Estimates for Character Sums with Various Convolutions

We provide estimates for sums of the form $$\mid \sum _{a\in A}\sum _{b\in B}\sum _{c\in C}\chi (a+b+c)\mid$$ and $$\mid \sum _{a\in A}\sum _{b\in B}\sum _{c\in C}\sum _{d\in D}\chi (a+b+cd)\mid$$ when $A,B,C,D\subset \mathbb F_p$,thefieldwith$p$elementsand$\chi$ is a non-trivial multiplicative character modulo $p$.