Regularity properties of singular integral operators

Abdellah Youssfi

Studia Mathematica (1996)

  • Volume: 119, Issue: 3, page 199-217
  • ISSN: 0039-3223

Abstract

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For s>0, we consider bounded linear operators from D ( ℝ n ) into D ' ( ℝ n ) whose kernels K satisfy the conditions | ∂ x γ K ( x , y ) | ≤ C γ | x - y | - n + s - | γ | for x≠y, |γ|≤ [s]+1, | ∇ y ∂ x γ K ( x , y ) | ≤ C γ | x - y | - n + s - | γ | - 1 for |γ|=[s], x≠y. We establish a new criterion for the boundedness of these operators from L 2 ( ℝ n ) into the homogeneous Sobolev space Ḣ s ( ℝ n ) . This is an extension of the well-known T(1) Theorem due to David and Journé. Our arguments make use of the function T(1) and the BMO-Sobolev space. We give some applications to the Besov and Triebel-Lizorkin spaces as well as some other potential spaces.

How to cite

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Youssfi, Abdellah. "Regularity properties of singular integral operators." Studia Mathematica 119.3 (1996): 199-217. <http://eudml.org/doc/216296>.

@article{Youssfi1996,
abstract = {For s>0, we consider bounded linear operators from $D(ℝ^n)$ into $D^\{\prime \}(ℝ^n)$ whose kernels K satisfy the conditions $|∂^\{γ\}_\{x\}K(x,y)| ≤ C_\{γ\}|x-y|^\{-n+s-|γ|\}$ for x≠y, |γ|≤ [s]+1, $|∇_\{y\} ∂^\{γ\}_\{x\} K(x,y)| ≤ C_\{γ\}|x-y|^\{-n+s-|γ|-1\}$ for |γ|=[s], x≠y. We establish a new criterion for the boundedness of these operators from $L^2(ℝ^n)$ into the homogeneous Sobolev space $Ḣ^s(ℝ^n)$. This is an extension of the well-known T(1) Theorem due to David and Journé. Our arguments make use of the function T(1) and the BMO-Sobolev space. We give some applications to the Besov and Triebel-Lizorkin spaces as well as some other potential spaces.},
author = {Youssfi, Abdellah},
journal = {Studia Mathematica},
keywords = {Besov spaces; regularity; singular integral operators; Theorem; BMO-Sobolev space; Triebel-Lizorkin spaces},
language = {eng},
number = {3},
pages = {199-217},
title = {Regularity properties of singular integral operators},
url = {http://eudml.org/doc/216296},
volume = {119},
year = {1996},
}

TY - JOUR
AU - Youssfi, Abdellah
TI - Regularity properties of singular integral operators
JO - Studia Mathematica
PY - 1996
VL - 119
IS - 3
SP - 199
EP - 217
AB - For s>0, we consider bounded linear operators from $D(ℝ^n)$ into $D^{\prime }(ℝ^n)$ whose kernels K satisfy the conditions $|∂^{γ}_{x}K(x,y)| ≤ C_{γ}|x-y|^{-n+s-|γ|}$ for x≠y, |γ|≤ [s]+1, $|∇_{y} ∂^{γ}_{x} K(x,y)| ≤ C_{γ}|x-y|^{-n+s-|γ|-1}$ for |γ|=[s], x≠y. We establish a new criterion for the boundedness of these operators from $L^2(ℝ^n)$ into the homogeneous Sobolev space $Ḣ^s(ℝ^n)$. This is an extension of the well-known T(1) Theorem due to David and Journé. Our arguments make use of the function T(1) and the BMO-Sobolev space. We give some applications to the Besov and Triebel-Lizorkin spaces as well as some other potential spaces.
LA - eng
KW - Besov spaces; regularity; singular integral operators; Theorem; BMO-Sobolev space; Triebel-Lizorkin spaces
UR - http://eudml.org/doc/216296
ER -

References

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