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Weighted norm inequalities for operators of potential type and fractional maximal functions. (English) Zbl 0873.42012

Let \((X,d,\mu)\) be a space of homogeneous type in the Coifman and Weiss sense, that is, \(X\) is a topological space equipped with a pseudo-metric \(d\) and a doubling measure \(\mu\). The type of operators considered is \[ T(fd\sigma)(x)=\int_{X}K(x,y)f(y)d\sigma(y). \]
The authors consider the following sort of two-weight norm estimate \[ \Biggl( \int_{X}T(fd\sigma)(x)^{q}d\omega (x)\Biggr)^{1/q} \leq C\Biggl(\int_{X}f(x)^{p}d\sigma(x)\Biggr)^{1/p} \] The main results are of the following type; the weighted inequality above holds for \(1<p\leq q<\infty\) and for all positive \(f\), if and only if \[ \Biggl( \int_{X}T(\chi_{Q}d\sigma)(x)^{q}d\omega (x) \Biggr)^{1/q}\leq C|Q|_{\sigma}^{1/p} \] and \[ \Biggl( \int_{X}T^{*}(\chi_{Q}d\sigma)(x)^{p'}d\omega (x) \Biggr)^{1/p'}\leq C|Q|_{\omega}^{1/q'} \] for a fixed dyadic decomposition \(Q\in\mathcal D\), and \(|Q|_{\mu}\) denotes the \(\mu\) measure of \(Q\).
The authors also prove some variants of this result, where they make further assumptions on the operator and instead can make the above integration conditions more local. They also get results for half-spaces \(X\times[0,\infty)\) as corollaries of the homogeneous space results.
Reviewer: O.Svensson (Lulea)

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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