Endpoint estimates for generalized commutators of hardy operators on H 1 space

We study the h 1 -boundedness of the generalized commutators of hardy operator with a homogeneous kernel as follows: ω,a,β m f (x) = (1 / | x | n - β) ∫ | y | < | x | (ω (x - y) / | x - y | m - 1) r m (a; x,y) f (y) d y,where r m (a; x,y) = a (x) - + | α | < m (1 / α !) d α a (y) (x - y) α with m z +,0 ≤ β < n and ω lip 1 (s n - 1). we prove that,when m ≥ 1,ω,a,β m is not bounded from h 1 to l n / (n - β) unless ω,a,β m ≡ 0. finally,we prove that ω,a,β m is bounded from h 1 to l n / (n - β),∞ with m ≥ 1. © 2013 xiao yu and shanzhen lu.