# A counter-example in singular integral theory

Studia Mathematica (2012)

• Volume: 213, Issue: 2, page 157-167
• ISSN: 0039-3223

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## Abstract

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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 $f\in C¹\left({ℝ}^{N}\setminus 0\right)$ 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 $F\left(x\right)={\int }_{{ℝ}^{N}}k\left(x-y\right)f\left(y\right)dy\forall x\in {ℝ}^{N}\setminus 0$. 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.

## How to cite

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Lawrence B. Difiore, and Victor L. Shapiro. "A counter-example in singular integral theory." Studia Mathematica 213.2 (2012): 157-167. <http://eudml.org/doc/285634>.

@article{LawrenceB2012,
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(|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.},
author = {Lawrence B. Difiore, Victor L. Shapiro},
journal = {Studia Mathematica},
keywords = {singular integrals; spherical harmonic kernels; singular in the lp-sense},
language = {eng},
number = {2},
pages = {157-167},
title = {A counter-example in singular integral theory},
url = {http://eudml.org/doc/285634},
volume = {213},
year = {2012},
}

TY - JOUR
AU - Lawrence B. Difiore
AU - Victor L. Shapiro
TI - A counter-example in singular integral theory
JO - Studia Mathematica
PY - 2012
VL - 213
IS - 2
SP - 157
EP - 167
AB - 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.
LA - eng
KW - singular integrals; spherical harmonic kernels; singular in the lp-sense
UR - http://eudml.org/doc/285634
ER -

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