Endpoint Estimates for Generalized Commutators of Hardy Operators onH1Space

摘要

We study theH1-boundedness of the generalized commutators of Hardy operator with a homogeneous kernel as follows:ℋΩ,A,βmf(x)=(1/|x|n-β)∫|y|<|x|(‍Ω(x-y)/|x-y|m-1)Rm(A;x,y)f(y)dy, whereRm(A;x,y)=A(x)-∑|α|<m‍(1/α!)DαA(y)(x-y)αwithm∈Z+,0≤β<nandΩ∈Lip1(Sn-1). We prove that, whenm≥1,ℋΩ,A,βmis not bounded fromH1toLn/(n-β)unlessℋΩ,A,βm≡0. Finally, we prove thatℋΩ,A,βmis bounded fromH1toLn/(n-β),∞withm≥1.