×
Arai, Ryutaro; Nakai, Eiichi

An extension of the characterization of CMO and its application to compact commutators on Morrey spaces. (English) Zbl 1437.42031

J. Math. Soc. Japan 72, No. 2, 507-539 (2020).
Summary: A. Uchiyama [Tohoku Math. J. (2) 30, 163–171 (1978; Zbl 0384.47023)] gave a proof of the characterization of \(\text{CMO}(\mathbb{R}^n)\) which is the closure of \(C^{\infty}_{\text{comp}}(\mathbb{R}^n)\) in \(\text{BMO}(\mathbb{R}^n)\). We extend the characterization to the closure of \(C^{\infty}_{\text{comp}}(\mathbb{R}^n)\) in the Campanato space with variable growth condition. As an application we characterize compact commutators \([b,T]\) and \([b,I_{\alpha}]\) on Morrey spaces with variable growth condition, where \(T\) is the Calderón-Zygmund singular integral operator, \(I_{\alpha}\) is the fractional integral operator and \(b\) is a function in the Campanato space with variable growth condition.

Cited in 4 Documents

MSC:

42B35 Function spaces arising in harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators

Keywords:

Campanato space; Morrey space; variable growth condition; singular integral; fractional integral; commutator

Citations:

Zbl 0384.47023
PDF BibTeX XML Cite
\textit{R. Arai} and \textit{E. Nakai}, J. Math. Soc. Japan 72, No. 2, 507--539 (2020; Zbl 1437.42031)
Full Text: DOI Euclid
  • logo of hbz
  • logo of WorldCat

References:

[1] D. R. Adams, A note on Riesz potentials, Duke Math. J., 42 (1975), 765-778. · Zbl 0336.46038
[2] R. Arai and E. Nakai, Commutators of Calderón-Zygmund and generalized fractional integral operators on generalized Morrey spaces, Rev. Mat. Complut., 31 (2018), 287-331. · Zbl 1391.42013
[3] R. Arai and E. Nakai, Compact commutators of Calderón-Zygmund and generalized fractional integral operators with a function in generalized Campanato spaces on generalized Morrey spaces, Tokyo J. Math. Advance publication., https://projecteuclid.org/euclid.tjm/1533520825. · Zbl 1391.42013
[4] L. Chaffee and R. H. Torres, Characterization of compactness of the commutators of bilinear fractional integral operators, Potential Anal., 43 (2015), 481-494. · Zbl 1337.42010
[5] S. Chanillo, A note on commutators, Indiana Univ. Math. J., 31 (1982), 7-16. · Zbl 0523.42015
[6] Y. Chen and Y. Ding, Compactness characterization of commutators for Littlewood-Paley operators, Kodai Math. J., 32 (2009), 256-323. · Zbl 1205.42012
[7] Y. Chen, Y. Ding and X. Wang, Compactness of commutators of Riesz potential on Morrey spaces, Potential Anal., 30 (2009), 301-313. · Zbl 1163.42004
[8] Y. Chen, Y. Ding and X. Wang, Compactness for commutators of Marcinkiewicz integrals in Morrey spaces, Taiwanese J. Math., 15 (2011), 633-658. · Zbl 1229.42015
[9] Y. Chen, Y. Ding and X. Wang, Compactness of commutators for singular integrals on Morrey spaces, Canad. J. Math., 64 (2012), 257-281. · Zbl 1242.42009
[10] Y. Chen and H. Wang, Compactness for the commutator of the parameterized area integral in the Morrey space, Math. Inequal. Appl., 18 (2015), 1261-1273. · Zbl 1334.42027
[11] Y. Chen, K. Zhu and Y. Ding, On the compactness of the commutator of the parabolic Marcinkiewicz integral with variable kernel, J. Funct. Spaces, (2014), art. ID 693534, 12 pp. · Zbl 1309.47037
[12] R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2), 103 (1976), 611-635. · Zbl 0326.32011
[13] G. Di Fazio and M. A. Ragusa, Commutators and Morrey spaces, Boll. Un. Mat. Ital. A (7), 5 (1991), 323-332. · Zbl 0761.42009
[14] Eridani, H. Gunawan and E. Nakai, On generalized fractional integral operators, Sci. Math. Jpn., 60 (2004), 539-550. · Zbl 1058.42007
[15] X. Fu, D. Yang and W. Yuan, Generalized fractional integrals and their commutators over non-homogeneous metric measure spaces, Taiwanese J. Math., 18 (2014), 509-557. · Zbl 1357.42016
[16] L. Grafakos, Modern Fourier Analysis, third edition, Grad. Texts in Math., 250, Springer, New York, 2014, xvi+624 pp. · Zbl 1304.42002
[17] T. Iida, Weighted estimates of higher order commutators generated by BMO-functions and the fractional integral operator on Morrey spaces, J. Inequal. Appl., (2016), paper no. 4, 23 pp. · Zbl 1331.26009
[18] S. Janson, On functions with conditions on the mean oscillation, Ark. Mat., 14 (1976), 189-196. · Zbl 0341.43005
[19] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat., 16 (1978), 263-270. · Zbl 0404.42013
[20] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426. · Zbl 0102.04302
[21] Y. Komori and T. Mizuhara, Notes on commutators and Morrey spaces, Hokkaido Math. J., 32 (2003), 345-353. · Zbl 1044.42011
[22] S. Mao and H. Wu, Characterization of functions via commutators of bilinear fractional integrals on Morrey spaces, Bull. Korean Math. Soc., 53 (2016), 1071-1085. · Zbl 1345.42018
[23] T. Mizuhara, Commutators of singular integral operators on Morrey spaces with general growth functions, In: Harmonic Analysis and Nonlinear Partial Differential Equations, Kyoto, 1998, Sûrikaisekikenkyûsho Kôkyûroku, 1102 (1999), 49-63. · Zbl 0951.42501
[24] E. Nakai, Pointwise multipliers for functions of weighted bounded mean oscillation, Studia Math., 105 (1993), 105-119. · Zbl 0812.42008
[25] E. Nakai, Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr., 166 (1994), 95-103. · Zbl 0837.42008
[26] E. Nakai, Pointwise multipliers on the Morrey spaces, Mem. Osaka Kyoiku Univ. III Natur. Sci. Appl. Sci., 46 (1997), 1-11.
[27] E. Nakai, On generalized fractional integrals, Taiwanese J. Math., 5 (2001), 587-602. · Zbl 0990.26007
[28] E. Nakai, On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type, Sci. Math. Jpn., 54 (2001), 473-487. · Zbl 1007.42013
[29] E. Nakai, On generalized fractional integrals on the weak Orlicz spaces, \( \text{BMO}_{\phi} \), the Morrey spaces and the Campanato spaces, In: Function Spaces, Interpolation Theory and Related Topics, Lund, 2000, de Gruyter, Berlin, 2002, 389-401. · Zbl 1021.42006
[30] E. Nakai, The Campanato, Morrey and Hölder spaces on spaces of homogeneous type, Studia Math., 176 (2006), 1-19. · Zbl 1121.46031
[31] E. Nakai, A generalization of Hardy spaces \(H^p\) by using atoms, Acta Math. Sin. (Engl. Ser.), 24 (2008), 1243-1268. · Zbl 1153.42011
[32] E. Nakai, Singular and fractional integral operators on Campanato spaces with variable growth conditions, Rev. Mat. Complut., 23 (2010), 355-381. · Zbl 1206.42015
[33] E. Nakai, Generalized fractional integrals on generalized Morrey spaces, Math. Nachr., 287 (2014), 339-351. · Zbl 1304.46027
[34] E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal., 262 (2012), 3665-3748. · Zbl 1244.42012
[35] E. Nakai and K. Yabuta, Pointwise multipliers for functions of bounded mean oscillation, J. Math. Soc. Japan, 37 (1985), 207-218. · Zbl 0546.42019
[36] S. Nakamura and Y. Sawano, The singular integral operator and its commutator on weighted Morrey spaces, Collect. Math., 68 (2017), 145-174. · Zbl 1375.42020
[37] U. Neri, Fractional integration on the space \(H^1\) and its dual, Studia Math., 53 (1975), 175-189. · Zbl 0269.44012
[38] T. Nogayama and Y. Sawano, Compactness of the commutators generated by Lipschitz functions and fractional integral operators, Mat. Zametki, 102 (2017), 749-760; translation in Math. Notes, 102 (2017), 687-697. · Zbl 1387.42018
[39] J. Peetre, On the theory of \(\mathcal{L}_{p,\lambda}\) spaces, J. Funct. Anal., 4 (1969), 71-87. · Zbl 0175.42602
[40] C. Pérez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J., 43 (1994), 663-683. · Zbl 0809.42007
[41] Y. Sawano and S. Shirai, Compact commutators on Morrey spaces with non-doubling measures, Georgian Math. J., 15 (2008), 353-376. · Zbl 1213.42082
[42] Y. Sawano, S. Sugano and H. Tanaka, Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces, Trans. Amer. Math. Soc., 363 (2011), 6481-6503. · Zbl 1229.42024
[43] S. Shirai, Notes on commutators of fractional integral operators on generalized Morrey spaces, Sci. Math. Jpn., 63 (2006), 241-246. · Zbl 1127.42018
[44] S. Shirai, Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on classical Morrey spaces, Hokkaido Math. J., 35 (2006), 683-696. · Zbl 1122.42007
[45] A. Uchiyama, On the compactness of operators of Hankel type, Tôhoku Math. J. (2), 30 (1978), 163-171. · Zbl 0384.47023
[46] K. Yabuta, Generalizations of Calderón-Zygmund operators, Studia Math., 82 (1985), 17-31. · Zbl 0585.42012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.