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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. It is known that in the case $t eq0$ the optimal constant for twisted convolution is the same as that for convolution, though no extremizers exist. Expanding about the manifold of triples of maximizers and $t=0$ we prove a sharpened inequality for twisted convolution with an arbitrary antisymmetric form in place of $sigma$.
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 fu
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
An inequality refining the lower bound for a periodic (Breitenberger) uncertainty constant is proved for a wide class of functions. A connection of uncertainty constants for periodic and non-periodic functions is extended to this class. A particular
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 show that twisting of an infinite straight three-dimensional tube with non-circular cross-section gives rise to a Hardy-type inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dir