The full T1 theorem for certain Triebel-Lizorkin spaces

Previous authors have considered generalizations of the David-Journe T1 the orem to the scale of Triebel-Lizorkin spaces (F) over dot(p)(alpha q) (IRn) under the T1 = 0 and T* 1 = 0 assumptions, where T is a (generalized) Cald eron-Zygmund operator. We prove boundedness on (F) over dot(p)(0q) (IRn) un der weaker assumptions on T1 and T* 1, which are related to the sharp T1, T *1 epsilon BMO assumptions for L-p approximate to (F) over dot(p)(02) in th e David-Journe theorem. In some cases we also show that our conditions are sharp. By similar techniques, we also obtain sharper conditions for "famili es of molecules" and "norming families" for these spaces.