To put this question into proper context, what I am asking is related to the construction of smooth function $H_m$ over the torus $\mathbb{T}$ such that
$$\left|\widehat{H_m}(k)\right|\leq C\log(\left|k\right|)\left|k\right|^{\frac{-1}{2 + \alpha}}$$
The full discussion can be found on Lectures on Harmonic Analysis Thomas H. Wolff, pp. 69-73 (but especially page 73; first hit on Google gives the notes). My specific question will be boldfaced at the end of this post.
On a high level my question is about bounding the Fourier transform of the product of some functions when we know bounds for the individual transformed functions (show a specific bound for $\widehat{fg}(k)$ when you know something about $\widehat{f}$ and $\widehat{g}$). Unfortunately I don't understand the source material/problem so well that I could leave a lot of details out. Hence I am forced to list a lot objects and results, so bear with me.
(Objects and results:) We have the following objects:
0.) $\alpha > 0$ which is a fixed parameter.
1.) $\mathcal{P}(M)$ which is the set of prime numbers on the interval $(M/2, M]$
2.) $\phi$ which is a fixed mollifier on $\mathbb{R}$ supported on $[-1,1]$
3.) $\phi^\epsilon(x) := \epsilon^{-1}\phi(x/\epsilon)$
4.) $\Phi^\epsilon$ which is the periodization (to be defined shortly) of $\phi^\epsilon$
5.) $\Phi^\epsilon_p(x) := \Phi^\epsilon(px)$ for a prime number $p$
6.) periodization of a function $f:\mathbb{R}\to\mathbb{R}$ defined by $f_{\mathrm{per}}(x) := \sum_{n\in\mathbb{Z}}f(x - n), x\in [0,1]$
7.) $F(x;M, \epsilon) = \frac{1}{\left|\mathcal{P}(M)\right|}\sum_{p\in\mathcal{P}(M)}\Phi_p^\epsilon(x)$
8.) $r_0 > 1$ chosen such that
$$g(r) = \begin{cases}r^{\frac{-1}{2 + \alpha}}\log(r) &: r\geq r_0\\\ r_0^{\frac{-1}{2 + \alpha}}\log(r_0) &: r\leq r_0\end{cases}$$
satisfies $g(r)\leq 1$ and $g(r)$ is nonincreasing for all $r$.
and then results (from the pages 69-73) that
i.) $\widehat{F(x;M, \epsilon)}(0) = 1$
ii.) $\widehat{F(x;M, \epsilon)}(k) = 0$ if $0 < |k| \leq \frac{M}{2}$
iii.) For every $N$ there exists $C_N$ such that $|\widehat{F(x;M,\epsilon)}(k)|\leq C_N\frac{\log(|k|)}{M}\left(1 + \frac{\epsilon |k|}{M}\right)^{-N}$
iv.) For any $\psi\in C^\infty(\mathbb{T})$ we have the bound
$$\left|\widehat{\psi(x)F(x;M,\epsilon)}(k) - \widehat{\psi(x)}(k)\right|\leq \begin{cases}C\frac{\log(|k|)}{M}\left(1 + \frac{|k|}{M^{2+\alpha}}\right)^{-100}&:|k|\geq \frac{M}{4}\\\ CM^{-100}&: |k|\leq \frac{M}{4}\end{cases}$$
for some $C$ independent of $k,M$
v.) For any $\psi \in C^\infty (\mathbb{T}), \epsilon > 0$ and $M_0 > 10r_0$ there exists $N$ large enough and a sequence $M_0 < M_1 < \cdots < M_N$ that
$$\left|\widehat{\psi(x)G(x)}(k) - \widehat{\psi(x)}(k)\right|\leq \epsilon g(|k|)$$
when $G(x) := N^{-1}\left(F_{M_1} + \cdots + F_{M_N}\right)(x)$
(Set up:) Given all the above, the author then claims that if we take $\epsilon = 2^{-2 - m}$, set $G_0\equiv 1$ and construct $G_m$ inductively with $\psi_m(x) = \prod_{i=0}^{m-1}G_i(x)$ and $M_0\geq 10r_0 + m$, that is choose large enough $N$ that the result v.) holds with $\psi_m$ and $\epsilon$, then the functions $H_m(x) = \prod_{i=1}^mG_i(x)$ satisfy for all $m=1,2,\dots$
a.) $$\frac{1}{2}\leq \widehat{H_m}(0)\leq \frac{3}{2}$$
b.) $$\left|\widehat{H_m}(k)\right|\leq C\log(\left|k\right|)\left|k\right|^{\frac{-1}{2 + \alpha}}$$
for some constant $C$.
(Question:) I am stuck at trying to understand how the claim a.) can be justified (I haven't thought about b.) but I am sure that it will make more sense after tackling a.)). Namely, I don't know how to bound such an average quantity (product of the sample mean of such functions $F_M$), and the author doesn't really illustrate how we get from the inductive construction of $H_m$ to the conclusion. One can try to prove a.) with induction as $H_1 = G_N = N^{-1}(F_{M_1} + \cdots + F_{M_N})$ for some $N$ and $M_k$s whence
$$\widehat{G_N}(0) = N^{-1}(\widehat{F_{M_1}}(0) + \cdots + \widehat{F_{M_N}}(0)) = 1 \in [1/2, 3/2]$$
But equipped with a rudimentary knowledge of convolution tricks, I find performing the same kind of reasoning to $H_m = \prod_{i=1}^m G_{N_i}$ quite puzzling. We could try to write
$$\widehat{H_{m+1}}(k) = \widehat{\left(G_{N_{m+1}}\cdot \left(\prod_{i=1}^{m}G_{N_i}\right)\right)}(k) = \widehat{G_{N_{m+1}}}\ast \widehat{\left(\prod_{i=1}^{m}G_{N_i}\right)}(k)$$
where the last term corresponds to $H_{m}$ satisfying the induction hypothesis.
Am I missing something obvious about the given results & objects or is there some trick I can use to bound the convolution of $\widehat{G_{N_{m+1}}}$ and $\widehat{H_{m}}$ when we know bounds for them both (and the induction hypothesis for $H_{m}$)?