Tokyo Journal of Mathematics

Compact Commutators of Calderón-Zygmund and Generalized Fractional Integral Operators with a Function in Generalized Campanato Spaces on Generalized Morrey Spaces

Ryutaro ARAI and Eiichi NAKAI

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Abstract

We consider the commutators $[b,T]$ and $[b,I_{\rho}]$, where $T$ is a Calderón-Zygmund operator, $I_{\rho}$ is a generalized fractional integral operator and $b$ is a function in Campanato spaces with variable growth condition. It is known that these commutators are bounded on generalized Morrey spaces with variable growth condition. In this paper we discuss the compactness of these commutators.

Article information

Source
Tokyo J. Math., Advance publication (2018), 26 pages.

Dates
First available in Project Euclid: 6 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1533520825

Subjects
Primary: 42B35: Function spaces arising in harmonic analysis
Secondary: 46E30: Spaces of measurable functions (Lp-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

Citation

ARAI, Ryutaro; NAKAI, Eiichi. 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, 6 August 2018. https://projecteuclid.org/euclid.tjm/1533520825


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References

  • \BibAuthorsD. R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), no. 4, 765–778.
  • \BibAuthorsR. Arai and E. Nakai, Commutators of Calderón-Zygmund and generalized fractional integral operators on generalized Morrey spaces, Rev. Mat. Complut. 31 (2018), no. 2, 287–331.
  • \BibAuthorsS. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), no. 1, 7–16.
  • \BibAuthorsR. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611–635.
  • \BibAuthorsG. Di Fazio and M. A. Ragusa, Commutators and Morrey spaces, Boll. Un. Mat. Ital. A (7) 5 (1991), no. 3, 323–332.
  • \BibAuthorsEridani, H. Gunawan and E. Nakai, On generalized fractional integral operators, Sci. Math. Jpn. 60 (2004), no. 3, 539–550.
  • \BibAuthorsEridani, H. Gunawan, E. Nakai and Y. Sawano, Characterizations for the generalized fractional integral operators on Morrey spaces, Math. Inequal. Appl. 17 (2014), no. 2, 761–777.
  • \BibAuthorsX. Fu, D. Yang and W. Yuan, Generalized fractional integrals and their commutators over non-homogeneous metric measure spaces, Taiwanese J. Math. 18 (2014), no. 2, 509–557.
  • \BibAuthorsL. Grafakos, Modern Fourier analysis, Third edition, Graduate Texts in Mathematics, 250. Springer, New York, 2014. xvi+624 pp.
  • \BibAuthorsH. Gunawan, A note on the generalized fractional integral operators, J. Indonesian Math. Soc. (MIHMI) 9(1) (2003), 39–43.
  • \BibAuthorsH. Gunawan and Eridani, Fractional integrals and generalized Olsen inequalities, Kyungpook Math. J. 49 (2009), no. 1, 31–39.
  • \BibAuthorsH. Gunawan, Y. Sawano and I. Sihwaningrum, Fractional integral operators in nonhomogeneous spaces, Bull. Aust. Math. Soc. 80 (2009), no. 2, 324–334.
  • \BibAuthorsT. 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.
  • \BibAuthorsS. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), no. 2, 263–270.
  • \BibAuthorsL. V. Kantorovich and G. P. Akilov, Functional analysis. Translated from the Russian by Howard L. Silcock. Second edition. Pergamon Press, Oxford-Elmsford, N.Y., 1982.
  • \BibAuthorsV. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko, Integral operators in non-standard function spaces Vol. 1, Variable exponent Lebesgue and amalgam spaces, Operator Theory: Advances and Applications, 248, Birkhäuser/Springer, 2016.
  • \BibAuthorsV. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko, Integral operators in non-standard function spaces Vol. 2, Variable exponent Hölder, Morrey-Campanato and grand spaces, Operator Theory: Advances and Applications, 249, Birkhäuser/Springer, 2016.
  • \BibAuthorsY. Komori and T. Mizuhara, Notes on commutators and Morrey spaces, Hokkaido Math. J. 32 (2003), no. 2, 345–353.
  • \BibAuthorsT. Mizuhara, Commutators of singular integral operators on Morrey spaces with general growth functions, Harmonic analysis and nonlinear partial differential equations (Kyoto, 1998), Sûrikaisekikenkyûsho Kôkyûroku No. 1102 (1999), 49–63.
  • \BibAuthorsE. Nakai, Pointwise multipliers for functions of weighted bounded mean oscillation, Studia Math. 105 (1993), no. 2, 105–119.
  • \BibAuthorsE. Nakai, Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr. 166 (1994), 95–103.
  • \BibAuthorsE. Nakai, Pointwise multipliers on the Morrey spaces, Mem. Osaka Kyoiku Univ. III Natur. Sci. Appl. Sci. 46 (1997), no. 1, 1–11.
  • \BibAuthorsE. Nakai, On generalized fractional integrals, Taiwanese J. Math. 5 (2001), no. 3, 587–602.
  • \BibAuthorsE. Nakai, On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type, Sci. Math. Jpn. 54 (2001), no. 3, 473–487.
  • \BibAuthorsE. Nakai, On generalized fractional integrals on the weak Orlicz spaces, $\BMO_{\phi}$, the Morrey spaces and the Campanato spaces, Function spaces, interpolation theory and related topics (Lund, 2000), 389–401, de Gruyter, Berlin, 2002.
  • \BibAuthorsE. Nakai, The Campanato, Morrey and Hölder spaces on spaces of homogeneous type, Studia Math. 176 (2006), no. 1, 1–19.
  • \BibAuthorsE. Nakai, Orlicz-Morrey spaces and the Hardy-Littlewood maximal function, Studia Math. 188 (2008), no. 3, 193–221.
  • \BibAuthorsE. Nakai, A generalization of Hardy spaces $H^p$ by using atoms, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 8, 1243–1268.
  • \BibAuthorsE. Nakai, Singular and fractional integral operators on Campanato spaces with variable growth conditions, Rev. Mat. Complut. 23 (2010), no. 2, 355–381.
  • \BibAuthorsE. Nakai, Generalized fractional integrals on generalized Morrey spaces, Math. Nachr. 287 (2014), no. 2–3, 339–351.
  • \BibAuthorsE. Nakai and K. Yabuta, Pointwise multipliers for functions of bounded mean oscillation, J. Math. Soc. Japan 37 (1985), 207–218.
  • \BibAuthorsS. Nakamura and Y. Sawano, The singular integral operator and its commutator on weighted Morrey spaces, Collect. Math. 68 (2017), no. 2, 145–174.
  • \BibAuthorsR. O'Neil, Fractional integration in Orlicz spaces. I., Trans. Amer. Math. Soc. 115 (1965), 300–328.
  • \BibAuthorsJ. Peetre, On interpolation functions II, Acta Sci. Math. (Szeged) 29 (1968), 91–92.
  • \BibAuthorsJ. Peetre, On the theory of $\mathcal L_{p,\lambda}$ spaces, J. Funct. Anal. 4 (1969), 71–87.
  • \BibAuthorsC. Pérez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J. 43 (1994), 663–683.
  • \BibAuthorsY. Sawano and S. Shirai, Compact commutators on Morrey spaces with non-doubling measures, Georgian Math. J. 15 (2008), no. 2, 353–376.
  • \BibAuthorsY. 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 (2012), no 12, 6481–6503.
  • \BibAuthorsS. Shirai, Notes on commutators of fractional integral operators on generalized Morrey spaces, Sci. Math. Jpn. 63 (2006), no. 2, 241–246.
  • \BibAuthorsS. Shirai, Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on classical Morrey spaces, Hokkaido Math. J. 35 (2006), no. 3, 683–696.
  • \BibAuthorsS. Sugano, Some inequalities for generalized fractional integral operators on generalized Morrey spaces, Math. Inequal. Appl. 14 (2011), no. 4, 849–865.
  • \BibAuthorsA. Uchiyama, On the compactness of operators of Hankel type, Tôhoku Math. J. (2) 30 (1978), No. 1, 163–171.
  • \BibAuthorsK. Yabuta, Generalizations of Calderón-Zygmund operators, Studia Math. 82 (1985), 17–31.