A counter-example in singular integral theory
Volume 213 / 2012
Abstract
An improvement of a lemma of Calderón and Zygmund involving singular spherical harmonic kernels is obtained and a counter-example is given to show that this result is best possible. In a particular case when the singularity is $O(\vert \!\log r\vert)$, let $f\in C^{1}(\mathbb{R}^{N}\backslash\{ 0\})$ and suppose $f$ vanishes outside of a compact subset of $\mathbb{R}^{N},N\geq2.$ Also, let $k(x) $ be a Calderón–Zygmund kernel of spherical harmonic type. Suppose $f(x) =O(\vert\! \log r\vert)$ as $r\rightarrow0$ in the $L^{p}$-sense. Set \[ F(x) =\int_{\mathbb{R}^{N}}k(x-y) f(y)\, dy \quad \forall x\in\mathbb{R}^{N}\backslash\{0\}. \] Then $F(x) ={O}(\log^{2}r) $ as $r\rightarrow0$ in the $L^{p}$-sense, $1< p< \infty.$ A counter-example is given in $\mathbb{R}^{2}$ where the increased singularity ${O}( \log^{2}r) $ actually takes place. This is different from the situation that Calderón and Zygmund faced.
