Sharp volume bounds for BRK-type sets of surfaces of revolution

We prove sharp volume bounds (up to dimensional constants) for the $\delta$-neighbourhoods of a general family of Besicovitch-Rado-Kinney (BRK) type sets associated with surfaces of revolution in ${\mathbb{R}}^n$, $n\ge 3$, which, in particular, include spheres, elliptic paraboloids, and cones. As a result, we obtain sharp $L^p$-$L^q$ bounds for their associated Wolff-type maximal operators for all $1\le p,q\le \mathrm{\infty }$. In ${\mathbb{R}}^3$, it is worth noting that our result necessitates a variant of Sogge’s cinematic curvature condition.