Riesz transforms for bi-Schrödinger operators on weighted Lebesgue spaces

Abstract

Let d∈{5,6,7,…} and a weight w∈A∞ρ. We consider the fourth-order Riesz transform T=∇4L−1 associated with the bi-Schrödinger operator L=Δ2+V2, where V∈RHσ∩G2 with σ>2d3 and G2 stands for a Gaussian class of potentials. We show that T is bounded on Lwp(Rd) for all p in a suitable range depending on σ. If more conditions are imposed on either σ or V, the range for p can be extended to (1,∞).