Inequality regarding L^6 norm of a trigonometric polynomial I am reading Bourgain's paper "Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations", and am stuck on the following inequality (p117):
Let $f=\sum_{-N}^{N}a_n e^{i(nx+n^2t)}$. Then by direct calculation one obtains
$$
\|f\|_6^6=\|f^3\|_2^2=\sum_{n,j}\left|\sum_{n_1^2+n_2^2+(n-n_1-n_2)^2=j} a_{n_1} a_{n_2} a_{n-n_1-n_2} \right|^2
$$
The bit I am confused by is he then claims that this is bounded by
$$
\max_{|n|<3N \\ |j|<3N^2} r_{n,j} \cdot \left( \sum_{m<N} |a_m|^2 \right)^3
$$
where $$r_{n,j} = \#\{(n_1,n_2): |n_i|<N \text{ and } n_1^2+n_2^2+(n-n_1-n_2)^2=j\}$$
Where does this last inequality come from? I had been trying to use some kind of bound of the form $abc \lesssim a^3+b^3+c^3$ and the embedding $l^3 \subset l^2$but I don't think the first inequality there is actually true and even with these I don't see what happens to the sum over $(n,j)$ out front.
Thanks in advance.

This took me a bit to figure out, but it's just Cauchy-Schwarz (once).
\begin{align*} \sum_{\substack{|n| \le 3N \\ j < 3N^2}} \left|\sum_{\substack{|x|,|y|,|z| \le N \\ x+y+z=n \\ x^2+y^2+z^2 = j}} a_xa_ya_z\right|^2 &\le \sum_{\substack{|n| \le 3N \\ j < 3N^2}} \left(\sum_{\substack{|x|,|y|,|z| \le N \\ x+y+z=n \\ x^2+y^2+z^2 = j}}a_x^2a_y^2a_z^2\right)\left(\sum_{\substack{|x|,|y|,|z| \le N \\ x+y+z=n \\ x^2+y^2+z^2 = j}}1^2\right) \\ &\le \left(\max_{\substack{|n| \le 3N \\ |j| \le 3N^2}} r_{n,j}\right)\sum_{\substack{|n| \le 3N \\ j < 3N^2}} \sum_{\substack{|x|,|y|,|z| \le N \\ x+y+z=n \\ x^2+y^2+z^2 = j}}a_x^2a_y^2a_z^2 \\ &= \left(\max_{\substack{|n| \le 3N \\ |j| \le 3N^2}} r_{n,j}\right)\sum_{|x|,|y|,|z| \le N} a_x^2a_y^2a_z^2 \\ &= \left(\max_{\substack{|n| \le 3N \\ |j| \le 3N^2}} r_{n,j}\right)\left(\sum_{|x| \le N} a_x^2\right)^3
\end{align*}
