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 $Pi_j={xin mathbb F_q^d: prod_{k=1}^d x_k=j}$ for $jin mathbb F_q^*,$ 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 $L^pto L^r$ 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 ${Pi_j}_{jin mathbb F_q^*}.$