Klainerman–Machedon space-time bound for the quantum BBGKY hierarchy

Let the interaction parameter satisfy $\beta\in(0,1]$, and let $\{\Gamma_N\}_{N=1}^{\infty}$ be a sequence of quantum BBGKY hierarchy solutions satisfying the energy condition

$$\sup_{t\in\mathbb{R}}\left\Vert S^{(k)}\gamma_N^{(k)}\right\Vert_{L^2_{\mathbf{x},\mathbf{x}'}}\leq C^k$$

for all sufficiently large $N$ depending on $k$, where $S^{(k)}$ is the product of the operators $\langle\nabla_{x_j}\rangle\langle\nabla_{x'_j}\rangle$. Let $\Gamma=\{\gamma^{(k)}\}_{k=1}^{\infty}$ be a limit point of $\{\Gamma_N\}_{N=1}^{\infty}$. Klainerman–Machedon conjecture. Every such limit point satisfies the space-time bound

$$\int_0^T\left\Vert R^{(k)}B_{j,k+1}\gamma^{(k+1)}(t)\right\Vert_{L^2_{\mathbf{x},\mathbf{x}'}}\,dt\leq C^k,$$

where $R^{(k)}$ is the product of $|\nabla_{x_j}||\nabla_{x'_j}|$ and $B_{j,k+1}$ is the collision operator obtained by taking the partial trace of the commutator with $\delta(x_j-x_{k+1})$. The bound is the key spacetime estimate needed for uniqueness and convergence to the nonlinear Schrödinger hierarchy; the supplied source establishes it as a theorem for $\beta\in(0,2/3)$, while the stated range up to $\beta=1$ is the conjectural formulation.