In his seminal paper (Acta Math. 1960), H"ormander established the $L^p$-$L^q$ boundedness of Fourier multipliers on $\mathbb{R}^n$ for the range $1<p \leq 2 \leq q<\infty.$ Recently, Ruzhansky and Akylzhanov (JFA, 2020) extended H"ormander’s theorem for general locally compact separable unimodular groups using group von Neumann algebra techniques and as a consequence, they obtained the $L^p$-$L^q$ boundedness of spectral multipliers for general positive unbounded invariant operators on locally compact separable unimodular groups.

In this talk, we will discuss the $L^p$-$L^q$ boundedness of global pseudo-differential operators and Fourier multipliers on smooth manifolds for the range $1<p\leq 2 \leq q<\infty$ using the nonharmonic Fourier analysis developed by Ruzhansky, Tokmagambetov, and Delgado. As an application, we obtain the boundedness of spectral multipliers, embedding theorems, and time asymptotic the heat kernels for the anharmonic oscillator.

This talk is based on my joint works with Duván Cardona (UGent), Marianna Chatzakou (Imperial College London), Michael Ruzhansky (UGent), and Niyaz Tokmagambetov (UGent).