Endpoint Estimates for Generalized Commutators of Hardy Operators on H1 Space

We study the H-1-boundedness of the generalized commutators of Hardy operator with a homogeneous kernel as follows: H-Omega,Lambda,beta(m) f(x) = (1/vertical bar x vertical bar(n-beta))integral (vertical bar y vertical bar vertical bar x vertical bar) (Omega(x - y)/vertical bar x - y vertical bar R-m-1(m) (A; x,y) f(y)dy, where R-m (A; x, y) = A(x) - Sigma(vertical bar alpha vertical bar<m) (1/alpha!D-alpha A(y) (x - y)(alpha) with m is an element of Z(+), 0 <= beta < n and Omega is an element of Lip(1)(Sn-1). We prove that, when m >= 1, H-Omega,Lambda,beta(m) is not bounded from H-1 to Ln/(n-beta) unless H-Omega,Lambda,beta(m) 0. Finally, we prove that H-Omega,Lambda,beta(m) is bounded from H-1 to L-n/(n-beta),L-infinity with m >= 1.