# Commutators with fractional integral operators

Studia Mathematica (2016)

• Volume: 233, Issue: 3, page 279-291
• ISSN: 0039-3223

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## Abstract

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We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $\mu ,\lambda \in {A}_{p,q}$ and α/n + 1/q = 1/p, the norm $||\left[b,{I}_{\alpha }\right]:{L}^{p}\left({\mu }^{p}\right)\to {L}^{q}\left({\lambda }^{q}\right)||$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $\nu =\mu {\lambda }^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.

## How to cite

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Irina Holmes, Robert Rahm, and Scott Spencer. "Commutators with fractional integral operators." Studia Mathematica 233.3 (2016): 279-291. <http://eudml.org/doc/286138>.

@article{IrinaHolmes2016,
abstract = {We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $μ,λ ∈ A_\{p,q\}$ and α/n + 1/q = 1/p, the norm $||[b,I_\{α\}]: L^\{p\}(μ^\{p\}) → L^\{q\}(λ^\{q\})||$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $ν = μλ^\{-1\}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.},
author = {Irina Holmes, Robert Rahm, Scott Spencer},
journal = {Studia Mathematica},
keywords = {fractional integral operator; commutator; weighted inequalities; Bloom BMO},
language = {eng},
number = {3},
pages = {279-291},
title = {Commutators with fractional integral operators},
url = {http://eudml.org/doc/286138},
volume = {233},
year = {2016},
}

TY - JOUR
AU - Irina Holmes
AU - Robert Rahm
AU - Scott Spencer
TI - Commutators with fractional integral operators
JO - Studia Mathematica
PY - 2016
VL - 233
IS - 3
SP - 279
EP - 291
AB - We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $μ,λ ∈ A_{p,q}$ and α/n + 1/q = 1/p, the norm $||[b,I_{α}]: L^{p}(μ^{p}) → L^{q}(λ^{q})||$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $ν = μλ^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.
LA - eng
KW - fractional integral operator; commutator; weighted inequalities; Bloom BMO
UR - http://eudml.org/doc/286138
ER -

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