Estimates for character sums in finite fields of order $p^2$ and $p^3$

Let $p$ be a prime number, ${\mathbb{F}}_{p^n}$ be the finite field of order $p^n$, and $\{{\omega }_1,\dots {\omega }_n\}$ be a basis of ${\mathbb{F}}_{p^n}$ over ${\mathbb{F}}_p$. Let, further, $N_i,H_i$ be integers such that $1\leq H_i\leq p$,$,,i=1,\ldots,n$\mathrm{.}Define$n$-dimensionalparallelepiped$B\subseteq$\mathbb{F}${p^n}$asfollows:$$B=\left{\sum{i=1}^nx_i\omega_i ,:, N_i+1\leq x_i\leq N_i+H_i, ,,, 1\leq i\leq n\right}. Let $n\in\{2,3\}$, $\chi$ be a nontrivial multiplicative character of $\mathbb{F}_{p^n}$ and $|B|\geq p^{n(1/4+\varepsilon)}$, and let us assume that $H_1\leq\ldots\leq H_n$. Then we prove that\left|\sum_{x\in B}\chi(x)\right|\ll_{\varepsilon} |B|p^{-\varepsilon^2/12}, if $\chi|_{\mathbb{F}_p}$ is not identical, and\left|\sum_{x\in B}\chi(x)\right|\ll_{\varepsilon} |B|p^{-\varepsilon^2/12}+|B\cap \omega_n$\mathbb{F}$_p| $$ otherwise.