A counter-example in singular integral theory

Lawrence B. Difiore¹ ; Victor L. Shapiro¹

¹ Department of Mathematics University of California Riverside, CA 92521, U.S.A.

Studia Mathematica, Tome 213 (2012) no. 2, pp. 157-167

Résumé

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.

Détail

DOI : 10.4064/sm213-2-3

Keywords: improvement lemma calder zygmund involving singular spherical harmonic kernels obtained counter example given result best possible particular singularity vert log vert mathbb backslash suppose vanishes outside compact subset mathbb geq calder zygmund kernel spherical harmonic type suppose vert log vert rightarrow sense set int mathbb x y quad forall mathbb backslash log rightarrow sense infty counter example given mathbb where increased singularity log actually takes place different situation calder zygmund faced