Sharp $L^p$-$L^q$ estimate for the spectral projection associated with the twisted Laplacian

Abstract

In this note we are concerned with estimates for the spectral projection operator ${\mathcal{P}}_{\mu }$ associated with the twisted Laplacian $L$. We completely characterize the optimal bounds on the operator norm of ${\mathcal{P}}_{\mu }$ from $L^p$ to $L^q$ when $1\le p\le 2\le q\le \infty$. As an application, we obtain uniform resolvent estimate for $L$.

20 pages, 1 figure