A counter-example in singular integral theory

Lawrence B. Difiore ; Victor L. Shapiro

Studia Mathematica, Tome 209 (2012), p. 157-167 / Harvested from The Polish Digital Mathematics Library

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(|log r|), let $f ∈ C¹(ℝ^{N}∖{0})$ and suppose f vanishes outside of a compact subset of $ℝ^{N}$, N ≥ 2. Also, let k(x) be a Calderón-Zygmund kernel of spherical harmonic type. Suppose f(x) = O(|log r|) as r → 0 in the $L^{p}$-sense. Set $F(x) = ∫_{ℝ^{N}} k(x-y)f(y)dy ∀x ∈ ℝ^{N}∖{0}$. Then F(x) = O(log²r) as r → 0 in the $L^{p}$-sense, 1 < p < ∞. A counter-example is given in ℝ² where the increased singularity O(log²r) actually takes place. This is different from the situation that Calderón and Zygmund faced.

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Publié le : 2012-01-01

Zbl 06136669

EUDML-ID : urn:eudml:doc:285634