22 | April | 2019 | Almost Originality

About 2 years ago, Müller Ricci and Wright published a paper that opened a new line of investigation in the field of Fourier restriction: that is, the study of the pointwise meaning of the Fourier restriction operators. Here is an account of a recent contribution to this problem that largely sorts it out.

1. Maximal Fourier Restriction
Recall that, given a smooth submanifold $\Sigma$ of $\mathbb{R}^d$ with surface measure $d\sigma$ , the restriction operator ${R}$ is defined (initially) for Schwartz functions as

$\displaystyle f \mapsto Rf:= \widehat{f}\Big|_{\Sigma};$

it is only after having proven an a-priori estimate such as $\|Rf\|_{L^q(\Sigma,d\sigma)} \lesssim \|f\|_{L^p(\mathbb{R}^d)}$ that we can extend ${R}$ to an operator over the whole of $L^p(\mathbb{R}^d)$ , by density of the Schwartz functions. However, it is no longer clear what the relationship is between this new operator that has been operator-theoretically extended and the original operator that had a clear pointwise definition. In particular, a non-trivial question to ask is whether for $d\sigma$ -a.e. point $\xi \in \Sigma$ we have

$\displaystyle \lim_{r \to 0} \frac{1}{|B(0,r)|} \int_{\eta \in B(0,r)} |\widehat{f}(\xi - \eta)| d\eta = \widehat{f}(\xi), \ \ \ \ \ (1)$

where $B(0,r)$ is the ball of radius ${r}$ and center ${0}$ . Observe that the Lebesgue differentiation theorem already tells us that for a.e. element of $\mathbb{R}^d$ in the Lebesgue sense the above holds; but the submanifold $\Sigma$ has Lebesgue measure zero, and therefore the differentiation theorem cannot give us any information. In this sense, the question above is about the structure of the set of the Lebesgue points of $\widehat{f}$ and can be reformulated as: