Research On The Boundedness Of Riesz-type Operators On Heisenberg Groups

In recent years,some problems about the boundedness of Riesz-type operators and their commutators on the Heisenberg group have become a research topic in harmonic analysis.Let Hn be the Heisenberg group,Q=2n+2,be its homogeneous dimension.The purpose of this paper is to study the Hardy type estimates for the operator T_?=V^?(-?_Hⁿ+V)^-? and the commutator[b,T?],where 0<?<Q/2.Let H_L^p(Hⁿ)be the Hardy type space associated with L=-?_Hⁿ+V,where ?_Hⁿ is the sub-Laplacian,0(?)V ? B_q1(q1? Q/2),B_q1 represents the reverse Holder class.In this paper,first,we give some basic knowledge,including auxiliary function ?(·),BMO type space BMO??(Hn),Hardy type space H_L¹(Hⁿ),and related conclusions;then,we obtain the L^p(Hⁿ)-boundedness of T?;subsequently,we prove that[b,T?]is bounded on L^p(Hⁿ);at last,with the help of the L^p(Hⁿ)-boundedness of Ta and[b,T?],we prove the following two conclusions respectively;when Q/(Q+?₀)<p?1,?₀=min{1,2-Q/q1},Ta is bounded from H_L^p(Hⁿ)into L^p(Hⁿ);if b is a BMO type function,then[b,T?]is bounded from H_L¹(Hⁿ)into weak L¹(Hⁿ).