Kyoto Journal of Mathematics

Mapping properties of the discrete fractional maximal operator in metric measure spaces

Toni Heikkinen, Juha Kinnunen, Juho Nuutinen, and Heli Tuominen

Full-text: Open access

Abstract

This work studies boundedness properties of the fractional maximal operator on metric measure spaces under standard assumptions on the measure. The main motivation is to show that the fractional maximal operator has similar smoothing and mapping properties as the Riesz potential. Instead of the usual fractional maximal operator, we also consider a so-called discrete maximal operator which has better regularity. We study the boundedness of the discrete fractional maximal operator in Sobolev, Hölder, Morrey, and Campanato spaces. We also prove a version of the Coifman– Rochberg lemma for the fractional maximal function.

Article information

Source
Kyoto J. Math., Volume 53, Number 3 (2013), 693-712.

Dates
First available in Project Euclid: 19 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1376917630

Digital Object Identifier
doi:10.1215/21562261-2265932

Mathematical Reviews number (MathSciNet)
MR3102566

Zentralblatt MATH identifier
1280.42012

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory 35J60: Nonlinear elliptic equations 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Citation

Heikkinen, Toni; Kinnunen, Juha; Nuutinen, Juho; Tuominen, Heli. Mapping properties of the discrete fractional maximal operator in metric measure spaces. Kyoto J. Math. 53 (2013), no. 3, 693--712. doi:10.1215/21562261-2265932. https://projecteuclid.org/euclid.kjm/1376917630


Export citation

References

  • [1] D. Aalto and J. Kinnunen, The discrete maximal operator in metric spaces, J. Anal. Math. 111 (2010), 369–390.
  • [2] D. R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), 765–778.
  • [3] D. R. Adams, Lecture Notes on $L^{p}$-Potential Theory, Dept. of Mathematics, Univ. of Umeå, Umeå, Sweden, 1981.
  • [4] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996.
  • [5] D. R. Adams and J. Xiao, Morrey spaces in harmonic analysis, Ark. Mat. 50 (2012), 201–230.
  • [6] H. Arai and T. Mizuhara, Morrey spaces on spaces of homogeneous type and estimates for $\square_{b}$ and the Cauchy–Szegő projection, Math. Nachr. 185 (1997), 5–20.
  • [7] A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts Math. 17, Eur. Math. Soc., Zürich, 2011.
  • [8] S. M. Buckley, Is the maximal function of a Lipschitz function continuous?, Ann. Acad. Sci. Fenn. Math. 24 (1999), 519–528.
  • [9] F. Chiarenza and M. Frasca, Morrey spaces and Hardy–Littlewood maximal function, Rend. Mat. Appl. (7) 7 (1987), 273–279.
  • [10] R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), 249–254.
  • [11] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certain espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971.
  • [12] D. Edmunds, V. Kokilashvili, and A. Meskhi, Bounded and Compact Integral Operators, Math. and its Applic. 543, Kluwer, Dordrecht, 2002.
  • [13] A. E. Gatto, C. Segovia, and S. Vági, On fractional differentiation and integration on spaces of homogeneous type, Rev. Mat. Iberoam. 12 (1996), 111–145.
  • [14] A. E. Gatto and S. Vági, “Fractional integrals on spaces of homogeneous type” in Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math. 122, Dekker, New York, 1990, 171–216.
  • [15] I. Genebashvili, A. Gogatishvili, V. Kokilashvili, and M. Krbec, Weight Theory for Integral Transforms on Spaces of Homogeneous Type, Pitman Monogr. and Surv. in Pure and Appl. Math. 92, Longman, Harlow, England, 1998.
  • [16] O. Gorosito, G. Pradolini, and O. Salinas, Boundedness of the fractional maximal operator on variable exponent Lebesgue spaces: A short proof, Rev. Un. Mat. Argentina 53 (2012), 25–27.
  • [17] S. Keith and X. Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. (2) 167 (2008), 575–599.
  • [18] J. Kinnunen and V. Latvala, Lebesgue points for Sobolev functions on metric spaces, Rev. Mat. Iberoam. 18 (2002), 685–700.
  • [19] J. Kinnunen and E. Saksman, Regularity of the fractional maximal function, Bull. London Math. Soc. 35 (2003), 529–535.
  • [20] J. Kinnunen and H. Tuominen, Pointwise behaviour of $M^{1,1}$ Sobolev functions, Math. Z. 257 (2007), 613–630.
  • [21] R. A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. in Math. 33 (1979), 271–309.
  • [22] R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), 257–270.
  • [23] N. G. Meyers, Mean oscillation over cubes and Hölder continuity, Proc. Amer. Math. Soc. 15 (1964), 717–721.
  • [24] Y. Mizuta, T. Shimomura, and T. Sobukawa, Sobolev’s inequality for Riesz potentials on functions in non-doubling Morrey spaces, Osaka J. Math. 46 (2009), 255–271.
  • [25] E. Nakai, The Campanato, Morrey and Hölder spaces on spaces of homogeneous type, Studia Math. 176 (2006), 1–19.
  • [26] E. Nakai, Singular and fractional integral operators on Campanato spaces with variable growth conditions, Rev. Mat. Complut. 23 (2010), 355–381.
  • [27] J. Peetre, On the theory of $\mathcal{L}_{p,\lambda}$ spaces, J. Funct. Anal. 4 (1969), 71–87.
  • [28] Y. Sawano, Sharp estimates of the modified Hardy-Littlewood maximal operator on the nonhomogenous space via covering lemmas, Hokkaido Math. J. 34 (2005), 435–458.
  • [29] Y. Sawano, T. Sobukawa, and H. Tanaka, Limiting case of the boundedness of fractional integral operators on nonhomogeneous space, J. Inequal. Appl. 2006, art. ID 92470, 1–16.
  • [30] Y. Sawano and H. Tanaka, Morrey spaces for non-doubling measures, Acta Math. Sinica 21 (2005), 1535–1544.
  • [31] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), 243–279.
  • [32] N. Shanmugalingam, Harmonic functions on metric spaces, Illinois J. Math. 45 (2001), 1021–1050.
  • [33] J. Xiao, Bounded functions of vanishing mean oscillation on compact metric spaces, J. Funct. Anal. 209 (2004), 444–467.