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

  • Detlef Müller

    Christian-Albrechts-Universität zu Kiel, Germany
  • Fulvio Ricci

    Scuola Normale Superiore, Pisa, Italy
  • James Wright

    University of Edinburgh, UK
A maximal restriction theorem and Lebesgue points of functions in $\mathcal F(L^p)$ cover
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Abstract

Fourier restriction theorems, whose study had been initiated by E.M. Stein, usually describe a family of a priori estimates of the LqL^q-norm of the restriction of the Fourier transform of a function ff in Lp(Rn)L^p(\mathbb R^n) to a given subvariety SS, endowed with a suitable measure. Such estimates allow to define the restriction Rf\mathcal{R} f of the Fourier transform of an LpL^p-function to SS in an operator theoretic sense. In this article, we begin to investigate the question what is the „intrinsic" pointwise relation between Rf\mathcal{R} f and the Fourier transform of ff, 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 ff restricted to SS.

Cite this article

Detlef Müller, Fulvio Ricci, James Wright, A maximal restriction theorem and Lebesgue points of functions in F(Lp)\mathcal F(L^p). Rev. Mat. Iberoam. 35 (2019), no. 3, pp. 693–702

DOI 10.4171/RMI/1066