A Sharpened Rearrangement Inequality for Convolution on the Sphere

Abstract

One may define a trilinear convolution form on the sphere involving two functions on the sphere and a monotonic function on the interval $[-1,1]$. A symmetrization inequality of Baernstein and Taylor states that this form is maximized when the two functions on the sphere are replaced with their nondecreasing symmetric rearrangements. In the case of indicator functions, we show that under natural hypotheses, the symmetric rearrangements are the only maximizers up to symmetry by establishing a sharpened inequality.

Error found in reduction to small perturbations, the proof that Prop 2.1 implies Theorem 1.2

References in corpus (2)

• A sharpened Riesz-Sobolev inequality
• Maximizers of Rogers-Brascamp-Lieb-Luttinger functionals in higher dimensions