Sobolev spaces related to Schrödinger operators with polynomial potentials

TLDR: In this article, 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 was proved.

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   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.