介绍了齐性群上的加权Hardy算子,获得其在中心Morrey空间与中心BMO空间上的最佳有界性。另外,也得出在中心Morrey空间中加权Hardy算子交换子有界的充要条件。
Let $ \mathbb{G}$ denote be a homogeneous group, the authors introduce a type of weighted Hardy operators on homogeneous group and obtain their sharp bounds on central Morrey spaces and λ-central BMO spaces in the setting of homogeneous group. In addition, the authors obtain sufficient and necessary conditions of the weight functions so that the commutators of the weighted Hardy operators (with symbols in central BMO space) are bounded on the central Morrey spaces in the setting of homogeneous group.
[1] Carton-Lebrun C, Fosset M.Moyennes et quotients de Taylor dans BMO. Bulletin de la Société Royaledes Sciences de Liège, 1984, 53(2):85-87.
[2] Xiao J.Lp and BMO bounds of weighted Hardy-Littlewood averages[J]. Journal of Mathematical Analysis and Applications, 2001, 262(2): 660-666. DOI:10.1006/jmaa.2001.7594.
[3] Fu Z W, Lu S Z, Yuan W.A weighted variant of Riemann-Liouville fractional integrals on $ \mathbb{R}^n$[J]. Abstract and Applied Analysis, 2012, 2012(1): 780132. DOI:10.1155/2012/780132.
[4] Fu Z W, Gong S L, Lu S Z, et al.Weighted multilinear Hardy operators and commutators[J]. Forum Mathematicum, 2015, 27(5): 2825-2851. DOI:10.1515/forum-2013-0064.
[5] Wu Q Y, Fu Z W.Weighted p-adic Hardy operators and their commutators on p-adic central Morrey spaces[J]. Bulletin of the Malaysian Mathematical Sciences Society, 2017, 40(2): 635-654. DOI:10.1007/s40840-017-0444-5.
[6] Zhao F Y, Fu Z W, Lu S Z.Endpoint estimates for n-dimensional Hardy operators and their commutators[J]. Science China Mathematics, 2012, 55(10): 1977-1990. DOI:10.1007/s11425-012-4465-0.
[7] Christ M, Grafakos L.Best constants for two nonconvolution inequalities[J]. Proceedings of the American Mathematical Society, 1995, 123(6): 1687-1693. DOI:10.1090/s0002-9939-1995-1239796-6.
[8] Lu S Z, Fu Z W, Zhao F Y, et al.Hardy operators on Euclidean spaces and related topics[M]. New Jersey: World Scientific, 2023.
[9] Ruzhansky M, Suragan D.Hardy Inequalities on Homogeneous Groups: 100 years of Hardy inequalities[M]. Cham,Switzerland: Birkhäuser, 2019.
[10] Fu Z W, Grafakos L, Lu S Z, et al.Sharp bounds for m-linear Hardy and Hilbert operators[J]. Houston Journal of Mathematics, 2023, 38: 225-244.
[11] He Q J, Li X, Yan D Y.Sharp bounds for Hardy type operators on higher-dimensional product spaces[J]. Frontiers of Mathematics in China, 2018, 13(6): 1341-1353. DOI:10.1007/s11464-018-0740-x.
[12] Zhao F Y, Lu S Z.The best bound for n-dimensional fractional Hardy operators[J]. Mathematical Inequalities & Applications, 1998(1): 233-240. DOI:10.7153/mia-18-17.
[13] Persson L E, Samko S.A note on the best constants in some Hardy inequalities[J]. Journal of Mathematical Inequalities, 2007(2): 437-447. DOI:10.7153/jmi-09-37.
[14] Ruzhansky M, Shriwastawa A, Tiwari B.A note on best constants for weighted integral Hardy inequalities on homogeneous groups[J]. Results in Mathematics,2023, 79(1): 29. DOI:10.1007/s00025-023-02058-3.
[15] Mo H X.Commutators of generalized Hardy operators on homogeneous groups[J]. Acta Mathematica Scientia, 2010, 30(3): 897-906. DOI:10.1016/S0252-9602(10)60087-2.
[16] Lakey J, Alvarez J, Guzmán-Partida M.Spaces of bounded λ-central mean oscillation, Morrey spaces,and λ-central Carleson measures[J]. Collectanea Mathematica, 2000, 51: 1-47.
[17] Coifman R R, Rochberg R, Weiss G.Factorization theorems for hardy spaces in several variables[J]. The Annals of Mathematics, 1976, 103(3): 611-635. DOI:10.2307/1970954.
