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

Abstract

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 K^α is a smooth fractional CZ kernel, we show there exists a bounded operator T^α: L²(σ) → L²(ω) associated with K^α if and only if there is a positive constant ${\mathfrak{A}_{{K^\alpha }}}(\sigma ,\omega )$ so that

$$\matrix{ \hfill {\int_{\left\| {x - {x_0}} \right\| < N} {{{\left| {\int_{\varepsilon < \left\| {x - y} \right\| < N} {{K^\alpha }(x,y)d\sigma (y)} } \right|}^2}d\omega (x) \le {\mathfrak{A}_{{K^\alpha }}}(\sigma ,\omega )\int_{\left\| {{x_0} - y} \right\| < N} {d\sigma (y),} } } \cr \hfill {{\rm{for}}\,{\rm{all}}\,0 < \varepsilon < N\,{\rm{and}}\,{x_0} \in {\mathbb{R}^n},} \cr } $$

where ‖y‖ ≡ max_1≤k≤n ∣y_k∣, along with a dual inequality. More generally this holds for measures σ and ω comparable in the sense of Coifman and Fefferman that satisfy a fractional Ask{∞/α} condition.

These results are deduced from the following theorem of T1 type, namely that if σ and ω are doubling measures, comparable in the sense of Coifman and Fefferman, and satisfying one-tailed Muckenhoupt conditions, then T^α: L²(σ) → L²(ω) if and only if the dual pair of testing conditions hold, as well as a strong form of the weak boundedness property,

$$\left| {\int_F {\left( {{T^\alpha }{{\bf{1}}_E}} \right)d\omega } } \right| \le {\cal B}{\cal J}{\cal C}{{\cal T}_{{T^\alpha }}}(\sigma ,\omega )\sqrt {{{\left| {{Q_\sigma }} \right|}_\sigma }{{\left| {{Q_\sigma }} \right|}_{\omega ,}}} \,\,\,\,{\rm{for}}\,{\rm{all}}\,{\rm{cubes}}\,Q \subset {\mathbb{R}^n},$$

where ${\cal B}{\cal J}{\cal C}{{\cal T}_{{T^\alpha }}}(\sigma ,\omega )$ is a positive constant called the bilinear cube/indicator testing constant. The comparability of measures and the bilinear cube/indicator testing condition can both be dropped if the stronger indicator/cube testing conditions are assumed.