Averages and maximal averages over Product j-varieties in finite fields

We study both averaging and maximal averaging problems for Product $j$-varieties defined by $\Pi_j=\{x\in \mathbb F_q^d: \prod_{k=1}^d x_k=j\}$ for $j\in \mathbb F_q^*,$ where $\mathbb F_q^d$ denotes a $d$-dimensional vector space over the finite field $\mathbb F_q$ with $q$ elements. We prove the sharp $L^p\to L^r$ boundedness of averaging operators associated to Product $j$-varieties. We also obtain the optimal $L^p$ estimate for a maximal averaging operator related to a family of Product $j$-varieties $\{\Pi_j\}_{j\in \mathbb F_q^*}.$