Define $\mathfrak{m}$ as the union of the minor arcs of the form $|\alpha-\frac{a}{q}|\leq 1/qQ$, with $(a,q)=1$ and $Q_0<q\leq Q$, with $Q_0\geq N/Q$, for a certain $N\geq Q$ large.

Is it possible to find a sharp upper bound for

$$\int_{\mathfrak{m}}|\sum_{n\leq N}\omega(n)e(n\beta)|^2d\beta$$

for the additive function $\omega(n)$, which counts the number of distinct prime factors of an integer $n$?

For sharp upper bound I mean nothing that uses Parseval's identity. The idea should be to insert the definition of $\omega(n)$ in the exponential sum and then use the structure of the minor arcs in question to obtain a saving compared to the straightforward application of Parseval's identity.

Thanks in advance!