Abstract
For a Young function Θ with 0 ≤ α < 1, let M α,Θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type (X, d, µ) by M α,Θ f(x) = sup x∈B µ(B)α‖f‖Θ,B , where ‖f‖Θ,B is the mean Luxemburg norm of f on a ball B. When α = 0 we simply denote it by M Θ. In this paper we prove that if Φ and Ψ are two Young functions, there exists a third Young function Θ such that the composition M α,Ψ ∘ M Φ is pointwise equivalent to M α,Θ. As a consequence we prove that for some Young functions Θ, if M α,Θ f < ∞ a.e. and δ ∈ (0, 1) then (M α,Θ f)δ is an A 1-weight.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Pérez, C.: Two weighted norm inequalities for Riesz potentials and uniform L p-weighted Sobolev inequalities. Indiana Univ. Math. J., 39(1), 31–44 (1990)
Coifman, R., Rochberg, R.: Another characterization of BMO. Proc. Amer. Math. Soc., 79, 249–254 (1980)
Pérez, C.: Weighted norm inequalities for singular integral operators. J. London Math. Soc., 49, 296–308 (1994)
Pérez, C.: Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function. J. Fourier Anal. Appl., 3(6), 743–756 (1997)
Pradolini, G., Salinas, O.: Commutators of singular integrals on spaces of homogeneous type. Czechoslovak Math. J., 57(132), 75–93 (2007)
Bernardis, A., Hartzstein, S., Pradolini, G.: Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type. J. Math. Anal. Appl., 322, 825–846 (2006)
Cruz-Uribe, D., Fiorenza, A.: Endpoint estimates and weighted norm inequalities for commutators of fractional integrals. Publ. Mat., 47, 103–131 (2003)
Gorosito, O., Pradolini, G., Salinas, O.: Weighted weak-type estimates for multilinear commutators of fractional integrals on spaces of homogeneous type. Acta Mathematica Sinica, English Series, 23(10), 1813–1826 (2007)
Carozza, M., Di Napoli, A. P.: Composition of maximal operators. Publ. Math., 40, 397–409 (1996)
Macías, R., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math., 33, 257–270 (1979)
Coifman, R., Weiss, G.: Analyse harmonique non-commutative sur certains spaces homogènes, Lecture Notes in Math., Vol. 242, Springer-Verlag, New York-Berlin, 1971
Aimar, H.: Singular integral and approximate identities on space of homogeneous type. Trans. Amer. Math. Soc., 292, 135–153 (1985)
Macías, R., Segovia, C.: A well behaved quasi-distance on spaces of homogeneous type. Trab. Mat. Inst. Argentina Mat., 32, 1–18 (1981)
Bagby, R., Parsons, J.: Orlicz spaces and rearranged maximal functions. Math. Nachr., 132, 15–27 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by Spanish government Grant MTM2005-8350-C03-02, Junta de Andalucía FQM-354, CONICET, ANPCyT, CAI+D-UNL and SECYT-UNC
Rights and permissions
About this article
Cite this article
Bernardis, A.L., Pradolini, G., Lorente, M. et al. Composition of fractional Orlicz maximal operators and A 1-weights on spaces of homogeneous type. Acta. Math. Sin.-English Ser. 26, 1509–1518 (2010). https://doi.org/10.1007/s10114-010-8445-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-010-8445-4