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For s>0, we consider bounded linear operators from $D\left({ℝ}^n\right)$ into $D^{\text{'}}\left({ℝ}^n\right)$ whose kernels K satisfy the conditions $|{\partial }_x^{\gamma }K(x,y)|\le C_{\gamma }{|x-y|}^{-n+s-\left|\gamma \right|}$ for x≠y, |γ|≤ [s]+1, $|{\nabla }_y{\partial }_x^{\gamma }K(x,y)|\le C_{\gamma }{|x-y|}^{-n+s-\left|\gamma \right|-1}$ for |γ|=[s], x≠y. We establish a new criterion for the boundedness of these operators from $L^2\left({ℝ}^n\right)$ into the homogeneous Sobolev space ${Ḣ}^s\left({ℝ}^n\right)$. 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...

In this paper we consider the regularity problem for the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ where $b$ is a locally integrable function and $(R_j{)}_{1\le j\le n}$ are the Riesz transforms in the $n$-dimensional euclidean space ${ℝ}^n$. More precisely, we prove that these commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded from $L^p$ into the Besov space ${\dot{B}}_p^{s,p}$ for $1\<p\<+\infty$ and $0\<s\<1$ if and only if $b$ is in the $BMO$-Triebel-Lizorkin space ${\dot{F}}_{\infty }^{s,p}$. The reduction of our result to the case $p=2$ gives in particular that the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded form $L^2$ into the Sobolev space ${\dot{H}}^s$ if and only if $b$ is in the $BMO$-Sobolev...

L'objet de ce travail est l'étude de la continuité des opérateurs d'intégrales singulières (au sens de Calderón-Zygmund) sur les espaces de Sobolev H. Il complète le travail fondamental de David-Journé [6], concernant le cas s = 0, et ceux de P. G. Lemarié [10] et M. Meyer [11] concernant le cas 0 < s < 1.