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February  2020, 25(2): 671-690. doi: 10.3934/dcdsb.2019260

Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials

Vagif S. Guliyev 1,2,3, , Ramin V. Guliyev 2, , Mehriban N. Omarova 3,4, and  Maria Alessandra Ragusa 1,5,,

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

* Corresponding author: maragusa@dmi.unict.it (Maria Alessandra Ragusa)

In honor of the 60-th birthday of Juan J. Nieto
The research of V.S. Guliyev and M. Omarova was partially supported by the grant of 1st Azerbaijan Russia Joint Grant Competition (Grant No. EIF-BGM-4-RFTF-1/2017-21/01/1).
The research of V.S. Guliyev and M.A. Ragusa are partially supported by the Ministry of Education and Science of the Russian Federation (Agreement number N. 02.03.21.0008)

Received  February 2018 Revised  October 2018 Published  November 2019

Fund Project: The research of V.S. Guliyev and M.A. Ragusa are partially supported by the Ministry of Education and Science of the Russian Federation (Agreement number N. 02.03.21.0008).

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.

Keywords: Fractional integral operator, Schrödinger differential operator, fractional calculus.
Mathematics Subject Classification: Primary: 42B20, 42B25, 42B35; Secondary: 35J10, 47H50.
Citation: Vagif S. Guliyev, Ramin V. Guliyev, Mehriban N. Omarova, Maria Alessandra Ragusa. Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 671-690. doi: 10.3934/dcdsb.2019260
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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl., 7 (1987), 273-279.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

V. S. Guliyev, Local generalized Morrey spaces and singular integrals with rough kernel, Azerb. J. Math., 3 (2013), 79-94.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

J. Zhong, Harmonic Analysis for some Schrödinger Type Operators, Ph.D. thesis, Princeton University, 1993.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl., 7 (1987), 273-279.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

V. S. Guliyev, Local generalized Morrey spaces and singular integrals with rough kernel, Azerb. J. Math., 3 (2013), 79-94.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

J. Zhong, Harmonic Analysis for some Schrödinger Type Operators, Ph.D. thesis, Princeton University, 1993.  Google Scholar

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