Restriction estimates for space curves with respect to general measures

Abstract

Abstract. In this paper we consider adjoint restriction estimates for space curves with respect to general measures and obtain optimal estimates when the curves satisfy a finite type condition. The argument here is new in that it doesn't rely on the offspring curve method, which has been extensively used in the previous works. Our work was inspired by the recent argument due to Bourgain and Guth which was used to deduce linear restriction estimates from multilinear estimates for hypersurfaces.

Remarks on Estimates for the Adjoint Restriction Operator to Curves Over the Sphere

“…As mentioned in the above, for d = 2 the optimal result including the end line cases was obtained in [19] and [9]. For d ≥ 3, the sufficiency part of Theorem 1.1 follows from Theorem 1.2 below which is a special case of [20,Theorem 1.1]. In [20], the estimates with respect to general α-dimensional measure (see Definition 3.1) were obtained and those results are sharp in that there are α-dimensional measures for which the estimate fails outside of the asserted region.…”

“…For the purpose, we actually prove more than what we need by replacing the (d − 1)-dimensional measure σ ℓ with general α-dimensional measure. We basically follow the argument in [20]. For ν ∈ R we denote by ⌈ν⌉ the smallest integer which is not less than ν.…”

“…For d ≥ 3, the sufficiency part of Theorem 1.1 follows from Theorem 1.2 below which is a special case of [20,Theorem 1.1]. In [20], the estimates with respect to general α-dimensional measure (see Definition 3.1) were obtained and those results are sharp in that there are α-dimensional measures for which the estimate fails outside of the asserted region. Clearly, since the surface measure is (d − 1)-dimensional, from Theorem 3.2 ([20, Theorem 1.1] with α = d − 1) we have Theorem 1.2.…”

“…It is rather surprising that Theorem 1.2 gives the sharp results since the result in [20] does not rely on specific geometric properties of the associated measures but only on the dimensional condition of the measure. Thus our main contribution here is to show the failure of the estimate (1.4) for the cases (1.6) or (1.7).…”

“…In fact, we can apply Theorem 3.2 for the curves on each dyadic interval via rescaling. For the purpose we will consider L p − L q estimate for T γ λ with respect to general α-dimensional measures, which was considered in [20] (also see [23,6] for the Stein-Tomas restriction theorem with respect to general measures).…”