An elementary approach to the operator method in additive combinatorics

The additive energy $E(A)$ of a finite set $A$ from an abelian group is the number of solvings
$$ a+b=c+d, a, b, c, d \in A. $$
Finding the upper bounds for the additive energy of sets from some given classes is very popular subject of additive combinatorics. In 2012 Shkredov provided operator method that allows to estimate $E(A)$ via bounding of so-called higher energy
$$ E_3(A):=\# \{a_1-b_1=a_2-b_2=a_3-b_3:a_i, b_i \in A, i=1,2,3\} $$
and the common energy
$$ E(A,D):=\# \{a_1-d_1=a_2-d_2:a_i \in A, d_i \in D, i=1,2\} $$
for an arbitrary $D\subset A-A$. He uses the properties of the operator
$$ T_A(x,y)=(A \circ A)(x-y)=\#\{a-b=x-y:a,b \in A \} $$
for obtain a certain very general inequality to the number of the solutions of two linear equation systems.

We suggest an elementary approach to prove this inequality which gives an elementary proof of the corresponding bounds of $E(A)$ (in particular, for the family of convex sets and the collection of sets having few products).

Also we discuss a generalization of this inequality (which is elementary as well) connecting with the Sidorenko's conjecture from graph theory. Sidorenko's conjecture states that for any bipartite graph $H$ and any graph $G$ we have
$$ t_H \le t_{K_2} (G), $$
where $t_H(G)=\frac{h_H (G)}{|G|^{|V(H)|}}$ and $h_H(G)$ is the number of homomorphisms from $H$ to $G$.

We consider a special case of the graph $|G|$ which is defined in the terms of convolutions of $A$ and prove the required bound for many non-bipartite graphs $H$.