$L^p$-$L^q$ Boundedness of Spectral Multipliers of the Anharmonic Oscillator

Abstract

In this note we study the $L^p-L^q$ boundedness of Fourier multipliers of anharmonic oscillators, and as a consequence also of spectral multipliers, for the range $1<p \leq 2 \leq q <\infty$. The underlying Fourier analysis is associated with the eigenfunctions of an anharmonic oscillator in some family of differential operators having derivatives of any order. Our analysis relies on a version of the classical Paley-type inequality, introduced by H\"ormander, that we extend in our nonharmonic setting.