The nonlinear Brascamp–Lieb conjecture

Let $(\mathbf{L},\mathbf{p})$ be a Brascamp–Lieb datum with finite Brascamp–Lieb constant $\mathbf{BL}(\mathbf{L},\mathbf{p})$. For each $1\leq j\leq k$, let $B_j:\mathbb{R}^{d}\to\mathbb{R}^{d_j}$ be smooth near $0$, a submersion there, and satisfy $dB_j(0)=L_j$. Nonlinear Brascamp–Lieb conjecture. There exists a neighborhood $U$ of $0$ such that

$$\int_U\prod_{j=1}^k(f_j\circ B_j)^{p_j}\lesssim\prod_{j=1}^k\left(\int_{\mathbb{R}^{d_j}}f_j\right)^{p_j}.$$

This is the nonlinear counterpart of the classical Brascamp–Lieb inequality and is presented as the nonlinear form of the preceding tentative oscillatory Brascamp–Lieb assertion.