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 ${\mathrm{\Pi }}_j=\{x\in {\mathbb{F}}_q^d:{\prod }_{k=1}^d{x}_k=j\}$ for $j\in {\mathbb{F}}_q^{\ast },$ 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 $\{{\mathrm{\Pi }}_j{\}}_{j\in {\mathbb{F}}_q^{\ast }}\mathrm{.}$