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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.
Consider the trilinear form for twisted convolution on $mathbb{R}^{2d}$: begin{equation*} mathcal{T}_t(mathbf{f}):=iint f_1(x)f_2(y)f_3(x+y)e^{itsigma(x,y)}dxdy,end{equation*} where $sigma$ is a symplectic form and $t$ is a real-valued parameter. I
Hadamards determinant inequality was refined and generalized by Zhang and Yang in [Acta Math. Appl. Sinica 20 (1997) 269-274]. Some special cases of the result were rediscovered recently by Rozanski, Witula and Hetmaniok in [Linear Algebra Appl. 532
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to charact
We study the Rellich inequalities in the framework of equalities. We present equalities which imply the Rellich inequalities by dropping remainders. This provides a simple and direct understanding of the Rellich inequalities as well as the nonexistence of nontrivial extremisers.
This paper is a companion to our prior paper arXiv:0705.4619 on the `Small Ball Inequality in All Dimensions. In it, we address a more restrictive inequality, and obtain a non-trivial, explicit bound, using a single essential estimate from our prior