Hardy space estimates for the parabolic maximal operator below $p=1$

We study the parabolic maximal operator $M_{\textrm{par}}$ along the moment curve $(t,t^2)$. In 1988, Christ proved that $M_{\textrm{par}}$ maps the parabolic Hardy space $H_{\textrm{par}}^1(\mathbb R^2)$ into $L^{1,\infty}(\mathbb R^2)$. We show that this result is sharp at $p=1$: for every $0<p<1$