Regularity properties of singular integral operators

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