Endpoint estimates and weighted norm inequalities for commutators of fractional integrals
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...
| Authors: | , |
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| Format: | article |
| Publication Date: | 2003 |
| Country: | España |
| Institution: | Universitat Autònoma de Barcelona |
| Repository: | Dipòsit Digital de Documents de la UAB |
| Language: | English |
| OAI Identifier: | oai:ddd.uab.cat:1999 |
| Online Access: | https://ddd.uab.cat/record/1999 https://dx.doi.org/urn:doi:10.5565/PUBLMAT_47103_05 |
| Access Level: | Open access |
| Keyword: | Fractional integrals Commutators BMO Orlicz spaces Maximal functions Norm inequalities |
| Summary: | 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). |
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