A counter-example in singular integral theory

• Autorzy: Lawrence B. Difiore , Victor L. Shapiro
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 32A55: Singular integrals
• 42B20: Singular and oscillatory integrals (Calder\'on-Zygmund, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2012
• Tom: 213
• Numer: 2
• Strony: 157-167

Daty

wydano

2012

Twórcy

autor

Lawrence B. Difiore

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

autor

Victor L. Shapiro

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

Identyfikatory

DOI

10.4064/sm213-2-3