Endpoint estimates and weighted norm inequalities for commutators of fractional integrals

By D. Cruz-Uribe, Alberto Fiorenza. 2003

71 citations recorded by the source corpus.

We prove that the commutator [b, Iα], b ∈ BMO, Iα the fractional integral operator, satisfies the sharp, modular weak-type inequality f(x) tdx, where B(t) = tlog(e + t) and Ψ(t)=[tlog(e + tα/n)]n/(n−α). These commutators were first considered by Chanillo, and our result complements his. The heart of our proof consists of the pointwise inequality, M#([b, Iα]f)(x) ≤ CbBMO [Iαf(x) + Mα,Bf(x)], where M# is the sharp maximal operator, and Mα,B is a generalization of the fractional maximal operator in the scale of Orlicz spaces. Using this inequality we also prove one-weight inequalities for the commutator; to do so we prove one and two-weight norm inequalities for Mα,B which are of interest in their own right.[b, Iα]f(x)

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4e20168eca4148a21b6a94a9a8f4a6a5002e62e0

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Citations in the corpus

Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent (2010, background): “Cruz-Uribe and Fiorenza [ 4 ] have also considered it for a generalized fractional maximal operator in the scale of Orlicz spaces.”
Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces (2007, methodology): “As for condition (11) the case = 0 and p1 = p2 was proved in Cruz-Uribe and Perez [7], the case 0 < n, 1 < p1 p2 < ∞ in Cruz-Uribe and Fiorenza [6]; in the case 0 < < n, 1 < p1 < p2 < ∞ under additional assumption that v1−p ′ 1 satisfies the reverse doubling condition, a proof is also given in [9, C”
Extrapolation and interpolation in generalized Orlicz spaces (2018, background): “These operators were introduced by Chanillo [9] and weighted inequalities analogous to those for Iα were proved in [16].”
Two weight norm inequalities for fractional integral operators and commutators (2014, background): “For a counter-example and a substitute inequality, see [15].”
Weighted norm inequalities for Schr\"odinger type operators (2011, background): “5, adapting the same arguments pages 21-23 of [5], we can obtain”