Sobolev spaces related to Schrödinger operators with polynomial potentials

Abstract

The aim of this note is to prove the following theorem. Let

$$Af(x)=P(D)f(x)+V(x)f(x),$$

where P(ix) is a nonnegative homogeneous elliptic polynomial on R ^d and V is a nonnegative polynomial potential. Then for every 1 < p < ∞ and every α > 0 there exist constants C ₁, C ₂ > 0 such that

$$\|P(D)^{\alpha}f\|_{L^p}+\|V^{\alpha}f\|_{L^p} \le C_1\|A^{\alpha}f\|_{L^p}$$

and

$$\|A^{\alpha}f\|_{L^p} \le C_2\|\left(P(D)^{\alpha}+V^{\alpha} \right)f\|_{L^p}$$

for f in the Schwartz class ${\mathcal{S}({\bf R}^d)}$ . We take advantage of the Christ inversion theorem for singular integral operators with a small amount of smoothness on nilpotent Lie groups, the maximal subelliptic L ²-estimates for the generators of stable semi-groups of measures, and the principle of transference of Coifman–Weiss.