  In a recent paper [Ann. of Math. 189 (2019), 837--861], Du and Zhang proved a
fractal Fourier restriction estimate and used it to establish the sharp $L^2$
estimate on the Schr\"{o}dinger maximal function in $\Bbb R^n$, $n \geq 2$. 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 $\Bbb R^n$ that is closely
related to the polynomial Wolff axioms. We also prove that all the estimates of
our family are true in $\Bbb R^2$.
