Higher-order Parabolic Schrodinger operators on Lebesgue Spaces

Abstract

Let k∈{1,2,3…}, d∈{2k+1,2k+2,…} and V∈RHσ∩G(−Δ)kk,d with σ≥d/2+k. Here RHσ is a reverse Holder class and G(−Δ)kk,d is a Gaussian class associated with (−Δ)k. Consider the parabolic Schrodinger operator L=∂t+(−Δ)k+Vk. We show that for certain k, α and V the Riesz transforms ∇2kxL−1 and VkαL−α are bounded on Lp(Rd+1+) for suitable values of p∈(1,∞).