Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

IMPACT FACTOR 2018: 0.536

CiteScore 2018: 0.83

SCImago Journal Rank (SJR) 2018: 1.041
Source Normalized Impact per Paper (SNIP) 2018: 0.801

Mathematical Citation Quotient (MCQ) 2018: 0.83

Open Access
See all formats and pricing
More options …

Fractional Maximal Functions in Metric Measure Spaces

Toni Heikkinen / Juha Lehrbäck / Juho Nuutinen / Heli Tuominen
Published Online: 2013-05-28 | DOI: https://doi.org/10.2478/agms-2013-0002


We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.

Keywords: Fractional maximal function; fractional Sobolev space; Campanato space; metric measure space

MSC: 42B25; 46E35

  • [1] D. R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), 765-778.Google Scholar

  • [2] D. R. Adams, Lecture Notes on Lp-Potential Theory, Dept. of Math., University of Umeå, 1981.Google Scholar

  • [3] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin Heidelberg, 1996.Google Scholar

  • [4] S. M. Buckley, Is the maximal function of a Lipschitz function continuous?, Ann. Acad. Sci. Fenn. Math. 24 (1999), 519-528.Google Scholar

  • [5] S. M. Buckley, Inequalities of John-Nirenberg type in doubling spaces, J. Anal. Math 79 (1999), 215-240.CrossrefGoogle Scholar

  • [6] D. Edmunds, V. Kokilashvili, and A. Meskhi, Bounded and Compact Integral Operators, Mathematics and its Applications, vol. 543, Kluwer Academic Publishers, Dordrecht, Boston, London, 2002.Google Scholar

  • [7] J. García-Cuerva and J. L.Rubio De Francia, Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, 116. Notas de Matemática, 104. North-Holland Publishing Co., Amsterdam, 1985.Google Scholar

  • [8] A. E. Gatto and S. Vági, Fractional integrals on spaces of homogeneous type, Analysis and Partial Differential Equations, C. Sadosky (ed.), Dekker, 1990, 171-216.Google Scholar

  • [9] A. E. Gatto, C. Segovia, and S. Vági, On fractional differentiation and integration on spaces of homogeneous type, Rev. Mat. Iberoamericana 12 (1996), no. 1, 111-145.Google Scholar

  • [10] I. Genebashvili, A. Gogatishvili, V. Kokilashvili and M. Krbec, Weight Theory for Integral Transforms on Spaces of Homogeneous Type, Addison Wesley Longman Limited, 1998.Google Scholar

  • [11] A. Gogatishvili, Two-weight mixed inequalities in Orlicz classes for fractional maximal functions defined on homogeneous type spaces, Proc. A. Razmadze Math. Inst. 112 (1997), 23-56.Google Scholar

  • [12] A.Gogatishvili, Fractional maximal functions in weighted Banach function spaces, Real Anal. Exchange 25 (1999/00), no. 1, 291-316.Google Scholar

  • [13] 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), no. 1, 25-27.Google Scholar

  • [14] P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), no.4, 403-415.Google Scholar

  • [15] P. Hajłasz, Sobolev spaces on metric-measure spaces, In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), pp. 173-218, Contemp. Math. 338, Amer. Math. Soc. Providence, RI, 2003.Google Scholar

  • [16] P. Hajłasz and P.Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688Google Scholar

  • [17] T. Heikkinen, J. Kinnunen, J. Nuutinen, and H. Tuominen, Mapping properties of the discrete fractional maximal operator in metric measure spaces, to appear in Kyoto J. Math.Google Scholar

  • [18] T. Heikkinen and H. Tuominen, Smoothing properties of the discrete fractional maximal operator on Besov and Triebel-Lizorkin spaces, http://arxiv.org/abs/1301.4819Google Scholar

  • [19] J. Kinnunen and E. Saksman, Regularity of the fractional maximal function, Bull. London Math. Soc. 35 (2003), no. 4, 529-535.Google Scholar

  • [20] N. Kruglyak and E. A.Kuznetsov, Sharp integral estimates for the fractional maximal function and interpolation, Ark. Mat. 44 (2006), no. 2, 309-326.Google Scholar

  • [21] M. T. Lacey, K. Moen, C. Pérez, and R. H. Torres, Sharp weighted bounds for fractional integral operators, J. Funct. Anal. 259 (2010), no. 5, 1073-1097.Web of ScienceGoogle Scholar

  • [22] P. MacManus, Poincaré inequalities and Sobolev spaces, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), Publ. Mat. 2002, 181-197.Google Scholar

  • [23] P. MacManus, The maximal function and Sobolev spaces, unpublished preprintGoogle Scholar

  • [24] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274.Google Scholar

  • [25] E. Nakai, The Campanato, Morrey and Hölder spaces on spaces of homogeneous type, Studia Math. 176 (2006), no. 1, 1-19.Google Scholar

  • [26] C. Pérez and R. Wheeden, Potential operators, maximal functions, and generalizations of A1, Potential Anal. 19 (2003), no. 1, 1-33.Google Scholar

  • [27] E. Routin, Distribution of points and Hardy type inequalities in spaces of homogeneous type, preprint (2012), http://arxiv.org/abs/1201.5449Google Scholar

  • [28] E. T. Sawyer, R. L. Wheeden, and S. Zhao, Weighted norm inequalities for operators of potential type and fractional maximal functions, Potential Anal. 5 (1996), no. 6, 523-580.Google Scholar

  • [29] R. L. Wheeden, A characterization of some weighted norm inequalities for the fractional maximal function, Studia Math. 107 (1993), 257-272.Google Scholar

  • [30] D. Yang, New characterizations of Hajłasz-Sobolev spaces on metric spaces, Sci. China Ser. A 46 (2003), no. 5, 675-689.CrossrefGoogle Scholar

About the article

Received: 2013-01-30

Accepted: 2013-05-10

Published Online: 2013-05-28

Citation Information: Analysis and Geometry in Metric Spaces, Volume 1, Pages 147–162, ISSN (Online) 2299-3274, DOI: https://doi.org/10.2478/agms-2013-0002.

Export Citation

©2013 Versita Sp. z o.o.. This content is open access.

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Toni Heikkinen, Juha Kinnunen, Janne Korvenpää, and Heli Tuominen
Arkiv för Matematik, 2015, Volume 53, Number 1, Page 127
Yufeng Lu, Dachun Yang, and Wen Yuan
Potential Analysis, 2014, Volume 41, Number 1, Page 215
Tomasz Adamowicz, Michał Gaczkowski, and Przemysław Górka
Revista Matemática Complutense, 2018
Krzysztof Stempak and Xiangxing Tao
Journal of Function Spaces, 2014, Volume 2014, Page 1

Comments (0)

Please log in or register to comment.
Log in