Sharp two-weight, weak-type norm inequalities for singular integral operators. (English) Zbl 0961.42013
The authors suggest the following analogue of the \(A_p\)-condition for the multi-dimensional two-weight case:
Theorem. Let \(T\) be a Caldón-Zygmund operator. Given a pair of weights \((u,v)\) and \(p, 1<p<\infty\), suppose that for some \(\delta>0\) and for all cubes \(Q\), \[ \|u\|_{(L\log L)^{p-1}+\delta,Q} \Biggl(\frac{1}{|Q|}\int_Q v^{-p'/p} dx \Biggr)^{p/p'} \leq K<\infty. \] Then for every \(t>0\) and \(f\in L^p(v)\) \[ u(\{x\in R^n:|Tf(x)|>t\}) \leq \frac{C}{t^p}\int_{R^n}|f|^p v dx. \] Further, this result is sharp since it does not hold in general when \(\delta=0\).
Theorem. Let \(T\) be a Caldón-Zygmund operator. Given a pair of weights \((u,v)\) and \(p, 1<p<\infty\), suppose that for some \(\delta>0\) and for all cubes \(Q\), \[ \|u\|_{(L\log L)^{p-1}+\delta,Q} \Biggl(\frac{1}{|Q|}\int_Q v^{-p'/p} dx \Biggr)^{p/p'} \leq K<\infty. \] Then for every \(t>0\) and \(f\in L^p(v)\) \[ u(\{x\in R^n:|Tf(x)|>t\}) \leq \frac{C}{t^p}\int_{R^n}|f|^p v dx. \] Further, this result is sharp since it does not hold in general when \(\delta=0\).
Reviewer: Maria Roginskaya (Göteborg)
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.) |
| 47G10 | Integral operators |
| 42B25 | Maximal functions, Littlewood-Paley theory |
