A two weight weak inequality for potential type operators. (English) Zbl 0746.42011
The authors consider weak type inequalities, involving two weight functions, for potential operators acting on Lorentz spaces. The underlying metric on \(R^ n\) is defined in terms of the quasi-norm
\[
\| x\|=\max_{i=1,\dots,n}| x_ i|^{\alpha_ i},\;\alpha_ i>0\text{ and } \sum_{i=1}^ n \alpha_ i=n.
\]
The necessary and sufficient condition on the weights is analogous to the one studied earlier by E. Sawyer [Trans. Am. Math. Soc. 281, 339-345 (1984; Zbl 0539.42008)].
Reviewer: D.S.Kurtz (Las Cruces)
MSC:
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.) |