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Commutators with Riesz potentials in one and several parameters. (English) Zbl 1138.42008
Let $$M_b$$ be the operator of pointwise multiplication by $$b$$, that is $$\text M_b f=bf$$. Set $$[A,B]= AB- BA$$. The Riesz potentials are the operators
$R_\alpha f(x)=\int f(x-y)\frac{dy}{abs y ^{\alpha}},\qquad 0<\alpha<1.$
They map $$L^p\mapsto L^q$$, for $$1-\alpha+\frac1q=\frac1p$$, a fact we shall take for granted in this paper. A Theorem of S. Chanillo [Indiana Univ. Math. J. 31, 7–56 (1982; Zbl 0523.42015)] states that one has the equivalence
$\|[M_b, R_\alpha]\|_{p\to q}\simeq \|b\|_{\text{BMO}}$
with the later norm being that of the space of functions of bounded mean oscillation. We discuss a new proof of this result in a discrete setting, and extend part of the equivalence above to the higher parameter setting.

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 42B35 Function spaces arising in harmonic analysis 31C45 Other generalizations (nonlinear potential theory, etc.)
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