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A two weight weak type inequality for fractional integrals. (English) Zbl 0539.42008
If \(1<p\leq q<\infty\), \(\omega\) and \(\mu\) are positive Borel measures on \(R^ n\), and if \(T(f\mu)(x):=\int_{R^ n}K(x-y)f(y)d\mu(y),\) where the kernel K(x) is a positive lower semicontinuous radial function decreasing in \(| x|\) and satisfying an additional condition if \(n\geq 2\), the author gives a necessary and sufficient condition in order that \(\int_{\{T(f\mu)>\lambda \}}d\omega \leq A \lambda^{- q}(\int_{R^ n}| f|^ pd\mu)^{q/p}\) for all \(f\geq 0\), \(\lambda>0\). In the special case \(K(x)=| x|^{\alpha -n} (0<\alpha<n)\), \(d\omega(x)=w(x)dx\), \(d\mu(x)=v(x)^{1-p'}dx\) (p’,q’ are the conjugate indices to p,q, and w,v are non-negative weight functions on \(R^ n)\), and replacing f by \(fv^{p'-1}\), \(T(f\mu)\) becomes the fractional integral operator \(T_{\alpha}\), where \(T_{\alpha}f(x):=\int_{R^ n}| x-y|^{\alpha -n}f(y)dy.\)
The theorem then shows that in order that \(\int_{\{T_{\alpha}f>\lambda \}}w(x)dx\leq A\lambda^{- q}(\int_{R^ n}| f(x)|^ pv(x)dx)^{q/p}\) for all \(f\geq 0\), \(\lambda>0\), it is necessary and sufficient that, for all cubes \(Q\subset R^ n\), \[ \int_{Q}[T_{\alpha}(\chi_ Qw)(x)]^{p'}v(x)^{1- p'}dx\leq B(\int_{Q}w(x)dx)^{p'/q'}<\infty. \]
Reviewer: D.C.Russell

MSC:
42B25 Maximal functions, Littlewood-Paley theory
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