, J. Differential Equations, 259 (2015), 2368-2387.
doi: 10.1016/j.jde.2015.03.032. |
[21] |
T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (Sendai, 1990), ICM 90 Satellite Proceedings, Springer-Verlag, Tokyo, 1991,183–189. |
[22] |
C. Morrey,
On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166.
doi: 10.1090/S0002-9947-1938-1501936-8. |
[23] |
E. Nakai,
Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr., 166 (1994), 95-103.
doi: 10.1002/mana.19941660108. |
[24] |
J. J. Nieto and C. C. Tisdell,
Existence and uniqueness of solutions to first-order systems of nonlinear impulsive boundary-value problems with sub-, super-linear or linear growth, Electron. J. Differential Equations, 2007 (2007), 1-14.
|
[25] |
M. Otelbaev,
Imbedding theorems for spaces with weight and their use in investigating the spectrum of the Schrödinger operator, Trudy Mat. Inst. Steklov., 150 (1979), 265-305.
|
[26] |
M. Otelbaev, Spectrum Estimates of the Sturm-Liouville Operator, Alma Ata, Gylym, 1990. Google Scholar |
[27] |
C. Perez,
Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., 128 (1995), 163-185.
doi: 10.1006/jfan.1995.1027. |
[28] |
S. Polidoro and M. A. Ragusa,
Hölder regularity for solutions of ultraparabolic equations in divergence form, Potential Anal., 14 (2001), 341-350.
doi: 10.1023/A:1011261019736. |
[29] |
M. A. Ragusa,
Local Hölder regularity for solutions of elliptic systems, Duke Math. J., 113 (2002), 385-397.
doi: 10.1215/S0012-7094-02-11327-1. |
[30] |
M. A. Ragusa and A. Tachikawa,
On continuity of minimizers for certain quadratic growth functionals, J. Math. Soc. Japan, 57 (2005), 691-700.
doi: 10.2969/jmsj/1158241929. |
[31] |
Z. Shen,
$L_p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble), 45 (1995), 513-546.
doi: 10.5802/aif.1463. |
[32] |
L. Softova,
Singular integrals and commutators in generalized Morrey spaces, Acta Math. Sin. (Engl. Ser.), 22 (2006), 757-766.
doi: 10.1007/s10114-005-0628-z. |
[33] |
E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993. |
[34] |
L. Tang and J. Dong,
Boundedness for some Schrödinger type operator on Morrey spaces related to certain nonnegative potentials, J. Math. Anal. Appl., 355 (2009), 101-109.
doi: 10.1016/j.jmaa.2009.01.043. |
[35] |
R. Wheeden and A. Zygmund, Measure and Integral, An Introduction to Real Analysis, Pure and Applied Mathematics, 43, Marcel Dekker, Inc., New York-Basel, 1977. |
[36] |
H. K. Xu and J. J. Nieto,
Extremal solutions of a class of nonlinear integro-differential equations in Banach spaces, Proc. Amer. Math. Soc., 125 (1997), 2605-2614.
doi: 10.1090/S0002-9939-97-04149-X. |
[37] |
J. Zhong, Harmonic Analysis for some Schrödinger Type Operators, Ph.D. thesis, Princeton University, 1993. |
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