On weighted multilinear polynomial averages in finite fields

We study the weighted multilinear polynomial averages in finite fields. The essential ingredient is the $u^s$-norm control of the corresponding weighted multilinear polynomial averages in finite fields, which is motivated by Teräväinen \cite{T24}. As an application, we prove an asymptotic formula for the number of the following multidimensional rational function progressions in the subsets of ${\mathbb{F}}_p^D$: $$\mathbf{\text{x}},\mathbf{\text{x}}+P_1(\phi (y))v_1,\cdots ,\mathbf{\text{x}}+P_k(\phi (y))v_k,$$ where $\mathbb{V}=\{v_1,\cdots ,v_k\in {\mathbb{Z}}^D\}$ is a collection of nonzero vectors, $\mathbb{P}=\{P_1,\cdots ,P_k\in \mathbb{Z}[y]\}$ is a collection of linearly independent polynomials with zero constant terms, and $\phi (y)\in \mathbb{Q}(y)$ is a nonzero rational function.