Restriction estimates using decoupling theorems and two-ends Furstenberg inequalities

We propose to study the restriction conjecture using decoupling theorems and two-ends Furstenberg inequalities. Specifically, we pose a two-ends Furstenberg conjecture, which implies the restriction conjecture. As evidence, we prove this conjecture in the plane by using the Furstenberg set estimate. Moreover, we use this planar result to prove a restriction estimate for $p>22\mathrm{/}7$ in three dimensions, which implies Wolff’s $5\mathrm{/}2$-hairbrush bound for Kakeya sets in ${\mathbb{R}}^3$. Our approach also makes improvements for the restriction conjecture in higher dimensions.