Revista Matemática Iberoamericana


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Volume 10, Issue 3, 1994, pp. 467–505
DOI: 10.4171/RMI/159

Published online: 1994-12-31

On singular integrals of Calderón-type in $\mathbb R^n$, and BMO

Dorina Mitrea[1]

(1) University of Missouri, Columbia, United States

We prove $L^p$ (and weighted $L^p$) bounds for singular integrals of the form $$\rm p.v. \int_{\mathbb R^n} E \lgroup \frac{A(x)–A(y)}{|x–y} \rgroup \frac{\Omega(x–y)}{|x–y|^n} f(y)dy,$$ where $E(t) =$ cos $t$ if $\Omega$ is odd, and $E(t) =$ sin $t$ if $\Omega$ is even, and where $\bigtriangledown A \in$ BMO. Even in the case that $\Omega$ is smooth, the theory of singular integrals with "rough" kernels plays a key role in the proof. By standard techniques, the trigonometric function $E$ can then be replaced by a large class of smooth functions $F$. Some related operators are also considered. As a further application, we prove a compactness result for certain layer potentials.

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Mitrea Dorina: On singular integrals of Calderón-type in $\mathbb R^n$, and BMO. Rev. Mat. Iberoam. 10 (1994), 467-505. doi: 10.4171/RMI/159