A Fourier restriction theorem for hypersurfaces which are graphs of certain real polynomials
Kei Morii
TL;DR
The paper addresses Fourier restriction for hypersurfaces S in $\,\mathbb{R}^n$ that are graphs of real polynomials, extending Strichartz's restriction results from quadratic surfaces. The main approach uses decay estimates of one-dimensional oscillatory integrals in a Strichartz-type framework, introducing $G_z(\xi)=\frac{(\tilde{R}(\xi)-r)_+^z}{\Gamma(z+1)}$ and proving boundedness of its inverse Fourier transform under a growth condition to obtain the restriction inequality for $p=\frac{2\lambda}{\lambda+1}$, with scaling via nonisotropic dilations; auxiliary integrals satisfy explicit forms $|A_2(x)|=2\sqrt{\pi}$ and $A_3(x)=\frac{2\pi}{\sqrt[3]{3}}\mathrm{Ai}\left(\frac{x}{\sqrt[3]{3}}\right)$. The results are then applied to partial differential equations: by duality, Fourier restriction, and dispersive bounds, they obtain spacetime Lebesgue estimates for solutions, culminating in the a priori bound $\|u\|_{L^{p/(p-1)}(\mathbb{R}^{1+n})} \le C\big(\|\phi\|_{L^2(\mathbb{R}^n)}+\|f\|_{L^p(\mathbb{R}^{1+n})}\big)$. Overall, the work connects oscillatory decay on polynomial-graph hypersurfaces to restriction theory and dispersive PDE bounds, with corollaries and extensions to noninteger $k$ and the homogeneous case.
Abstract
We will extend the Fourier restriction inequality for quadratic hypersurfaces obtained by Strichartz. We will consider the case where the hypersurface is a graph of a certain real polynomial which is a sum of one-dimensional monomials. It is essential to examine the decay of a one-dimensional oscillatory