A Family of Fractal Fourier Restriction Estimates with Implications on the Kakeya Problem

In a recent paper, Du and Zhang (Ann Math 189:837–861, 2019) proved a fractal Fourier restriction estimate and used it to establish the sharp \(L^2\) $L^2$ estimate on the Schrödinger maximal function in \(\mathbb R^n\) ${\mathbb{R}}^n$ , \(n \ge 2\) $n\ge 2$ . In this paper, we show that the Du–Zhang estimate is the endpoint of a family of fractal restriction estimates such that each member of the family (other than the original) implies a sharp Kakeya result in \(\mathbb R^n\) ${\mathbb{R}}^n$ that is closely related to the polynomial Wolff axioms. We also prove that all the estimates of our family are true in \(\mathbb R^2\) ${\mathbb{R}}^2$ .