Herz - Type Hardy Space Related To Operator And Its Characterization

Suppose L is a nonnegative, self-adjoint differential operator. And L has H∞-calculus on L2(Rn). The kernel pt(x,y) of operator e-tL satisfies the Gaus-sian uppper bound on Rn x Rn. Firstly, we use the area integral function SL associated with the operator L to define the Herz-type Hardy spaces Hq,La,p(Mn). And the (M, L) atom is introduced. Then, by the method of the atomic decom-position of classical Hardy spaces and the Hardy space HL1(Rn) associated with operator L, we characterize HKq,La,p(Rn) spaces for atoms, that is, we proved the atomic decomposition of Herz-type Hardy spaces associated with operator L. Secondly,as an application, the boundedness of some operators from HKq,La,p(Rn) to Kqa,p(Rn) is obtained. Finally, we defined the Herz type spaces Kq,La,p (Rn) as-sociated with operator L, and gave the characterization of the spaces Kq,a,p(Rn) by "L-blocks".