The Second-Order Riesz Transforms Related to Schrödinger Operators Acting on BMO Type Spaces on the Stratified Lie Group

Abstract

Let \(\mathcal {L}=-{\Delta } + V\) be a Schrodinger operator on stratified Lie group G, where V is a nonnegative potential satisfying the suitable reverse Holder’s inequality. In this paper, we study the boundedness of the second-order Riesz transforms such as \(\mathcal {L}^{-1} \nabla ^{2}\) and \(\mathcal {L}^{-1}(-{\Delta })\) on the spaces of BMO type for weighted case. We generalized the known results to the weighted case and the case of the stratified Lie group. So our results are new, even in the case \(\mathbb {R}^{n}\), and they have some interest in its own right.