zbMATH — the first resource for mathematics

Boundedness of the Hardy-Littlewood maximal operator along the orbits of contractive similitudes. (English) Zbl 1279.28012
Summary: We obtain results regarding the preservation of homogeneity properties along the whole orbit of a given iterated function system (IFS). We have essentially two types of results. The first class of them contains negative results: it is possible for a classical IFS to have a complete non-homogeneous sequence of spaces along the orbit, starting from very classical homogeneous spaces such as those defined by Muckenhoupt weights. The second class contains positive results which can be summarized here by saying that the sequence of spaces defined by the orbit of contractive similitudes starting at a normal space in the sense of Ahlfors, Macías, and Segovia, preserves doubling. As a consequence of these results we conclude boundedness properties of the Hardy-Littlewood maximal operator along the orbits.

28A80 Fractals
42B25 Maximal functions, Littlewood-Paley theory
60B10 Convergence of probability measures
Full Text: DOI
[1] Aimar, H., Macías, R.A.: Weighted norm inequalities for the Hardy-Littlewood maximal operator on spaces of homogeneous type. Proc. Am. Math. Soc. 91(2), 213–216 (1984) · Zbl 0539.42007
[2] Aimar, H.A., Carena, M., Iaffei, B.: Discrete approximation of spaces of homogeneous type. J. Geom. Anal. 19(1), 1–18 (2009) · Zbl 1178.28002
[3] Assouad, P.: Étude d’une dimension métrique liée à la possibilité de plongements dans R n . C. R. Acad. Sci. Paris Sér. A–B 288(15), A731–A734 (1979) · Zbl 0409.54020
[4] Coifman, R.R., Weiss, G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogènes. Lecture Notes in Mathematics, vol. 242. Springer, Berlin (1971). Étude de certaines intégrales singulières · Zbl 0224.43006
[5] Falconer, K.: Techniques in Fractal Geometry. Wiley, Chichester (1997) · Zbl 0869.28003
[6] Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969) · Zbl 0176.00801
[7] García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, vol. 116. North-Holland, Amsterdam (1985). Notas de Matemática [Mathematical Notes], 104 · Zbl 0578.46046
[8] Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981) · Zbl 0598.28011
[9] Macías, R. A., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33(3), 257–270 (1979) · Zbl 0431.46018
[10] Moran, P.A.P.: Additive functions of intervals and Hausdorff measure. Proc. Camb. Philos. Soc. 42, 15–23 (1946) · Zbl 0063.04088
[11] Mosco, U.: Variational fractals. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25(3–4), 683–712 (1998). Dedicated to Ennio De Giorgi · Zbl 1016.28010
[12] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972) · Zbl 0236.26016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.