A maximal restriction theorem and Lebesgue points of functions in $\mathcal{F}(L^p)$

Abstract

Fourier restriction theorems, whose study had been initiated by E.M. Stein, usually describe a family of a priori estimates of the $L^q$-norm of the restriction of the Fourier transform of a function $f$ in $L^p({\mathbb{R}}^n)$ to a given subvariety $S$, endowed with a suitable measure. Such estimates allow to define the restriction $\mathcal{R}f$ of the Fourier transform of an $L^p$-function to $S$ in an operator theoretic sense. In this article, we begin to investigate the question what is the „intrinsic" pointwise relation between $\mathcal{R}f$ and the Fourier transform of $f$, by looking at curves in the plane, for instance with non-vanishing curvature. To this end, we bound suitable maximal operators, including the Hardy–Littlewood maximal function of the Fourier transform of $f$ restricted to $S$.