fourier analysis - Estimation of DFT

The answer is no, if you mean an uniform bound in $j$. Here is the example:

Fix $j$ and define
$$
a_r = begin{cases} frac{1}{N}, & Re(exp(-2pi i j r/ K)) geq 0;\
0, & otherwise.end{cases}
$$
It is than easy to estimate that the number of $a_r = frac{1}{N}$ is comparable to $K$. Even more is true, one has that the number of $a_r = frac{1}{N}$ such that $Re(exp(-2pi i j r/ K)) geq sigma$ is comparable to $K$ for any $sigma > 0$.

This implies that
$$
Re(hat{a}_j) geq c N^{1 - alpha}
$$
for some different $c$.