[Paper Review] A maximal restriction theorem and Lebesgue points of functions in F(L^p)

Detlef Müller, Fulvio Ricci+1 more|arXiv (Cornell University)|Dec 14, 2016

Advanced Harmonic Analysis Research5 references14 citations

TL;DR

This paper establishes a maximal restriction theorem for the Fourier transform on $C^2$ curves in $\mathbb{R}^2$, showing that for $f \in L^p(\mathbb{R}^2)$ with $p < \frac{8}{7}$, the regularized averages of $\widehat{f}$ converge pointwise $\mu$-a.e. to the restriction operator $\mathcal{R}f$ with respect to affine arclength measure. It further proves that almost every non-degenerate point on such a curve is a Lebesgue point of $\widehat{f}$, and the restriction coincides with the Lebesgue value.

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 (say, on Euclidean space) to a given subvariety S, endowed with a suitabel measure. Such estimates allow to define the restriction Rf 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 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.

Motivation & Objective

• To understand the pointwise relationship between the restriction operator $\mathcal{R}f$ and the Fourier transform $\widehat{f}$ for $L^p(\mathbb{R}^2)$ functions.
• To determine under what conditions the regularized averages of $\widehat{f}$ converge pointwise $\mu$-a.e. to $\mathcal{R}f$ along a $C^2$ curve with nonvanishing curvature.
• To characterize the set of Lebesgue points of $\widehat{f}$ along such curves and relate them to the restriction operator.
• To establish $L^p$-$L^q$ boundedness of truncated two-parameter maximal functions associated with the Fourier transform on curves.

Proposed method

• The authors introduce a two-parameter maximal function $\mathcal{M}^+$ defined via averaging $\widehat{f}$ over rectangles in the frequency plane aligned with the tangent and normal directions of the curve.
• They prove $L^p$-$L^q$ boundedness of this maximal operator for $1 \leq p < \frac{8}{7}$ and $p' \geq 3q$, using the Kolmogorov-Seliverstov-Plessner linearization method.
• The proof relies on uniform estimates for a family of linear operators derived from the restriction problem, adapting techniques from Carleson and Sjölin for $C^2$ curves with nonvanishing curvature.
• The authors use the Hardy-Littlewood-Sobolev inequality to control $L^r$ norms of truncated maximal functions in terms of $L^s$ norms of the original function.
• They relate the maximal function to the restriction operator via duality and the identity $\|\mathcal{M}^+f\|_{L^q(I,\mu)} \lesssim \|f\|_{L^p(\mathbb{R}^2)}$ for $p < \frac{8}{7}$.
• The argument is localized to graphs of $C^2$ convex functions $\varphi$, with the affine arclength measure $\mu = \kappa^{1/3} dt$ used to weight the curve.

Experimental results

Research questions

• RQ1 For $f \in L^p(\mathbb{R}^2)$ with $p < \frac{8}{7}$, is the restriction $\mathcal{R}f$ equal to the Lebesgue value of $\widehat{f}$ at $\mu$-a.e. point on a $C^2$ curve with nonvanishing curvature?
• RQ2 Does the limit of $\chi_\varepsilon$-averages of $\widehat{f}$ exist $\mu$-a.e. along such curves, and does it equal $\mathcal{R}f$?
• RQ3 Can the truncated strong maximal function of $\widehat{f}$ be bounded from $L^p(\mathbb{R}^2)$ to $L^q(S,\mu)$ for the full range of exponents where the restriction inequality holds?
• RQ4 What is the role of affine arclength measure in ensuring pointwise convergence and Lebesgue point properties?

Key findings

• For $1 \leq p < \frac{4}{3}$, the $\chi_\varepsilon$-averages of $\widehat{f}$ converge pointwise $\mu$-a.e. to $\mathcal{R}f$ at points of nonvanishing curvature on a $C^2$ curve.
• For $1 \leq p < \frac{8}{7}$, almost every point of nonvanishing curvature on a $C^2$ curve is a Lebesgue point of $\widehat{f}$, and the Lebesgue value equals $\mathcal{R}f$ at such points.
• The truncated two-parameter maximal function $\mathcal{M}^+$ satisfies $\|\mathcal{M}^+f\|_{L^q(I,\mu)} \leq C_p \|f\|_{L^p(\mathbb{R}^2)}$ for $1 \leq p < \frac{8}{7}$ and $p' \geq 3q$.
• The $L^p$-$L^q$ boundedness of the maximal operator is established via the Kolmogorov-Seliverstov-Plessner method and uniform estimates for a family of linear operators.
• The result extends to the curve $\Gamma(t) = (t, t^2, \dots, t^d)$ in higher dimensions, though the method is currently limited to $\mathbb{R}^2$.
• The proof shows that $\|\mathcal{M}^+f\|_{L^q(I,\mu)} \lesssim \|f\|_{L^p(\mathbb{R}^2)}$ by relating $\mathcal{M}^+f$ to the maximal function of $|\widehat{f}|^2$, leveraging duality and the Hardy-Littlewood-Sobolev inequality.