Let $w$ belong to the class of Muckenhoupt weight $A_p$ for some $1<p<\infty$ and define the weighted Lebsgue space $$ L^p(\Omega,w):=\left\{u:\Omega\to\mathbb{R} \text{ measurable }: ||u||=\left(\int_{\Omega}|u|^pw(x)\,dx\right)^\frac{1}{p}<\infty\right\}. $$ One can prove that for any cube $Q$ in $\mathbb{R}^N$ ($N\geq 2$), there exist $s=s(A_p(w))>1$ such that for $q=\frac{ps}{p+s-1}$, we have $$ \left(\frac{1}{|Q|}\int_{Q}|f|^q\,dx\right)^\frac{1}{q}\leq C(A_p(w))\left(\frac{1}{w(Q)}\int_{Q}|f|^p w\,dx\right)^\frac{1}{p}, $$ for every $f\in L^p(Q,w)$ and for some constant $C=C(A_p(w))$ depending on the $A_p$ constant $A_p(w)$ of $w$.
My question is that does the above inequality imply that for any bounded smooth domain $\Omega$ in $\mathbb{R}^N$, one has $$ \left(\int_{\Omega}|f|^q\,dx)^\frac{1}{q}\leq C(\int_{\Omega}|f|^p w\,dx\right)^\frac{1}{p} $$ for every $f\in L^p(Q,w)$ with some constant $C$ independent of $f$.
Can you please help me regarding this.
This is mentioned in the paper (page 7) in the link : https://arxiv.org/pdf/1810.05061.pdf