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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  February 2020 Early access  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 and 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.

[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.

[13]

V. S. Guliyev, Function Spaces, Integral Operators and two Weighted Inequalities on Homogeneous Groups. Some Applications, Baku, Elm, 1999.

[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.

[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.

[13]

V. S. Guliyev, Function Spaces, Integral Operators and two Weighted Inequalities on Homogeneous Groups. Some Applications, Baku, Elm, 1999.

[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.

[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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