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 .