On the parabolic $H^p$ theory generated by $(p,\infty)$-atoms for $0<p<1$

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 $H_{\textrm{par}}^1(\mathbb R^2)$, formulated using $(1,\infty)$-atoms, into $L^{1,\infty}(\mathbb R^2)$. Working directly with parabolic $(p,\infty)$-atoms, we show that this result is sharp at $p=1$: for every $0<p<1$, the natural extension from $H_{\textrm{par}}^p(\mathbb R^2)$ to $L^{p,\infty}(\mathbb R^2)$ fails even for the corresponding single-scale operator. We then introduce a curvature-adapted modified Hardy space $H_{\textrm{par}}^{p,*}(\mathbb R^2)$ and a weak tendril space $\mathcal T^{p,\infty}(\mathbb R^2)$, and prove that $$M_{\textrm{par}}: H_{\textrm{par}}^{p,*}(\mathbb R^2) \longrightarrow \mathcal T^{p,\infty}(\mathbb R^2), \qquad 0<p<1,$$ is bounded. At $p=1$, these spaces recover those in Christ's theorem: $H_{\textrm{par}}^{1,*}(\mathbb R^2)=H_{\textrm{par}}^1(\mathbb R^2)$ and $\mathcal T^{1,\infty}(\mathbb R^2)=L^{1,\infty}(\mathbb R^2)$. Thus, our result provides a natural extension of Christ's work to the range $0<p<1$.