AbstractPrevious authors have considered generalizations of the David‐JournéT1 theorem to the scale of Triebel ‐ Lizorkin spaces (IRn) under theT1 = 0 andT= 0 assumptions, whereTis a (generalized) Calderon ‐ Zygmund operator. We prove boundedness on (IRn) under weaker assumptions onT1 andT1, which are related to the sharpT1,TBMOassumptions for in the David‐Journé 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”