zbMATH — the first resource for mathematics

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.

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.)
Full Text: DOI