A T1 theorem for general Calderón—Zygmund operators with comparable doubling weights, and optimal cancellation conditions

We begin an investigation into extending the T1 theorem of David and Journé, and the corresponding optimal cancellation conditions of Stein, to more general pairs of distinct doubling weights. For example, when 0 < α < n, and σ and ω are A∞ weights satisfying the one-tailed Muckenhoupt conditions, and Kα is a smooth fractional CZ kernel, we show there exists a bounded operator Tα: L2(σ) → L2(ω) associated with Kα if and only if there is a positive constant AKα(σ,ω)$${\mathfrak{A}_{{K^\alpha }}}(\sigma ,\omega )$$ so that ∫‖x−x0‖