Onsager conjecture and properties of Besov Spaces

I am currently studying the result in Peter Constantin 1, Weinan E, and E. S. Titi, Onsager’s Conjecture on the Energy Conservation for Solutionsof Euler’s Equation

link:https://web.math.princeton.edu/~weinan/papers/misc1.pdf

The result of the paper is basically proof of the weak part of Onsager conjecture in the periodic domain. In the first part of paper authors give bounds on $L^3$ norm of several terms in terms of Besov norms. More specifically:

I believe that some of the inequalities could be somehow shown form the definition of norm in $B^{\alpha,\infty}_3$ and Minkowski inequality. However especially in (7) and (8), I could not find a way to do this on my own. Any help as to showing why these inequalities hold is greatly appreciated.