A comparison principle for convolution measures with applications

We establish the general form of a geometric comparison principle for $n$-fold convolutions of certain singular measures in ${\mathbb{R}}^d$ which holds for arbitrary $n$ and $d$. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in a companion paper.