Carleson Measure Of Campanato-type Space Related To Schr(?)dinger Operator On Hierarchical Lie Groups

Let L=-?G+V be a Schrodinger operator on the stratified Lie group G,where?G is the sub-Laplacian and the nonnegative potential V belongs to the reverse Holder class Bq0 with q0>2/2 and Q is the homogeneous dimension of G.In this article,by Campanato type spaces AL?(G),we introduce Hardy type spaces associated with L denoted by HLp(G)and prove the atomic characterization of HLp(G).Further we prove the following duality relation:?L2(1/p-1)(G=(HLP(G))*,2/(2+?)<p<1 for ?=min{1,2-2/q0}.The above relation enables us to characterize AL?(G)via two families of Carleson measures generated by the heat semigroup and the Poisson semigroup,respectively.Also,we obtain two classes of perturbation formulas associated with the semigroups related to L.As applications,we obtain the boundedness of Littlewood-Paley g-function and the Lusin area function on AL?(G).