$L^{p} \rightarrow L^{q}$ estimates for maximal functions associated with nonisotropic dilations of hypersurfaces in $\mathbb{R}^3$

TL;DR

This paper establishes new $L^{p} ightarrow L^{q}$ estimates for maximal functions linked to nonisotropic dilations of hypersurfaces in $R^3$, including cases with vanishing Gaussian curvature, extending previous results and addressing challenges with Fourier integral operators.

Contribution

It provides novel $L^{p} ightarrow L^{q}$ estimates for maximal functions associated with nonisotropic dilations of hypersurfaces, including cases with vanishing curvature, and introduces new analysis for Fourier integral operators failing classical curvature conditions.

Findings

Derived $L^{p} ightarrow L^{q}$ estimates for nonisotropic dilations.

Extended estimates to hypersurfaces with vanishing Gaussian curvature.

Analyzed Fourier integral operators without uniform cinematic curvature.

Abstract

The goal of this article is to establish $L^{p} \rightarrow L^{q}$ estimates for maximal functions associated with nonisotropic dilations $\delta_t(x)=(t^{a_1}x_1,t^{a_2}x_2,t^{a_3}x_3)$ of hypersurfaces $(x_{1}, x_{2},\Phi(x_1,x_2))$ in $\mathbb{R}^3$, where the Gaussian curvatures of the hypersurfaces are allowed to vanish. When $2 \alpha_{2} = \alpha_{3}$, this problem is reduced to study of the $L^{p} \rightarrow L^{q}$ estimates for maximal functions along the curve $\gamma(x)=(x,x^2(1+\phi(x)))$ and associated dilations $\delta_t(x)=(tx_1,t^2x_2)$. The corresponding maximal function shows features related to the Bourgain circular maximal function, whose $L^{p} \rightarrow L^{q}$ estimate has been considered by [Schlag, JAMS, 1997], [Schlag-Sogge, MRL, 1997] and [Lee, PAMS, 2003]. However, in the study of the maximal function related to the mentioned curve $\gamma(x)$ and associated dilations, we get the $L^{p} \rightarrow L^{q}$ regularity properties for a family of corresponding Fourier integral operators which fail to satisfy the "cinematic curvature condition" uniformly, which means that classical local smoothing estimates could not be directly applied to our problem. What's more, the $L^{p} \rightarrow L^{q}$ estimates are also new for maximal functions associated with isotropic dilations of hypersurfaces $(x_{1}, x_{2},\Phi(x_1,x_2))$ mentioned before.