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.
@article{bwmeta1.element.doi-10_2478_agms-2013-0002,
author = {Toni Heikkinen and Juha Lehrb\"ack and Juho Nuutinen and Heli Tuominen},
title = {Fractional Maximal Functions in Metric Measure Spaces},
journal = {Analysis and Geometry in Metric Spaces},
volume = {1},
year = {2013},
pages = {147-162},
zbl = {1275.42032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_agms-2013-0002}
}
Toni Heikkinen; Juha Lehrbäck; Juho Nuutinen; Heli Tuominen. Fractional Maximal Functions in Metric Measure Spaces. Analysis and Geometry in Metric Spaces, Tome 1 (2013) pp. 147-162. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_agms-2013-0002/
[1] D. R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), 765-778. | Zbl 0336.46038
[2] D. R. Adams, Lecture Notes on Lp-Potential Theory, Dept. of Math., University of Umeå, 1981.
[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.
[5] S. M. Buckley, Inequalities of John-Nirenberg type in doubling spaces, J. Anal. Math 79 (1999), 215-240.[Crossref] | Zbl 0990.46019
[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. | Zbl 1023.42001
[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. | Zbl 0695.43006
[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. | Zbl 0921.43005
[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. | Zbl 0955.42001
[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. | Zbl 0881.42016
[12] A.Gogatishvili, Fractional maximal functions in weighted Banach function spaces, Real Anal. Exchange 25 (1999/00), no. 1, 291-316. | Zbl 1015.42014
[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. | Zbl 1256.42030
[14] P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), no.4, 403-415. | Zbl 0859.46022
[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. | Zbl 1048.46033
[16] P. Hajłasz and P.Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688 | Zbl 0954.46022
[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. | Zbl 1280.42012
[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 | Zbl 1295.42004
[19] J. Kinnunen and E. Saksman, Regularity of the fractional maximal function, Bull. London Math. Soc. 35 (2003), no. 4, 529-535. | Zbl 1021.42009
[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. | Zbl 1156.42308
[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] | Zbl 1196.42014
[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. | Zbl 0289.26010
[25] E. Nakai, The Campanato, Morrey and Hölder spaces on spaces of homogeneous type, Studia Math. 176 (2006), no. 1, 1-19. | Zbl 1121.46031
[26] C. Pérez and R. Wheeden, Potential operators, maximal functions, and generalizations of A1, Potential Anal. 19 (2003), no. 1, 1-33. | Zbl 1027.42015
[27] E. Routin, Distribution of points and Hardy type inequalities in spaces of homogeneous type, preprint (2012), http://arxiv.org/abs/1201.5449 | Zbl 1305.42018
[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. | Zbl 0873.42012
[29] R. L. Wheeden, A characterization of some weighted norm inequalities for the fractional maximal function, Studia Math. 107 (1993), 257-272. | Zbl 0809.42009
[30] D. Yang, New characterizations of Hajłasz-Sobolev spaces on metric spaces, Sci. China Ser. A 46 (2003), no. 5, 675-689.[Crossref] | Zbl 1092.46026