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

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 $\delta\geq (2n-1)\left(1/p-1/2\right)$.