[HG_AP] A quasimode approach to spectral multipliers
ABSTRACT
A central question of Euclidean harmonic analysis is; when does a multiplier
$$Mf=\mathcal{F}^{-1}\left[m(\cdot)\mathcal{F}[f](\cdot)\right]$$
defined as an operator $L^{2}\to L^{2}$ extend to a bounded operator $L^{p}\to L^{q}$? The Bochner-Riesz multipliers where
$$m_{R}(\xi)=\left(1-\frac{|\xi|^{2}}{R}\right)_{+}^{\delta}$$
are one well-known example of these type of operators. On manifolds we can consider analogous questions about whether spectral multipliers $M(\sqrt{\Delta})$ are bounded. Such questions have been long known to be connected to the growth properties of quasimodes (approximate solutions) to the eigenfunction equation $\sqrt{\Delta}u=\lambda u$. In this talk we will see how we can formalise the relationship between growth properties of quasimodes and boundedness of spectral multipliers and use the relationship to obtain new results about the latter.