Let $p$ be a prime, $g$ is a primitive root modulo $p$, and
$$ \chi_r(t)=\exp\left(\frac{2\pi i r \,\,ind_g t}{p-1}\right), r = 0, ..., p-2 $$
is a multiplicative character modulo $p$ with respect to $r$.
I have a set $ A \subseteq F^*_p $ and a Fourier transform $\hat{\mathbb{1}}_A $ of its indicator function, and I want to estimate the sum below for a fixed $r$. I have an estimation (basically, $(p-1)|A|$ comes from estimating the Fourier transform and $p-1$ is because of the multiplicative character):
$$ \bigg| \sum_{x\in (Z/pZ)^\times} \hat{\mathbb{1}}_A(x)^2 \chi_r(x)^2 \bigg| \le (p- 1) |A| \cdot (p-1) = (p-1)^2 |A|$$
I think it's possible to have a better estimate, namely
$$\bigg| \sum_{x\in (Z/pZ)^\times} \hat{\mathbb{1}}_A(x)^2 \chi_r(x)^2 \bigg| \le (p-1) \cdot \max_{x}|\mathbb{1}_A(x)^2| = (p-1)|A|^2$$
But I'm not sure whether those are correct, or are there any better obvious estimates that I'm missing?