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FIELD
Math: HCMC
DATE
Apr 23 (Tue), 2024
TIME
17:00 ~ 18:00
PLACE
ONLINE
SPEAKER
Tacy, Melissa
HOST
Ryu, Jaehyeon
INSTITUTE
University of Auckland
TITLE
[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.
FILE
70061713320761824_1.pdf