The Trilinear Estimate for Cyclic Gradients of Differences of Unitary Laws

In the setting of Unitary Laws with Free Unitary Noise: Standing Data, with the conventions of Unitary Laws with Free Unitary Noise: Standing Data §spaces, let $[d]$ be the initial segment determined by $d$, let $\Xi^{i}_{p,n}$ be the truncated cyclic gradients and $\lambda(X)$ the evaluation of a word polynomial; for $\mu,\nu\in\mathcal{L}_{d}$ the length-weighted gauge $\lVert\iota_{d}(\mu)-\iota_{d}(\nu)\rVert_{d,1}$ is defined by The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §sobolev.

(Trilinear estimate)¶ There is a real number $K>0$ such that for all $\mu,\nu\in\mathcal{L}_{d}$ and every $n\in\mathbb{N}$, with $x=\iota_{d}(\mu)-\iota_{d}(\nu)$,

$$\Bigl|\sum_{i\in[d]}\Bigl(\mu\bigl(\Xi^{i}_{x,n}\Xi^{i}_{x,n}\bigr)-\nu\bigl(\Xi^{i}_{x,n}\Xi^{i}_{x,n}\bigr)\Bigr)\Bigr|\le K\,\lVert x\rVert_{d}\,\lVert x\rVert_{d,1}^{2}.$$

Such a $K$ is a trilinear constant.