A maximal restriction theorem and Lebesgue points of functions in \(\mathcal{F}(L^p)\) (Q2313559)

Summary: 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 \(S\).