Characterizations Of Pseudo-differential Operators For Hardy Spaces

Hardy space theory is one of the most important parts in harmonic analysis, it can be characterized by different kinds of operators which is defined by the heat semigroup et△and the Poisson semigroup e-t(?)-△generated by Laplace operator.In this paper, we will prove that the classic Hardy space Hp(Rn)(0<p≤1) is characterized by the area integral, square function and the maximal function defined by semigroup e-tσ(D) and e-t(?)-σ(D) where σ(D) is pseudo-differential operator and its symbol σ(ζ) is nonnegative homogeneous function of degree γ.This paper will be divided into four parts:In the first part, we will introduce prior knowledge, including the convolution, Fourier transform and Fourier multipli-ers and so on. In the second part, we will introduce the real variable characterization of Hardy space. In the third part, we will prove that the classic Hardy space can be characterized by pseudo-differential σ operator. In the final part, we will introduce BMO (the dual of H1), and give its characterization related to σ(D).