Fourier restriction implies maximal and variational Fourier restriction

Abstract

We give an abstract argument that an a priori Fourier restriction estimate for a certain choice of exponents automatically implies maximal and variational Fourier restriction estimates. These, in turn, provide pointwise and quantitative interpretations of restriction of the Fourier transform to the hypersurface in question.

A variational restriction theorem

“…In [10], one of the present authors showed analogues of (1.3) and (1.6) when χ is replaced by a measure μ whose Fourier transform μ is C ∞ and satisfies the decay condition…”

“…Subsequent papers [10] and [14], which appeared after the first version of the present paper, generalize the averaging procedure even further, by convolving f with certain averaging measures μ. In light of this more recent research, we now take the opportunity to both generalize (1.6) and complete our short survey of maximal and variational Fourier restriction theories with papers that appeared in the meantime.…”

“…If one takes μ to be the normalized spherical measure, i.e., μ = σ/σ(S 2 ), then the decay of |∇ μ(x)| as |x| → ∞ is only O(|x| −1 ); see [1]. Consequently, the results from [10] do not apply. This was one of the sources of motivation for Ramos [14], who reused Vitturi's argument from [20] to conclude that the maximal estimate…”

“…Alternatively, one can invoke more general results from the subsequent literature, such as the work of Jones, Seeger, and Wright [9], which covered higher-dimensional convolutions, more general dilation structures, and both strong and weak-type variational estimates in a range of L p -spaces. Theorem 1 under condition (b) was covered by the paper [10], up to minor technicalities, such as the fact that here we do not need μ to be smoother than C 2 . However, [10] was concerned with more general surfaces and more general measures σ on them, while here we are able to give a more direct proof that is specific to the sphere and to the stated choice of the Lebesgue space exponents.…”

“…Theorem 1 under condition (b) was covered by the paper [10], up to minor technicalities, such as the fact that here we do not need μ to be smoother than C 2 . However, [10] was concerned with more general surfaces and more general measures σ on them, while here we are able to give a more direct proof that is specific to the sphere and to the stated choice of the Lebesgue space exponents. In fact, as we have already noted, the present proof predates [10].…”