Restriction of Fractional Derivatives of the Fourier Transform

TL;DR

This paper investigates how fractional derivatives of Fourier transforms behave when restricted to curved surfaces, extending previous results to more general conditions and function spaces.

Contribution

It generalizes existing theorems on fractional derivatives of Fourier transforms by considering broader conditions and surface restrictions.

Findings

Fractional derivatives of Fourier transforms are in Sobolev spaces after restriction to curved surfaces.

The results extend previous theorems to more general fractional orders and function spaces.

Provides conditions under which the restriction of Fourier transforms maintains certain regularity properties.

Abstract

In this paper, we showed that for suitable $(\beta,p, s,\ell)$ the $\beta$-order fractional derivative with respect to the last coordinate of the Fourier transform of an $L^p(\mathbb{R}^n)$ function is in $H^{-s}$ after restricting to a graph of a function with non-vanishing Gaussian curvature provided that the restriction of the Fourier transform of such function to the surface is in $H^{\ell}$. This is a generalization of the result in \cite{GoldStol}*{Theorem 1.12}.