Multipliers for eigenfunction expansions of some Schrödinger operators

Abstract

Let $G$ be a graded nilpotent Lie group and let $L$ be a positive Rockland operator on $G$. Let ${E_\lambda }$ denote the spectral resolution of $L$ on ${L^2}(G)$. A sufficient condition is given under which a function $m$ on ${{\mathbf {R}}^ + }$ is a ${L^p}$-multiplier for $L$, $1 < p < \infty$; that is ${\left \| {\int _0^\infty {m(\lambda )d{E_\lambda }f} } \right \|_p} \leqslant {C_p}{\left \| f \right \|_p}$ for a constant ${C_p}$, $f \in {L^p}(G) \cap {L^2}(G)$. Then the same is done for an operator $\pi (L)$, where $\pi$ is a unitary representation of $G$ induced from a unitary character of a normal connected subgroup $H$ of $G$. Hence the case of the Hermite operator $- {d^2}/d{x^2} + {x^2}$ is covered and an ${L^p}$-multiplier theorem for classical Hermite expansions is obtained.