The Full T1 Theorem for Certain Triebel - Lizorkin Spaces

Abstract

Previous authors have considered generalizations of the David-Journe T1 theorem to the scale of Triebel - Lizorkin spaces (IRn) under the T1 = 0 and T = 0 assumptions, where T is a (generalized) Calderon - Zygmund operator. We prove boundedness on (IRn) under weaker assumptions on T1 and T1, which are related to the sharp T1, TBMO assumptions for in the David-Journe theorem. In some cases we also show that our conditions are sharp. By similar techniques, we also obtain sharper conditions for “families of molecules” and “norming families” for these spaces.