A comparison principle for convolution measures with applications

Abstract

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.

arXiv comment17 pages, v2: updated reference to companion paper

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• Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid
• On extremizing sequences for the adjoint restriction inequality on the cone
• A sharp trilinear inequality related to Fourier restriction on the circle
• Extremizers for Fourier restriction on hyperboloids
• Sharp Strichartz inequalities for fractional and higher order Schrödinger equations
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Cited by in corpus (1)

• Sharp Strichartz inequalities for fractional and higher order Schrödinger equations