# A counter-example in singular integral theory

Lawrence B. Difiore; Victor L. Shapiro

Studia Mathematica (2012)

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

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topLawrence 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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