Is the exponent $2$ sharp in the Balog-Szemerédi-Gowers Theorem?

Question

The Balog-Szemerédi-Gowers theorem can be stated in the following form: let $A,B$ be subsets of $\mathbb{Z}/n\mathbb{Z}$ (say) with equal cardinality, such that $$ \|1_A*1_B\|_2 \ge K^{-1} \|1_A\|_1 \|1_B\|_2. $$ Then there exist $A'\subset A$, $B'\subset B$ with $|A'|\ge K^{-C}|A|$, $|B'|\ge K^{-C}|B|$ such that $|A'+B'|\le K^C |B'|$.

Here $C$ is an absolute constant.

My questions is: what happens if the $2$-norm is replaced by a $q$-norm? If $q\in (1,2)$, then it follows from Hölder that $$ \|1_A*1_B\|_q \ge K^{-1} \|1_A\|_1 \|1_B\|_q $$ implies $$ \|1_A*1_B\|_2 \ge K^{-\frac{q}{2(q-1)}} \|1_A\|_1 \|1_B\|_2, $$ so the conclusion still holds, with a constant $C$ that depends on $q$ (blowing up as $q\downarrow 1$).

What about $q>2$? More precisely:

Is there any $q>2$ such that the conclusion of the B-S-G Theorem continues to hold if one assumes that $\|1_A*1_B\|_q \ge K^{-1} \|1_A\|_1 \|1_B\|_q $ (possibly with a constant $C$ that depends on $q$)?

Answer 1

Sorry to answer my own question, but I realized the answer is easy, and similar to the case $q\in (1,2)$: yes, the B-S-G theorem holds for any $q\in (2,+\infty)$: if $q>2$, $$ \|1_A*1_B\|_q^q \le \|1_A*1_B\|_2^2 \|1_A*1_B\|_\infty^{q-2} \le \|1_A*1_B\|_2^2 |A|^{q-2}, $$ so if $\|1_A*1_B\|_q \ge K^{-1} \|1_A\|_1 \|1_B\|_q$, it follows that $$ |A|^q |B| K^{-q} \le \|1_A*1_B\|_q^q \le \|1_A*1_B\|_2^2 |A|^{q-2}, $$ and then $$ \|1_A*1_B\|_2 \ge K^{-q/2} \|1_A\|_1 \|1_B\|_2, $$ and we can apply the Balog-Szemerédi-Gowers Theorem.

This is simpler than I expected, but since the question got a few upvotes it might be useful to someone.