# On Hardy spaces associated with the twisted Laplacian and sharp estimates for the corresponding wave operator

## TL;DR

This paper characterizes Hardy spaces linked to the twisted Laplacian for 0<p<1 and establishes sharp boundedness of the associated wave operator, extending previous results and providing new estimates.

## Contribution

It provides new equivalent characterizations of Hardy spaces for the twisted Laplacian and proves sharp boundedness results for the wave operator on these spaces.

## Key findings

- Characterizations of Hardy spaces for 0<p<1 associated with the twisted Laplacian.
- Sharp boundedness of the wave operator from Hardy space to L^p.
- Extension of classical results to the setting of the twisted Laplacian.

## Abstract

We prove various equivalent characterisations of the Hardy space $H^p_{\mathcal{L}}(\mathbb{C}^n)$ for $0<p<1$ associated with the twisted Laplacian $\mathcal{L}$ which generalises the result of [MPR81] for the case $p=1$. Using the atomic characterisation of $H^p_{\mathcal{L}}(\mathbb{C}^n)$ corresponding to the twisted convolution, we prove sharp boundedness result for the wave operator $\mathcal{L}^{-\delta/2}e^{\pm it\sqrt{\mathcal{L}}}$ for a fixed $t>0$ on $H^p_{\mathcal{L}}(\mathbb{C}^n)$. More precisely we prove that it is a bounded operator from $H^p_{\mathcal{L}}(\mathbb{C}^n)$ to $L^p(\mathbb{C}^n)$ for $ 0<p\leq 1$ and $\delta\geq (2n-1)\left(1/p-1/2\right)$.