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.
Reviewer: A.Seeger (Madison)


42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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