The Generalization of Paraproducts and the Full T1 Theorem for Sobolev and Triebel]Lizorkin Spaces

We study the boundedness of generalized Calder´on]Zygmund operators acting on Sobolev and, more generally, Triebel]Lizorkin spaces of arbitrary order of smoothness. We are able to relax the assumptions T xg.s0 andror T* xg .s0, which have been required in earlier results by other authors. To do this, we consider generalized paraproduct operators. We obtain the sharpness of our assumptions for some special cases. Using the same technique, we also obtain some sharper results for ‘‘norming’’ and ‘‘molecular’’ families.