A maximal restriction theorem and Lebesgue points of functions in F(Lp)

Fourier restriction theorems, whose study had been initiated by E. M. Stein, usually describe a family of a priori estimates of the Lq-norm of the restriction of the Fourier transform of a function f in Lp(Rn) to a given subvariety S, endowed with a suitable measure. Such estimates allow to define the restriction Rf of the Fourier transform of an Lp-function to S in an operator theoretic sense. In this article, we begin to investigate the question what is the “intrinsic” pointwise relation between Rf 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.