Being \(p(\cdot ) \le q(\cdot )\) , by [1]}, Lemma REF  and (i), we have
\(\Vert  \chi _{Q} \Vert _{L^{p(\cdot )/p_{\ast }}_{\omega ^{p_{\ast }}}}^{q_{\ast }/p_{\ast }} =\Vert  \chi _Q \, \omega ^{p_{\ast }} \Vert _{L^{p(\cdot )/p_{\ast }}}^{q_{\ast }/p_{\ast }} \le (1+|Q|)^{q_{\ast }/p_{\ast }}\Vert  \chi _Q \, \omega ^{p_{\ast }} \Vert _{L^{q(\cdot )/p_{\ast }}}^{q_{\ast }/p_{\ast }}\) 
\(= (1+|Q|)^{q_{\ast }/p_{\ast }} \Vert  \chi _{Q} \Vert _{L^{q(\cdot )/q_{\ast }}_{\omega ^{q_{\ast }}}} < \infty .\) 

Имея \(p(\cdot) \leq q(\cdot)\), по [1], Лемме REF и (i) у нас есть

\(\Vert \chi_Q \Vert_{L^{p(\cdot)/p_{\ast}}_{\omega^{p_{\ast}}}}^{q_{\ast}/p_{\ast}} = \Vert \chi_Q \, \omega^{p_{\ast}} \Vert_{L^{p(\cdot)/p_{\ast}}}^{q_{\ast}/p_{\ast}} \leq (1+|Q|)^{q_{\ast}/p_{\ast}} \Vert \chi_Q \, \omega^{p_{\ast}} \Vert_{L^{q(\cdot)/p_{\ast}}}^{q_{\ast}/p_{\ast}}\)

\(= (1+|Q|)^{q_{\ast}/p_{\ast}} \Vert \chi_Q \Vert_{L^{q(\cdot)/q_{\ast}}_{\omega^{q_{\ast}}}} < \infty.\)