Weighted boundedness of the Hardy-Littlewood maximal and Calderón-Zygmund operators on Orlicz-Morrey and weak Orlicz-Morrey spaces. (English) Zbl 1493.46047
Summary: For the Hardy-Littlewood maximal and Calderón-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 42B35 | Function spaces arising in harmonic analysis |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
References:
| [1] | R. R. COIFMAN,Distribution function inequalities for singular integrals, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 2838-2839. · Zbl 0243.44006 |
| [2] | R. R. COIFMAN ANDC. FEFFERMAN,Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. · Zbl 0291.44007 |
| [3] | G. P. CURBERA, J. GARC´IA-CUERVA, J. M. MARTELL ANDC. P ´EREZ,Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals, Adv. Math. 203 (2006), no. 1, 256-318. · Zbl 1098.42017 |
| [4] | F. DERINGOZ, V.S. GULIYEV, E. NAKAI, Y. SAWANO ANDM. SHI,Generalized fractional maximal and integral operators on Orlicz and generalized Orlicz-Morrey spaces of the third kind, Positivity 23 (2019), no. 3, 727-757. · Zbl 1440.42076 |
| [5] | F. DERINGOZ, V.S. GULIYEV ANDS. SAMKO,Boundedness of the maximal and singular operators on generalized Orlicz-Morrey spaces, Operator theory, operator algebras and applications, 139-158, Oper. Theory Adv. Appl., 242, Birkh¨auser/Springer, Basel, 2014. · Zbl 1318.42015 |
| [6] | X. FU, D. YANG ANDW. YUAN,Boundedness of multilinear commutators of Calder´on-Zygmund operators on Orlicz spaces over non-homogeneous spaces, Taiwanese J. Math. 16 (2012), no. 6, 2203- 2238. · Zbl 1275.47079 |
| [7] | S. GALA, Y. SAWANO ANDH. TANAKA,A remark on two generalized Orlicz-Morrey spaces, J. Approx. Theory 198 (2015), 1-9. · Zbl 1322.42027 |
| [8] | J. GARC´IA-CUERVA ANDJ. RUBIO DEFRANCIA,Weighted norm inequalities and related topics, North-Holland Mathematics Studies, 116. Notas de Matem´atica [Mathematical Notes], 104. NorthHolland Publishing Co., Amsterdam, 1985. x+604 pp. ISBN: 0-444-87804-1. |
| [9] | L. GRAFAKOS,Classical Fourier analysis, Third edition, Graduate Texts in Mathematics, 249, Springer, New York, 2014. · Zbl 1304.42001 |
| [10] | V. S. GULIYEV, S. G. HASANOV, Y. SAWANO ANDT. NOI,Non-smooth atomic decompositions for generalized Orlicz-Morrey spaces of the third kind, Acta Appl. Math. 145 (2016), no. 1, 133-174. · Zbl 1362.42028 |
| [11] | J. GUSTAVSSON ANDJ. PEETRE,Interpolation of Orlicz spaces, Studia Math. 60 (1977), no. 1, 33- 59. · Zbl 0353.46019 |
| [12] | K.-P. HO,Vector-valued maximal inequalities on weighted Orlicz-Morrey spaces, Tokyo J. Math. 36 (2013), no. 2, 499-512. · Zbl 1295.42005 |
| [13] | K.-P. HO,Weak type estimates of singular integral operators on Morrey-Banach spaces, Integral Equations Operator Theory 91 (2019), no. 3, Paper no. 20, 18 pp. · Zbl 1418.42018 |
| [14] | K.-P. HO,Definability of singular integral operators on Morrey-Banach spaces, J. Math. Soc. Japan 72 (2020), no. 1, 155-170. · Zbl 1437.42016 |
| [15] | R. HUNT, B. MUCKENHOUPT ANDR. WHEEDEN,Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251. · Zbl 0262.44004 |
| [16] | R. KAWASUMI ANDE. NAKAI,Pointwise multipliers on weak Orlicz spaces, Hiroshima Math. J. 50 (2020), no. 2, 169-184. · Zbl 1464.46032 |
| [17] | R. KAWASUMI, E. NAKAI ANDM. SHI,Characterization of the boundedness of generalized fractional integral and maximal operators on Orlicz-Morrey and weak Orlicz-Morrey spaces, to appear in Math. Nachr.,https://arxiv.org/abs/2107.10553. |
| [18] | H. KITA,Orlicz spaces and their applications(Japanese), Iwanami Shoten, Publishers. Tokyo, 2009. WEIGHTED BOUNDEDNESS ON WEAKORLICZ-MORREY SPACES1187 |
| [19] | V. KOKILASHVILI ANDM. KRBEC,Weighted inequalities in Lorentz and Orlicz spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. · Zbl 0751.46021 |
| [20] | Y. KOMORI-FURUYA,Weak Lp-boundedness of the Hardy-Littlewood maximal function(in Japanese), Report collection of Harmonic Analysis Seminar 2011 at Osaka University, 39-42, printed in 2012. |
| [21] | Y. KOMORI ANDS. SHIRAI,Weighted Morrey spaces and a singular integral operator, Math. Nachr. 282 (2009), no. 2, 219-231. · Zbl 1160.42008 |
| [22] | P. LIU ANDM. WANG,Weak Orlicz spaces: Some basic properties and their applications to harmonic analysis, Sci. China. Math. 56 (2013), 789-802. · Zbl 1284.46024 |
| [23] | M. A. KRASNOSELSKY ANDY. B. RUTITSKY,Convex functions and Orlicz spaces, Translated from thefirst Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen 1961. · Zbl 0095.09103 |
| [24] | L. MALIGRANDA,Indices and interpolation, Dissertationes Math. (Rozprawy Mat.) 234 (1985), 49 pp. · Zbl 0566.46038 |
| [25] | L. MALIGRANDA,Orlicz spaces and interpolation, Seminars in mathematics 5, Departamento de Matem´atica, Universidade Estadual de Campinas, Brasil, 1989. · Zbl 0874.46022 |
| [26] | B. MUCKENHOUPT,Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. · Zbl 0236.26016 |
| [27] | E. NAKAI,Generalized fractional integrals on Orlicz-Morrey spaces, Banach and Function Spaces (Kitakyushu, 2003), Yokohama Publishers, Yokohama, 2004, 323-333. · Zbl 1118.42005 |
| [28] | E. NAKAI,Orlicz-Morrey spaces and the Hardy-Littlewood maximal function, Studia Math. 188 (2008), no. 3, 193-221. · Zbl 1163.46020 |
| [29] | E. NAKAI,Calder´on-Zygmund operators on Orlicz-Morrey spaces and modular inequalities, Banach and function spaces II, 393-410, Yokohama Publ., Yokohama, 2008. · Zbl 1163.46021 |
| [30] | R. O’NEIL,Fractional integration in Orlicz spaces. I., Trans. Amer. Math. Soc. 115 (1965), 300-328. · Zbl 0132.09201 |
| [31] | Y. SAWANO,Singular integral operators acting on Orlicz-Morrey spaces of thefirst kind, Nonlinear Stud. 26 (2019), no. 4, 895-910. · Zbl 1444.42015 |
| [32] | Y. SAWANO, S. SUGANO ANDH. TANAKA,Orlicz-Morrey spaces and fractional operators, Potential Anal. 36 (2012), no. 4, 517-556. · Zbl 1242.42017 |
| [33] | M. SHI, R. ARAI ANDE. NAKAI,Generalized fractional integral operators and their commutators with functions in generalized Campanato spaces on Orlicz spaces, Taiwanese J. Math. 23 (2019), no. 6, 1339-1364. · Zbl 1491.47038 |
| [34] | M. SHI, R. ARAI ANDE. NAKAI,Commutators of integral operators with functions in Campanato spaces on Orlicz-Morrey spaces, Banach J. Math. Anal. 15 (2021), no. 1, Paper no. 22, 41 pp. · Zbl 1455.42021 |
| [35] | A. TORCHINSKY,Interpolation of operations and Orlicz classes, Studia Math. 59 (1976), no. 2, 177- 207. · Zbl 0348.46027 |
| [36] | K. YABUTA,Generalizations of Calder´on-Zygmund operators, Studia Math. 82 (1985), 17-31 · Zbl 0585.42012 |
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