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Weighted norm inequalities for fractional maximal operators. (English) Zbl 0546.42018
Harmonic analysis, Semin. Montréal/Qué. 1980, CMS Conf. Proc. 1, 283-309 (1981).
[For the entire collection see Zbl 0538.00010.]
The author gives a necessary and sufficient condition in order that \(\| M_{\mu,\alpha}f\|_{L^{q,p}(\omega)}\leq C \| f\|_{L^ p(\nu)},\quad 0\leq\alpha \leq p\alpha <n,\quad 1/q\geq 1/p- \alpha /n.\) Here \(\mu\),\(\nu\),\(\omega\) are positive measures on \({\mathbb{R}}^ n\) and \(M_{\mu,\alpha}f(x)=\sup| Q|_{\mu}^{\alpha /n-1}\int_{Q}| f| d\mu\) where sup is taken over all cubes containing x. If \(p=q\), \(\mu =\nu =Lebesgue\) measure the result is reduced to the fact that \[ \int_{{\mathbb{R}}^ n}| M_{\alpha}f(x)|^ p w(x) dx\leq C \int_{{\mathbb{R}}^ n}| f(x)|^ p dx \] if and only if \(M_{p\alpha}w\) is bounded, where \(M_{\alpha}\) is \(M_{\mu,\alpha}\) with \(\mu =Lebesgue\) measure. The main result also contains that of D. R. Adams as a special case where \(p<q\), \(\mu =\nu =Lebesgue\) measure.
Reviewer: H.Tanabe

MSC:
42B25 Maximal functions, Littlewood-Paley theory
42B30 \(H^p\)-spaces