On restriction of the Fourier transform to hypersufaces

TL;DR

This paper proves the sharp boundedness of the Fourier restriction operator for convex analytic hypersurfaces in four-dimensional space, establishing precise $L_p$ to $L_2$ bounds.

Contribution

It establishes the sharp $(L_p, L_2)$ boundedness of the Fourier restriction operator for convex analytic hypersurfaces in $\,\mathbb{R}^4$, extending previous results.

Findings

The Fourier restriction operator is bounded for $1 \le p \le \frac{2h+2}{h+2}$.

The result is proven to be sharp.

The analysis applies to convex analytic hypersurfaces in four dimensions.

Abstract

It is considered Fourier transform of convex analytic hypersufaces on $R^{4} $. We prove that the Fourier restriction operator associated to convex analytic hypersufaces is \textit{$(L_{p}, L_{2})$} bounded whenever $1\le p\le \frac{2h+2}{h+2}$. The result is sharp.