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Tytuł artykułu

Fractional Maximal Functions in Metric Measure Spaces

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Rocznik
Tom
1
Strony
147-162
Opis fizyczny
Daty
otrzymano
2013-01-30
zaakceptowano
2013-05-10
online
2013-05-28
Twórcy
  • Department of Mathematics and Statistics, P.O. Box 35,
    FI-40014 University of Jyväskylä, Finland, juha.lehrback@jyu.fi
  • Department of Mathematics and Statistics, P.O. Box 35,
    FI-40014 University of Jyväskylä, Finland, juho.nuutinen@jyu.fi
  • Department of Mathematics and Statistics, P.O. Box 35,
    FI-40014 University of Jyväskylä, Finland, heli.m.tuominen@jyu.fi
Bibliografia
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  • [3] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin Heidelberg, 1996.
  • [4] S. M. Buckley, Is the maximal function of a Lipschitz function continuous?, Ann. Acad. Sci. Fenn. Math. 24 (1999), 519-528.
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  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [12] A.Gogatishvili, Fractional maximal functions in weighted Banach function spaces, Real Anal. Exchange 25 (1999/00), no. 1, 291-316.
  • [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.
  • [14] P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), no.4, 403-415.
  • [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.
  • [16] P. Hajłasz and P.Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688
  • [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.
  • [18] T. Heikkinen and H. Tuominen, Smoothing properties of the discrete fractional maximal operator on Besov and Triebel-Lizorkin spaces, http://arxiv.org/abs/1301.4819
  • [19] J. Kinnunen and E. Saksman, Regularity of the fractional maximal function, Bull. London Math. Soc. 35 (2003), no. 4, 529-535.
  • [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.
  • [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.[WoS]
  • [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.
  • [23] P. MacManus, The maximal function and Sobolev spaces, unpublished preprint
  • [24] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274.
  • [25] E. Nakai, The Campanato, Morrey and Hölder spaces on spaces of homogeneous type, Studia Math. 176 (2006), no. 1, 1-19.
  • [26] C. Pérez and R. Wheeden, Potential operators, maximal functions, and generalizations of A1, Potential Anal. 19 (2003), no. 1, 1-33.
  • [27] E. Routin, Distribution of points and Hardy type inequalities in spaces of homogeneous type, preprint (2012), http://arxiv.org/abs/1201.5449
  • [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.
  • [29] R. L. Wheeden, A characterization of some weighted norm inequalities for the fractional maximal function, Studia Math. 107 (1993), 257-272.
  • [30] D. Yang, New characterizations of Hajłasz-Sobolev spaces on metric spaces, Sci. China Ser. A 46 (2003), no. 5, 675-689.[Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_agms-2013-0002
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