A sharpened inequality for twisted convolution

Abstract

Consider the trilinear form for twisted convolution on R 2 d \mathbb {R}^{2d} : T t ( f ) ≔ ∬ f 1 ( x ) f 2 ( y ) f 3 ( x + y ) e i t σ ( x , y ) d x d y , \begin{equation*} \mathcal {T}_t(\mathbf {f})≔\iint f_1(x)f_2(y)f_3(x+y)e^{it\sigma (x,y)}dxdy, \end{equation*} where σ \sigma is a symplectic form and t t is a real-valued parameter. It is known that in the case t ≠ 0 t\neq 0 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 t=0 we prove a sharpened inequality for twisted convolution with an arbitrary antisymmetric form in place of σ \sigma .