# Revista Matemática Iberoamericana

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**Volume 35, Issue 3, 2019, pp. 693–702**

**DOI: 10.4171/rmi/1066**

Published online: 2019-04-01

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

Detlef Müller^{[1]}, Fulvio Ricci

^{[2]}and James Wright

^{[3]}(1) Christian-Albrechts-Universität zu Kiel, Germany

(2) Scuola Normale Superiore, Pisa, Italy

(3) University of Edinburgh, UK

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 $S$.

*Keywords: *Fourier restriction, maximal functions

Müller Detlef, Ricci Fulvio, Wright James: A maximal restriction theorem and Lebesgue points of functions in $\mathcal F(L^p)$. *Rev. Mat. Iberoam.* 35 (2019), 693-702. doi: 10.4171/rmi/1066