There is a basic question regrading the mixed 3-dimensional hyper-Kloosterman sum:

For any positive integer $n$, how to prove
$$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p} \left( \frac{z}{p}\right) e\left (\frac{\overline{z} x+ny+\overline{xy} }{p}\right)\ll_\varepsilon p^{\frac{3}{2}+\varepsilon}?$$ Here, $p$ is a prime and $\left( \frac{z}{p}\right)$ denotes the Jacobi symbol which is a Dirichlet character modulo $p$.

I searched P. Michel's book-Lectures on applied p-adic cohomology (https://arxiv.org/pdf/1712.03173v1.pdf), and E. Bombieri-S. Sperger's paper-On the estimation of certain exponential sums (Acta Arith. 69 (1995), 329-358). It seems that there is no a direct record regarding this type of sum with the character $\left( \frac{z}{p}\right)$ contained.

Perhaps, one can find the theory in, for example, L. Fu's paper-Weights of twisted exponential sums (Math. Z. 262 (2009), 449-472). However, it seems very difficult to discuss how the associated Laurent polynomial is non-degenerate with respect to its Newton polyhedron.

So, if any expert here know some knoledge on this question, please show some guides or corresponding references, many many thanks.

Thanks in advance!

There is a basic question regrading the mixed 3-dimensional hyper-Kloosterman sum:

For any positive integer $n$ not divisible by $p$, how to prove
$$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p} \left( \frac{z}{p}\right) e\left (\frac{\overline{z} x+ny+\overline{xy} }{p}\right)\ll_\varepsilon p^{\frac{3}{2}+\varepsilon}?$$ Here, $p$ is a prime and $\left( \frac{z}{p}\right)$ denotes the Jacobi symbol which is a Dirichlet character modulo $p$.

I searched P. Michel's book-Lectures on applied p-adic cohomology (https://arxiv.org/pdf/1712.03173v1.pdf), and E. Bombieri-S. Sperger's paper-On the estimation of certain exponential sums (Acta Arith. 69 (1995), 329-358). It seems that there is no a direct record regarding this type of sum with the character $\left( \frac{z}{p}\right)$ contained.

Perhaps, one can find the theory in, for example, L. Fu's paper-Weights of twisted exponential sums (Math. Z. 262 (2009), 449-472). However, it seems very difficult to discuss how the associated Laurent polynomial is non-degenerate with respect to its Newton polyhedron.

So, if any expert here know some knoledge on this question, please show some guides or corresponding references, many many thanks.

Thanks in advance!