Hamiltonians on Phase-Space Noncommutative Laws that are Bounded at Zero Momentum at Bounded Positions

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$, with lifts $\mathcal{H}_{M}$ as in Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts; real multiples of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing, laws $\mathrm{law}(X)$ of $L^{2}$ $d$-tuples are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples, and $\kappa_{d}(\Sigma_{d,r})$ is the image of the set $\Sigma_{d,r}$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws under the canonical map $\kappa_{d}$ of Square-Integrable Noncommutative Laws: Standing Notation §laws. For an $L^{2}$ $d$-tuple $X$ of a tracial W*-probability space, $0=0X=(0,\dots,0)$ is the zero $L^{2}$ $d$-tuple, a real multiple of $X$.

¶$\mathcal{H}$ is bounded at zero momentum at bounded positions if for every real $r>0$ there is a real $K$ such that, for every tracial W*-probability space $(H,M,\Omega)$ and every $L^{2}$ $d$-tuple $X$ of $(H,M,\Omega)$ with $\mathrm{law}(X)\in\kappa_{d}(\Sigma_{d,r})$,

$$\bigl|\mathcal{H}_{M}(X,0)\bigr|\le K.$$