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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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