As the subject says, I have some questions about some Ruzsa-type inequalities for additive-energy. It would be wonderful if anyone could shed some light.

First, the definition: let $E(A,B)$ be the number of all quadruples $(a,a',b,b') \in A \times A \times B \times B$ such that $a+b = a' + b'$ denote the additive energy of the sets $A$, $B$, and let $F(A,B)= \frac{|A|^{2}|B|^{2}}{E(A,B)}$.

Now let $A, B, C \subset G$ be subsets of an additive group $G$ so that $|A|=|B|=|C|=N$. Do the following inequalities hold?

1) If $F(A,B), F(A,C) \leq K\cdot N$, then $\min{(F(A, B+C), F(B, C+A), F(C, A+B))} \leq K^{c} N$, for some constant $c$.

2) If $F(A,A) \leq K\cdot N$, then for all non-negative integers $k,l$ there's a constant $c$ depending on them such that $F(k \cdot A, l \cdot A) \leq K^{c} N$.