Hardy Space Associated With The Schrodinger Operator And Its Applications

The research of singular integral operator about the boundedness in a function space is the central content of harmonic analysis. We know that the classical singular integral operator on Lebesgue spaces Lp (1<p<∞) is bounded, but they are not necessarily bounded in L1space. Hardy space is the L1space of alternatives, we can prove that the classical singular integral operator is bounded from H1to L1. Similarly, the classical Riesz transform can be used to characterize the classical Hardy space, but Riesz transform associated with the Schrodinger operator cannot be used to characterize the classical Hardy space. They are not bounded from H1to L1. So we define a new Hardy space:Hardy space associated with Schrodinger operator, and we also research the boundedness about the operator singular integral operator associated with the Schrodinger operator on its space.On Euclidean space, we will similarly extend to the Heisenberg group based on the classical-theory of Hardy space. Classical Hardy space and the Hardy space associated with Schrodinger operator has important links. In this paper, we focused on the research about Hardy space associated with the Schrodinger operator. We also have proved that if a function F satisfies a Mihlin condition with an exponent a>Q/2, then the operator F(L)=∫0∞F(λ)dEL(λ) is bounded on Lp(H")(1<p<β) and Hardy space HL1(H"), where HL1(H") is the space of functions f∈L1(H") such that Mf(x)=sup|e-sLf(x)|∈L1(H").Firstly, This paper introduce the meaning and status of researching Hardy space, we also introduce the content and the structure arrangement of this paper.Secondly, The paper give the correlative definition of the classical Hardy space and some equivalent characterizations.Thirdly, We give some theories about Hardy space associated with Schrodinger operator on the Heisenberg group.Next, We have extended to the Heisenberg group according to some results on the classical Hardy space, and we have proved the boundedness of the spectrum multiplier associated with the Schrodinger operator. Lastly, we sum up this paper and give some views of the following-up research.