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

2026/07/16 by Yongsheng Han, Bingyang Hu
#math.CA

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Abstract

We study the parabolic maximal operator M_\textrmpar along the moment curve (t,t²). In 1988, Christ proved that M_\textrmpar maps the parabolic Hardy space H_\textrmpar¹(\mathbb R²), formulated using (1,∞)-atoms, into L^1,∞(\mathbb R²). Working directly with parabolic (p,∞)-atoms, we show that this result is sharp at p=1: for every 0<p<1, the natural extension from H_\textrmpar^p(\mathbb R²) to L^p,∞(\mathbb R²) fails even for the corresponding single-scale operator. We then introduce a curvature-adapted modified Hardy space H_\textrmpar^p,*(\mathbb R²) and a weak tendril space \mathcal T^p,∞(\mathbb R²), and prove that M_\textrmpar: H_\textrmpar^p,*(\mathbb R²) \longrightarrow \mathcal T^p,∞(\mathbb R²), 0<p<1, is bounded. At p=1, these spaces recover those in Christ's theorem: H_\textrmpar^1,*(\mathbb R²)=H_\textrmpar¹(\mathbb R²) and \mathcal T^1,∞(\mathbb R²)=L^1,∞(\mathbb R²). Thus, our result provides a natural extension of Christ's work to the range 0<p<1.

Citations

• Parabolic maximal functions associated with a distribution, II

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