Estimates for Character Sums with Various Convolutions

We provide estimates for sums of the form $\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}χ(a+b+c)\right|$ and $\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\sum_{d\in D}χ(a+b+cd)\right|$ when $A,B,C,D\subset \mathbb F_p$, the field with $p$ elements and $χ$ is a non-trivial multiplicative character modulo $p$.

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