Regularity properties of singular integral operators

Volume 119 / 1996

Abdellah Youssfi Studia Mathematica 119 (1996), 199-217 DOI: 10.4064/sm-119-3-199-217

Abstract

For s>0, we consider bounded linear operators from $D(ℝ^n)$ into $D'(ℝ^n)$ 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.

Authors

  • Abdellah Youssfi

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