Endpoint estimates and weighted norm inequalities for commutators of fractional integrals. (English) Zbl 1035.42015
Summary: We prove that the commutator \([b,I_\alpha]\), \(b\in \text{BMO}\), \(I_\alpha\) the fractional integral operator, satisfies the sharp, modular weak-type inequality
\[
\Bigl|\biggl\{ x\in\mathbb{R}^n: \bigl| [b,I_\alpha]f(x) \bigr|< t\biggr\}\Bigr|\leq C\Psi\left( \int_{\mathbb{R}^n} B\left(\| b \|_{\text{BMO}} \frac{\bigl| f(x) \bigr|} {t}\right)dx\right),
\]
where \(B(t)=t\log(e+t)\) and \(\Psi(t) =[t\log(e+ t^{\alpha/n})]^{n/(n-\alpha)}\). These commutators were first considered by Chanillo, and our result complements his. The heart of our proof consists of the pointwise inequality,
\[
M^\#\bigl( [b, I_\alpha] f\bigr)(x)\leq C\| b\|_{\text{BMO}} \bigl[I_\alpha f(x)+ M_{\alpha, B} f(x)\bigr],
\]
where \(M^\#\) is the sharp maximal operator, and \(M_{\alpha,B}\) 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 \(M_{\alpha, B}\) which are of interest in their own right.
MSC:
42B25 | Maximal functions, Littlewood-Paley theory |
42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |