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February 2007 Commutators with Reisz potentials in one and several parameters
Michael T. LACEY
Hokkaido Math. J. 36(1): 175-191 (February 2007). DOI: 10.14492/hokmj/1285766657


Let $M_b$ be the operator of pointwise multiplication by $b$, that is $M_b f=bf$. Set $[A,B]=AB-BA$. The Reisz potentials are the operators $R_\alpha f(x)=\int f(x-y)\frac{dy}{\abs y ^{\alpha} }, 0<\alpha<1.$ They map $L^p\mapsto L^q$, for $1-\alpha+\frac{1}{q}=\frac{1}{p}$, a fact we shall take for granted in this paper. A Theorem of Chanillo [6] states that one has the equivalence $||[M_b,R_\alpha]||_{p\to q}\simeq ||b||_{BMO}$ with the later norm being that of the space of functions of bounded mean oscillation. We discuss a proof of this result in a discrete setting, and extend part of the equivalence above to the higher parameter setting.


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Michael T. LACEY. "Commutators with Reisz potentials in one and several parameters." Hokkaido Math. J. 36 (1) 175 - 191, February 2007.


Published: February 2007
First available in Project Euclid: 29 September 2010

zbMATH: 1138.42008
MathSciNet: MR2309828
Digital Object Identifier: 10.14492/hokmj/1285766657

Primary: 42B35
Secondary: 42B20 , 42B25

Keywords: bounded mean oscillation , commutator , fractional integral , Multiparameter , paraproduct , Reisz potential

Rights: Copyright © 2007 Hokkaido University, Department of Mathematics


Vol.36 • No. 1 • February 2007
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