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

We study the parabolic maximal operator $M_{\text{par}}$ along the moment curve $(t,t^2)$. In 1988, Christ proved that $M_{\text{par}}$ maps the parabolic Hardy space $H_{\text{par}}^1({\mathbb{R}}^2)$, formulated using $(1,\mathrm{\infty })$-atoms, into $L^{1,\mathrm{\infty }}({\mathbb{R}}^2)$. Working directly with parabolic $(p,\mathrm{\infty })$-atoms, we show that this result is sharp at $p=1$: for every $0<p<1$, the natural extension from $H_{\text{par}}^p({\mathbb{R}}^2)$ to $L^{p,\mathrm{\infty }}({\mathbb{R}}^2)$ fails even for the corresponding single-scale operator. We then introduce a curvature-adapted modified Hardy space $H_{\text{par}}^{p,\ast }({\mathbb{R}}^2)$ and a weak tendril space ${\mathcal{T}}^{p,\mathrm{\infty }}({\mathbb{R}}^2)$, and prove that $$M_{\text{par}}:H_{\text{par}}^{p,\ast }({\mathbb{R}}^2)⟶{\mathcal{T}}^{p,\mathrm{\infty }}({\mathbb{R}}^2),0<p<1,$$ is bounded. At $p=1$, these spaces recover those in Christ’s theorem: $H_{\text{par}}^{1,\ast }({\mathbb{R}}^2)=H_{\text{par}}^1({\mathbb{R}}^2)$ and ${\mathcal{T}}^{1,\mathrm{\infty }}({\mathbb{R}}^2)=L^{1,\mathrm{\infty }}({\mathbb{R}}^2)$. Thus, our result provides a natural extension of Christ’s work to the range $0<p<1$.