Multilinear averages on curves
We classify for which exponents $p=(p_1,\ldots,p_k)$ operators of the form $$\mathcal{M}(f_1,\ldots,f_k) := \int_{\mathbb{R}^n} \prod_{j=1}^k f_j\circ\pi_j(x) \, a(x) \, dx,$$ where each $\pi_j : \mathbb{R}^n \rightarrow \mathbb{R}^{n-1}$ is a "polynomial-like" smooth submersion and $a$ is a cutoff function, exhibit the following bound: $$ |\mathcal{M}(f_1,\ldots,f_k)| \le C \prod_{j=1}^k \lVert{f_j}\rVert_{L^{p_j}(\mathbb{R}^{n-1})}, $$ with $C$ being a finite constant allowed to depend on $a$, the $\pi_j$, and the $p_j$, but not on the $f_j$.