A T1 theorem for general smooth Calderón-Zygmund operators with doubling weights, and optimal cancellation conditions, II

Abstract

We extend the T1 theorem of David and Journé, and the corresponding optimal cancellation conditions of Stein, to pairs of doubling measures, completing the weighted theory begun in [20]. For example, when σ and ω are doubling measures satisfying the classical Muckenhoupt condition, and Kλ is a smooth λ-fractional CZ kernel, we show there exists a bounded operator Tλ:L2(σ)→L2(ω) associated with Kλif and only if there is a positive ‘cancellation’ constant AKλ(σ,ω) so that∫|x−x0|<N|∫ε<|x−y|<NKλ(x,y)dσ(y)|2dω(x)≤AKλ(σ,ω)∫|x0−y|<Ndσ(y),for all 0<ε<N and x0∈Rn, along with the dual inequality. These ‘cancellation’ conditions measure the classical L2 norm of integrals of the kernel over shells, and were shown in [20] to be equivalent to the associated T1 conditions,∫Q|Tλ(1Qσ)|2dω≤TTλ(σ,ω)∫Qdσ,for all cubes Q, and its dual, for doubling measures in the presence of the classical Muckenhoupt condition on a pair of measures.The cancellation conditions can be taken with respect to either cubes or balls, the latter resulting in a direct extension of Stein's characterization. More generally, this is extended to a weak form of Tb theorem.