On certain nonstandard Calderón-Zygmund operators.(English)Zbl 0826.42012

The author considers rough variants of the Calderón commutator in $$\mathbb{R}^n$$, defined by $Tf(x) = \text{ p.v. } \int \Omega \left( {x - y \over |x - y |} \right) |x - y |^{- n - 1} \biggl[ A(x) - A(y) - \bigl \langle \nabla A(y), x - y \bigr \rangle \biggr] f(y) dy$ where $$\Omega$$ belongs to $$L^r (S^{n - 1})$$, $$r > 1$$, and satisfies the moment condition $\int_{S^{n - 1}} \theta \Omega (\theta)d \theta = 0;$ furthermore it is assumed that $$\nabla A \in \text{BMO} (\mathbb{R}^n)$$. Under these conditions it is shown that $$T$$ is bounded on $$L^2$$, and the author points out that by routine but technically messy arguments one can also show the $$L^p$$-boundedness for $$1 < p < \infty$$. He proves that under the stronger assumption $$\Omega \in L^\infty (\mathbb{R}^n)$$ the operator is also bounded on the weighted $$L^p$$ space, provided the weight belongs to the Muckenhoupt class $$A_p$$. As an important tool in his proof he develops a variant of the $$T(1)$$ theorem by David and Journé, which is suitable to handle rough operators.