On maximal inequalities

We construct some counterexamples to the statements [1, 2.3.6] claiming maximal inequalities for the spaces $B_{p,q}^s(\Bbb{R}^n)$ and $F_{p,q}^s(\Bbb{R}^n)$ and propose a condition for these inequalities to hold. We consider some weighted inequality on a bounded interval $I$ of the real axis that involves $f\in C_0^\infty(I)$ and the derivative of $f$.