A Sharpened Rearrangement Inequality for Convolution on the Sphere

Understand

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.