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

Abstract

We prove various equivalent characterisations of the Hardy space $H_{\mathcal{L}}^p({\mathbb{C}}^n)$ 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 $H_{\mathcal{L}}^p({\mathbb{C}}^n)$ 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 $H_{\mathcal{L}}^p({\mathbb{C}}^n)$. More precisely we prove that it is a bounded operator from $H_{\mathcal{L}}^p({\mathbb{C}}^n)$ to $L^p({\mathbb{C}}^n)$ for $0<p\le 1$ and $\delta \ge (2n-1)\left(1/p-1/2\right)$.

43 pages