The Boundedness Of Riesz Transform And Square Function On Hardy Space H_L~p

Classical Hardy space is one of core contents of harmonic analysis, which plays a fundamental role in function space, interpolation of operator and the boundedness of operators. In recent years, Hardy spaces HLp(Rn) associated with an operator L and their dual spaces have been developed greatly. Furthermore, with these devel-opment of researches, the boundedness of operator function associated with L has arose people’s much interests. In this paper, the author fist introduces the definition of Hardy space HLp(Rn) (0< p< 1) associated with the L, where L is homogenous high order elliptic operator, and then show that the new Hardy space HLp(Rn) can be characterized by area integrals. On the basis of these works, the author studies the characterization and boundedness of the Riesz transform (?)mL-1/2 on the Hardy space and proves that the vertical square function ghL,k associated with L is the bound operator from HLp to LP.