Boundedness of High Order Commutators of Riesz Transforms Associated with Schrodinger Type Operators

Let L2=(-△)^2+ V^2 be the Schrödinger type operator, where V≠0 is a nonnegative potential and belongs to the reverse Holder class RHq1 for q1> n/2, n ≥5. The higher Riesz transform associated with L2 is denoted by R=△^2L2^-1/2and its dual is denoted by R^*=L2^-1/2△^2. In this paper, we consider the m-order commutators b^m, R] and bm, R^*], and establish the(L^p, L^q)-boundedness of these commutators when b belongs to the new Campanato space Λβ^θ(ρ) and 1/q = 1/p-mβ/n.

Boundedness of High Order Commutators of Riesz Transforms Associated with Schrödinger Type Operators

Let $\mathcal{L}_2=(-\Delta)^2+V^2$ be the Schrödinger type operator, where $V\neq 0$ is a nonnegative potential and belongs to the reverse Hölder class $RH_{q_1}$ for $q_1> n/2, n\geq 5.$ The higher Riesz transform associated with $\mathcal{L}_2$ is denoted by $\mathcal{R}=\nabla^2 \mathcal{L}_2^{-\frac{1}{2}}$ and its dual is denoted by $\mathcal{R}^*=\mathcal{L}_2^{-\frac{1}{2}} \nabla^2.$ In this paper, we consider the $m$-order commutators $b^m,\mathcal{R}]$ and $b^m,\mathcal{R}^*],$ and establish the $(L^p,L^q)$-boundedness of these commutators when $b$ belongs to the new Campanato space $\Lambda_\beta^\theta(\rho)$ and $1/q=1/p-m\beta/n.$