Vertical Littlewood-Paley-Stein functions related to Schrödinger operators on stratified Lie groups

Let L = − Δ G + ϒ be a Schrödinger operator with a nonnegative potential ϒ ∈ L l o c 1 ( G ) , where G is a stratified Lie group and − Δ G is the sub-Laplacian on G . It is known that the horizontal Littlewood-Paley-Stein functions associated to L are bounded on L p ( G ) for all p ∈ ( 1 , ∞ ) . In this paper, inspired by the recent work of Ouhabaz et al., we will study the boundedness of the vertical Littlewood-Paley-Stein function, defined by H L f ( x ) : = ( ∫ 0 ∞ | ∇ G e − s L f ( x ) | 2 d s ) 1 / 2 . We show that for some ϒ > 0 , H L is unbounded on L p ( G ) for any p > Q , where Q ≥ 3 is the homogeneous dimension of the group G . On the other hand, we prove the boundedness of H L on L p ( G ) for all p ∈ ( 1 , 2 ] . Under the additional assumption that ϒ ∈ R H q , the Reverse-Hölder class, for some q ∈ ( Q / 2 , Q ) , we can show that H L is weak type ( 1 , 1 ) and strong type ( p , p ) for all 1 < p < p 0 , where 1 p 0 = 1 q − 1 Q . Moreover, we prove that this p 0 is optimal in the sense that there exists a ϒ ∈ ∪ r < q R H r such that H L is not bounded on L p 0 ( G ) . Finally, under a stronger assumption ϒ ∈ R H Q , we show that H L is bounded from H L 1 ( G ) into L 1 ( G ) and bounded on B M O L ( G ) , which is new even in Euclidean spaces.