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

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 K λ is a smooth λ-fractional CZ kernel, we show there exists a bounded operator T λ : L 2 ( σ ) → L 2 ( ω ) associated with K λ if and only if there is a …