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The boundedness of fractional maximal operators on variable Lebesgue spaces over spaces of homogeneous type

Volume 242 / 2018

D. Cruz-Uribe, OFS, P. Shukla Studia Mathematica 242 (2018), 109-139 MSC: Primary 42B25; Secondary 46A80, 46E30. DOI: 10.4064/sm8556-6-2017 Published online: 24 November 2017

Abstract

Given a space of homogeneous type $(X,d,\mu )$, we present sufficient conditions on a variable exponent ${p(\cdot )}$ so that the fractional maximal operator $M_\eta $, $0\leq \eta \lt 1$, maps $L^{p(\cdot )}(X)$ to $L^{q(\cdot )}(X)$, where $1/{p(\cdot )}-1/{q(\cdot )}=\eta $. In the endpoint case we also prove the corresponding weak type inequality. As an application we prove norm inequalities for the fractional integral operator $I_\eta $. Our proof for the fractional maximal operator uses the theory of dyadic cubes on spaces of homogeneous type, and even in the Euclidean setting it is simpler than existing proofs. For the fractional integral operator we extend a pointwise inequality of Welland to spaces of homogeneous type. Our work generalizes results of Capone et al. (2007) and Cruz-Uribe et al. (2009) from the Euclidean case, and extends recent work by Adamowicz et al. (2015) on the Hardy–Littlewood maximal operator on spaces of homogeneous type.

Authors

  • D. Cruz-Uribe, OFSDepartment of Mathematics
    University of Alabama
    Tuscaloosa, AL 35487, U.S.A.
    e-mail
  • P. ShuklaNew Delhi, India
    e-mail

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