×

Regularity properties of singular integral operators. (English) Zbl 0857.42008

Summary: For \(s>0\) we consider bounded linear operators from \({\mathcal D}(\mathbb{R}^n)\) into \({\mathcal D}'(\mathbb{R}^n)\) whose kernels \(K\) satisfy the conditions \[ \begin{alignedat}{2}2|\partial^\gamma_x K(x,y)|&\leq C_\gamma|x-y|^{-n+s-|\gamma|} &&\quad\text{for } x\neq y,\;|\gamma|\leq [s]+1,\\ |\nabla_y \partial^\gamma_x K(x,y)|&\leq C_\gamma|x-y|^{-n+s-|\gamma|-1} &&\quad\text{for } |\gamma|=[s],\;x\neq y.\end{alignedat} \] We establish a new criterion for the boundedness of these operators from \(L^2(\mathbb{R}^n)\) into the homogeneous Sobolev space \(\dot H^s(\mathbb{R}^n)\). 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.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDFBibTeX XMLCite
Full Text: DOI EuDML