Restriction of Fourier transforms to curves and related oscillatory integrals

Abstract

Abstract. We prove sharp endpoint results for the Fourier restriction operator associated to nondegenerate curves in R d , d ≥ 3, and related estimates for oscillatory integral operators. Moreover, for some larger classes of curves in R d we obtain sharp uniform L p → L q bounds with respect to affine arclength measure, thereby resolving a problem of Drury and Marshall.

Sparse bounds for the bilinear spherical maximal function

“…Since the region of known $$L^p$$ improving estimate $$1/r<1/p+1/q<m(d,r)$$ is a convex domain in $$\mathbb {R}^3$$ (being the intersection of half‐spaces) and $$p, q>1$$, the generalized restricted type bounds obtained by bilinear interpolation automatically upgrade to the desired $$L^p$$ space bounds. We refer to [44, chapter 3] and [2, proposition 2.2] for details of multilinear interpolation.…”

“…We start by making a change of variables 𝑦 ′ = 𝑦 + ℎ 𝜅(𝑥) in the integral involving the translation of g, from which we get If |𝑦| > 1 + |ℎ|, then 𝜒 𝐵 2 (0,1) (𝑦) = 0 and 𝜒 𝐵 2 (0,1) (𝑦 − ℎ 𝜅(𝑥) ) = 0 (because 𝜅(𝑥) ∈ [1,2]).…”

“…The operator  𝑗 * ,𝑄 (𝑓, g) is bi-sublinear but we can define bilinear operators to control it. Namely for a given measurable function 𝜅 ∶ ℝ 𝑑 → [1,2] define…”

“…𝜅,𝑄 (𝑓, g)(𝑥) =  2 𝑚−4 𝜅(𝑥) (𝑓𝜒 1 2 𝑄 , g𝜒 Q(𝑗) )(𝑥).It is enough to prove that for all measurable 𝜅 ∶ ℝ 𝑑 →[1,2], one has| | | | | | ∑ 𝑄∈ 0…”

The integral along curves by residue theorem

“…Recently, the inequalities for generalized parametric integrals along polynomial curves have been studied [1]. Furthermore, the restriction of fourier transforms to curves and related oscillatory integrals has been used by researchers [2]. Move to author showed the restriction of Fourier transforms to curves II.…”