since \(v \in \mathcal {A}_1\)  implies that \(v^{1/q_0} \in \mathcal {A}_{p_0, q_0}\)  (for the definition of the \(\mathcal {A}_{p,q}\)  class, see [1]}). On the other hand, by Proposition REF , we have
\( \Vert  F \Vert ^{q_0}_{L^{rq(\cdot )}_{\omega ^{1/r}}} = \Vert  F^{q_0} \Vert _{L^{sq(\cdot )}_{\omega ^{1/s}}}\le C \sup \left\lbrace  \int _{\mathbb {R}^{n}} \left| [F(x)]^{q_0} g(x) \right| dx : \Vert  g \Vert _{L^{(sq(\cdot ))^{\prime }}_{\omega ^{-1/s}}} \le 1 \right\rbrace \) 

поскольку \(v \in \mathcal{A}_1\) означает, что \(v^{1/q_0} \in \mathcal{A}_{p_0, q_0}\) (для определения класса \(\mathcal{A}_{p,q}\) см. [1]). С другой стороны, по Proposition REF имеем

\( \Vert F \Vert^{q_0}_{L^{rq(\cdot)}_{\omega^{1/r}}} = \Vert F^{q_0} \Vert_{L^{sq(\cdot)}_{\omega^{1/s}}} \le C \sup \left\lbrace \int_{\mathbb{R}^n} |[F(x)]^{q_0} g(x)| dx: \Vert g \Vert_{L^{(sq(\cdot))'}_{\omega^{-1/s}}} \le 1 \right\rbrace\)