Consider $h(n)$ to be an arithmetical function. 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 $Q_0<q\leq Q$.

Fix $N\geq Q$ large. Do we know a lower bound for $$\int_{\mathfrak{m}}|\sum_{n\leq N}h(n)e(n\beta)|^2d\beta$$

for the additive function $h(n)=\omega(n)$, where $\omega(n)$ is the number of prime factors counting function?

Do we instead have an upper bound for the same quantity but for the multiplicative characteristic function of the sum of two squares?

Thanks in advance!