Remarks on estimates for the adjoint restriction operator to curves over the sphere

TL;DR

This paper discusses sharp estimates for the adjoint restriction operator to curves on the sphere, demonstrating their optimality and exploring related surface measure estimates.

Contribution

It provides new sharp estimates for the adjoint restriction operator to curves on the sphere and constructs examples to demonstrate their optimality.

Findings

Some estimates are sharp, especially with surface measures.

Constructed examples confirm the sharpness of these estimates.

Discussed related estimates over different surfaces.

Abstract

Recently, two of the authors obtained estimates for the adjoint restriction operator to finite type curves with respect to general measures. Strikingly, it turns out that some of such estimates are sharp, especially when the measures are given by surface measures under certain condition. A typical example is the surface measure on the sphere. We demonstrate sharpness of such estimates by constructing an example and, also, discuss related estimates over different type of surfaces.