This paper establishes the boundedness of fractional integral operators associated with Schr?dinger operators and their commutators with
functions on generalized fractional mixed Morrey spaces. This is achieved by employing function decomposition and stratification techniques in these spaces, while drawing on known boundedness results in the context of mixed Lebesgue spaces.
References
[1]
Fefferman, C.L. (1983) The Uncertainty Principle. Bulletin of the American MathematicalSociety, 9, 129-206. https://doi.org/10.1090/s0273-0979-1983-15154-6
[2]
Sen, Z.W. (1995) Estimates for Schrödinger Operators with Certain Potentials. Annales de l’Institut Fourier, 45, 513-546.
[3]
Zhong, J.P. (1999) The Sobolev Estimates for Some Schrödinger Type Operators. Mathematical Sciences Research Hot-Line, 3, 1-48.
[4]
Yang, D.C., Yang, D.Y. and Zhou, Y. (2009) Endpoint Properties of Localized Riesz Transforms and Fractional Integrals Associated to Schrödinger Operators. PotentialAnalysis, 30, 271-300. https://doi.org/10.1007/s11118-009-9116-x
[5]
Guliyev, V.S., Akbulut, A. and Celik, S. (2024) Fractional Integral Related to Schrödinger Operator on Vanishing Generalized Mixed Morrey Spaces. Boundary Value Problems, 2024, 1-17. https://doi.org/10.1186/s13661-024-01950-3
[6]
Xue, Q.Y. and Yu, J.L. (2025) Marcinkiewicz Integrals Associated with Schrödinger Operator on Campanato Type Spaces. Frontiers of Mathematics, 20, 1201-1237. https://doi.org/10.1007/s11464-024-0062-0
[7]
Morrey, C.B. (1938) On the Solutions of Quasi-Linear Elliptic Partial Differential Equations. TransactionsoftheAmericanMathematicalSociety, 43, 126-166. https://doi.org/10.1090/s0002-9947-1938-1501936-8
[8]
Benedek, A. and Panzone, R. (1961) The Space , with Mixed Norm. Duke Mathematical Journal, 28, 301-324.
[9]
Bagby, R.J. (1975) An Extended Inequality for the Maximal Function. ProceedingsoftheAmericanMathematicalSociety, 48, 419-419. https://doi.org/10.1090/s0002-9939-1975-0370171-x.
[10]
Stefanov, A. and Torres, R.H. (2004) Calderón-Zygmund Operators on Mixed Lebesgue Spaces and Applications to Null Forms. Proceedings of the London Mathematical Society, 70, 447-462.
[11]
Ward, E.L. (2010) New Estimates for Harmonic Analysis for Mixed Lebesgue Spaces. University of Kansas.
[12]
Nogayama, T. (2019) Mixed Morrey Spaces. Positivity, 23, 961-1000. https://doi.org/10.1007/s11117-019-00646-8
[13]
Nogayama, T., Ono, T., Salim, D. and Sawano, Y. (2021) Atomic Decomposition for Mixed Morrey Spaces. The Journal of Geometric Analysis, 31, 9338-9365. https://doi.org/10.1007/s12220-020-00513-z
[14]
Wei, M. (2022) Boundedness Criterion for Some Integral Operators on Generalized Mixed Morrey Spaces and Generalized Mixed Hardy-Morrey Spaces. Banach Journal of Mathematical Analysis, 16, Article No. 8. https://doi.org/10.1007/s43037-021-00160-8
[15]
Lu, G.H. and Tao, S.P. (2023) Estimate for Some Integral Operators and Their Commutators on Generalized Fractional Mixed Morrey Spaces. Bulletin des Sciences Mathématiques, 187, Article 103314. https://doi.org/10.1016/j.bulsci.2023.103314
[16]
Bongioanni, B., Harboure, E. and Salinas, O. (2011) Commutators of Riesz Transforms Related to Schrödinger Operators. Journal of Fourier Analysis and Applications, 17, 115-134. https://doi.org/10.1007/s00041-010-9133-6
[17]
Akbulut, A., Guliyev, V.S. and Omarova, M.N. (2017) Marcinkiewicz Integrals Associated with Schrödinger Operator and Their Commutators on Vanishing Generalized Morrey Spaces. Boundary Value Problems, 2017, Article No. 121. https://doi.org/10.1186/s13661-017-0851-4
[18]
Tang, L. and Dong, J.X. (2009) Boundedness for Some Schrödinger Type Operators on Morrey Spaces Related to Certain Nonnegative Potentials. Journal of Mathematical Analysis and Applications, 355, 101-109. https://doi.org/10.1016/j.jmaa.2009.01.043