This paper offers a novel portrayal of the generalized Campanato space with variable exponents. By introducing the position-dependent weight function ω ( x , r ) , the generalized variable exponent Campanato space significantly increases its flexibility and power to characterize complex phenomena. The boundedness of the fractional integrals and their commutators is obtained on the generalized Campanato space with variable exponents.
Campanato, S. (1963) Propriet´a di Holderianit´a di alcune classi di funzioni. Annali Della Scuola Normale Superiore Di Pisa - Classe Di Scienze , 17, 175-188.
Stampacchia, G. (1964) L p,λ -Spaces and Interpolation. Communications on Pure and Applied Mathematics , 17, 293-306. https://doi.org/10.1002/cpa.3160170303
John, F. and Nirenberg, L. (1961) On Functions of Bounded Mean Oscillation. Communications on Pure and Applied Mathematics , 14, 415-426. https://doi.org/10.1002/cpa.3160140317
Campanato, S. and Murthy, M.K.V. (1965) Una generalizzazione del teorema di Riesz-Thorin. Annali Della Scuola Normale Superiore Di Pisaclasse Di Scienze , 19, 87-100.
Stampacchia, G. (1965) The Spaces L p,λ and N p,λ and Interpolation. Annali Della Scuola Normale Superiore Di Pisa - Classe Di Scienze , 19, 443-462.
Afif, A., Omer. A. and Tao, S. (2016) Boundedness of Fractional Integral with Variable Kernel and Their Commutators on Variable Exponent Herz Spaces. Applied Mathematics , 7, 1165-1182.
Fu, Z., Gong, R., Pozzi, E. and Wu, Q. (2023) Cauchy–Szegö Commutators on Weighted Morrey Spaces. Mathematische Nachrichten, 296, 1859-1885. https://doi.org/10.1002/mana.202000139
Hu, P., Tao, J. and Yang, D. (2023) New John-Nirenberg-Campanato-Type Spaces Related to Both Maximal Functions and Their Commutators. Mathematical Methods in the Applied Sciences , 46, 5937-5963. https://doi.org/10.1002/mma.8879
Khalil, O., Tao, S. and Mahamat, B. (2021) Commutator of Marcinkiewicz Integral Operators on Herz-Morrey-Hardy Spaces with Variable Exponents. Applied Mathematics , 12, 421-448. https://doi.org/10.4236/am.2021.125030
Liu, F., Fu, Z. and Wu, Y. (2024) Variation Operators for Commutators of Rough Singular Integrals on Weighted Morrey Spaces. Journal of Applied Analysis & Computation , 14, 263-282. https://doi.org/10.11948/20230210
Shi, S. and Lu, S. (2015) Characterization of the Central Campanato Space via the Commutator Operator of Hardy Type. Journal of Mathematical Analysis and Applications , 429, 713-732. https://doi.org/10.1016/j.jmaa.2015.03.083
Wang, D. and Shu, L. (2022) New Function Classes of Morrey–Campanato Type and Their Applications. Banach Journal of Mathematical Analysis , 16, Article No. 39. https://doi.org/10.1007/s43037-022-00193-7
Coifman, R.R., Rochberg, R. and Weiss, G. (1976) Factorization Theorems for Hardy Spaces in Several Variables. The Annals of Mathematics , 103, 611-635. https://doi.org/10.2307/1970954
Chen, P., Duong, X.T., Li, J. and Wu, Q. (2019) Compactness of Riesz Transform Commutator on Stratified Lie Groups. Journal of Functional Analysis , 277, 1639-1676. https://doi.org/10.1016/j.jfa.2019.05.008
Duong, X.T., Lacey, M., Li, J., Wick, B.D. and Wu, Q. (2021) Commutators of Cauchy-Szego Type Integrals for Domains in C n with Minimal Smoothness. Indiana University Mathematics Journal , 70, Article 15051541.
Fu, Z., Pozzi, E. and Wu, Q. (2022) Commutators of Maximal Functions on Spaces of Homogeneous Type and Their Weighted, Local Versions. Frontiers of Mathematics in China , 17, 625-652. https://doi.org/10.1007/s11464-021-0912-y
Fu, Z., Lu, S. and Shi, S. (2021) Two Characterizations of Central BMO Space via the Commutators of Hardy Operators. Forum Mathematicum , 33, 505-529. https://doi.org/10.1515/forum-2020-0243
Janson, S. (1978) Mean Oscillation and Commutators of Singular Integral Operators. Arkiv för Matematik , 16, 263-270. https://doi.org/10.1007/bf02386000
Paluszynski, M. (1995) Characterization of the Besov Spaces via the Commutator Operator of Coifman, Rochberg and Weiss. Indiana University Mathematics Journal , 44, 1-17. https://doi.org/10.1512/iumj.1995.44.1976
Shi, S., Fu, Z. and Lu, S. (2020) On the Compactness of Commutators of Hardy Operators. Pacific Journal of Mathematics , 307, 239-256. https://doi.org/10.2140/pjm.2020.307.239
Chiarenza, F., Frasca, M. and Longo, P. (1993) W 2 ,p -Solvability of the Dirichlet Problem for Nondivergence Elliptic Equations with VMO Coefficients. Transactions of the American Mathematical Society , 336, 841-853.
Difazio, G. and Ragusa, M.A. (1993) Interior Estimates in Morrey Spaces for Strong Solutions to Nondivergence Form Equations with Discontinuous Coefficients. Journal of Functional Analysis , 112, 241-256. https://doi.org/10.1006/jfan.1993.1032
Ragusa, M.A. (2004) Cauchy–Dirichlet Problem Associated to Divergence form Parabolic Equations. Communications in Contemporary Mathematics , 6, 377-393. https://doi.org/10.1142/s0219199704001392
Caffarelli, L. A. and Peral, I. (1998) On W 1 ,p Estimates for Elliptic Equations in Divergence Form. Communications on Pure and Applied Mathematics : A Journal Issued by the Courant Institute of Mathematical Sciences , 51, 1-21.
Dong, H. and Kim, D. (2010) Elliptic Equations in Divergence Form with Partially BMO Coefficients. Archive for Rational Mechanics and Analysis , 196, 25-70. https://doi.org/10.1007/s00205-009-0228-7
Fan, D.S., Lu, S.Z. and Yang, D.C. (1998) Regularity in Morrey Spaces of Strong Solutions to Nondivergence Elliptic Equations with VMO. Georgian Mathematical Journal , 5, 425-440. https://doi.org/10.1515/gmj.1998.425
Chanillo, S. (1982) A Note on Commutators. Indiana University Mathematics Journal , 31, 7-16.
Shi, S.G. and Lu, S.Z. (2013) Some Characterizations of Campanato Spaces via Commutators on Morrey Spaces. Pacific Journal of Mathematics , 264, 221-234. https://doi.org/10.2140/pjm.2013.264.221
Kovcik, O. and Rákosník, J. (1991) On Spaces L p ( x ) and W k,p ( x ) . Czechoslovak Mathematical Journal , 41, 592-618.
Sharapudinov, I.I. (1979) Topology of the Space L p ( t ) ([0 , 1]). Mathematical Notes of the Academy of Sciences of the USSR , 26, 796-806. https://doi.org/10.1007/bf01159546
Cruz-Uribe, D. and Fiorenza, A. (2013) Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Springer.
Capone, C., Cruz-Uribe, D. and Fiorenza, A. (2007) The Fractional Maximal Operator and Fractional Integrals on Variable L p Spaces . 23, 743-770. https://doi.org/10.4171/rmi/511
Almeida, A., Hasanov, J. and Samko, S. (2008) Maximal and Potential Opera to Pacific Journal of Mathematicsrs in Variable Exponent Morrey Spaces. Georgian Mathematical Journal , 15, 195-208. https://doi.org/10.1515/gmj.2008.195
Kokilashvili, V. and Meskhi, A. (2008) Boundedness of Maximal and Singular Operators in Morrey Spaces with Variable Exponent. Armenian Journal of Mathematics , 1, 18-28.
Kokilashvili, V. and Meskhi, A. (2010) Maximal Functions and Potentials in Variable Exponent Morrey Spaces with Non-Doubling Measure. Complex Variables and Elliptic Equations , 55, 923-936. https://doi.org/10.1080/17476930903276068
Ohno, T. (2008) Continuity Properties for Logarithmic Potentials of Functions in Morrey Spaces of Variable Exponent. Hiroshima Mathematical Journal , 38, 363-383. https://doi.org/10.32917/hmj/1233152775
Rafeiro, H. and Samko, S. (2024) Fractional Operators of Variable Order from Variable Exponent Morrey Spaces to Variable Exponent Campanato Spaces on Quasi-Metric Measure Spaces with Growth Condition. Ricerche di Matematica , 73, 803-818. https://doi.org/10.1007/s11587-021-00639-4
Guliyev, V.S., Hasanov, J.J. and Samko, S.G. (2010) Boundedness of Nonlinear Analysis: Real World Applications the Maximal, Potential and Singular Operators in the Generalized Variable Exponent Morrey Spaces. Mathematica Scandinavica , 107, 285-304.
Shi, S. and Lu, S. (2014) A Characterization of Campanato Space via Commutator of Fractional Integral. Journal of Mathematical Analysis and Applications , 419, 123-137. https://doi.org/10.1016/j.jmaa.2014.04.040.
Eroglu, A., Guliyev, V.S. and Azizov, J.V. (2017) Characterizations for the Fractional Integral Operators in Generalized Morrey Spaces on Carnot Groups. Mathematical Notes , 102, 722-734. https://doi.org/10.1134/s0001434617110116
Fefferman, C.L. (1983) The Uncertainty Principle. Bulletin of the American Mathematical Society , 9, 129-206. https://doi.org/10.1090/s0273-0979-1983-15154-6
Gilbarg, D., Trudinger, N.S., Gilbarg, D. and Trudinger, N.S. (1977) Elliptic Partial Differential Equations of Second Order (Vol. 224, No. 2). Springer.
Cruz-Uribe, D. and Neugebauer, C.J. (1995) The Structure of the Reverse Hölder Classes. Transactions of the American Mathematical Society , 347, 2941-2960. https://doi.org/10.1090/s0002-9947-1995-1308005-6
Harboure, E., Salinas, O. and Viviani, B. (1998) Reverse-Holder Classes in the Orlicz Spaces Setting. Studia Mathematica , 130, 245-261.