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

TL;DR

This paper establishes optimal $L^p$-$L^q$ bounds for spectral projections of the twisted Laplacian, leading to uniform resolvent estimates, advancing understanding of spectral analysis for this operator.

Contribution

It provides a complete characterization of the optimal bounds for spectral projection operators associated with the twisted Laplacian in $L^p$-$L^q$ spaces.

Findings

Optimal bounds for spectral projection operators are characterized.

Uniform resolvent estimates are derived for the twisted Laplacian.

Results improve understanding of spectral properties of 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 $L$.