$L^p-L^q$ boundedness of Fourier multipliers associated with the anharmonic Oscillator

In this paper we study the $L^p$-$L^q$ boundedness of the Fourier multipliers in the setting where the underlying Fourier analysis is introduced with respect to the eigenfunctions of an anharmonic oscillator $A$. Using the notion of a global symbol that arises from this analysis, we extend a version of the Hausdorff-Young-Paley inequality that guarantees the $L^p$-$L^q$ boundedness of these operators for the range $1<p leq 2 leq q <infty$. The boundedness results for spectral multipliers acquired, yield as particular cases Sobolev embedding theorems and time asymptotics for the $L^p$-$L^q$ norms of the heat kernel associated with the anharmonic oscillator. Additionally, we consider functions $f(A)$ of the anharmonic oscillator on modulation spaces and prove that Linsku iis trace formula holds true even when $f(A)$ is simply a nuclear operator.