In this note, we prove Poincar\'e-Sobolev inequality in weighted spaces. Such inequality was first observed by Fabes-Kenig-Serapioni and later simplified by Chiarenza-Frasca.
Theorem. Let $1<p<\infty$, $w\in A_p$, and $R>0$. Then there exists a constant $N=N(d,p,[w]_{A_p})>0$ such that $$ \left(\frac{1}{w(B_R)} \int_{B_R} |u|^p w dx \right)^{1/p}\leq NR \left(\frac{1}{w(B_R)} \int_{B_R} |\nabla u|^p w dx \right)^{1/p}$$ for all $u\in C_0^\infty(B_R)$. Here $A_p$ denotes the Muckenhoupt $p$-admissible weight.
In this note, we follow the proof of Chiarenza-Frasca. The idea follows from an idea of Hedberg. It suffices to show that $$ \left(\frac{1}{w(B_R)} \int_{B_R} |I u|^p w dx \right)^{1/p}\leq NR \left(\frac{1}{w(B_R)} \int_{B_R} |u|^p w dx \right)^{1/p},$$ where $I f$ denotes the Riesz potential of order $1$: $$ I f(x)=c \int_{\mathbb{R}^d} \frac{f(y)}{|x-y|^{d-1}}dy. $$ Decompose $$ If(x)=I_\varepsilon + II_{\varepsilon},$$ where $$ I_\varepsilon =c\int_{|x-y|<\varepsilon} \frac{1}{|x-y|^{d-1}} f(y)dy $$ and $$ II_\varepsilon =c\int_{|x-y|\geq \varepsilon} \frac{1}{|x-y|^{d-1}} f(y)dy. $$ On the one hand, we decompose the integrand into dyadic shell to estimate that \begin{align} |I_\varepsilon|&\leq \sum_{j=0}^\infty \int_{\varepsilon 2^{-j-1} \leq |x-y|\leq \varepsilon 2^{-j}} \frac{|f(y)|}{|x-y|^{d-1}}dy\\ &\approx \frac{\varepsilon 2^{-j}}{(\varepsilon 2^{-j})^d}\sum_{j=0}^\infty \int_{|x-y|\leq \varepsilon 2^{-j}} |f(y)| dy\\ &\leq N \varepsilon Mf(x) \sum_{j=0}^\infty 2^{-j}\\ &\leq N \varepsilon Mf(x). \end{align} On the other hand, it follows from Holder's inequality that \begin{align} II_\varepsilon&\leq \Vert f \Vert_{L_{p,w}(B_R)} \left(\int_{\{|x-y|>\varepsilon\}\cap B_R} |x-y|^{(1-d)p'} w^{-1/(p-1)}dy\right)^{1/p'}. \end{align} By reverse Holder property of $A_p$ weight, there exists $1<q<p$ and $d>p/q$ so that $w\in A_q$. Hence it follows that $$ II_\varepsilon (x)\leq c \Vert f \Vert_{L_{p,w}(B_R)}\left(\int_{B_R} w^{-1/(q-1)} dy\right)^{(q-1)/p} \varepsilon^{1-dq/p}. $$ Hence we get $$ If(x)\leq N \varepsilon M f(x) + N \Vert f \Vert_{L_{p,w}(B_R)} \left(\int_{B_R} w^{-1/(q-1)} dy\right)^{(q-1)/p} \varepsilon^{1-dq/p}. $$ Now we minimize with respect to $\varepsilon$ to get $$ If(x)\leq N[Mf(x)]^{1-p/nq} \Vert f \Vert_{L_{p,w}(B_R)}^{p/dq} \left(\int_{B_R} w^{-1/(q-1)} \right)^{(q-1)/dq}. $$ By taking $L_{p,w}(B_R)$-norm to get $$ \Vert I f \Vert_{L_{pk,w}(B_R)} \leq N \Vert f \Vert_{L_{p,w}(B_R)} \left(\int_{B_R} w^{-1/(q-1)} dy \right)^{(q-1)/dq}, $$ where $k=dq/(dq-p)$. Then we have \begin{align} &\left(\frac{1}{w(B_R)} \int_{B_R} |If|^{pk} w dx \right)^{1/{pk}}\\ & \leq N\left(\frac{1}{w(B_R)} \int_{B_R} |f|^p w dx \right)^{1/p} w(B_R)^{\frac{1}{p}\left(1-\frac{1}{k}\right)} \left(\int_{B_R} w^{-\frac{1}{q-1}} dx \right)^{(q-1)/dq}. \end{align} By $A_q$-condition, we have \begin{align} &w(B_R)^{\frac{1}{p}\left(1-\frac{1}{k}\right)} \left(\int_{B_R} w^{-\frac{1}{q-1}} dx \right)^{(q-1)/dq}\\ &=|B_R|^{\frac{1}{dq}}|B_R|^{\frac{q-1}{dq}}\left(\frac{1}{|B_R|} \int_{B_R} w dx \right)^{\frac{1}{dq}} \left( \frac{1}{|B_R|} \int_{B_R} w^{-\frac{1}{q-1}} dx \right)^{\frac{q-1}{dq}}\\ &\leq NR [w]_{A_q}^{1/(dq)}. \end{align} This implies that \begin{align} \left(\frac{1}{w(B_R)} \int_{B_R} |If|^{pk} w dx \right)^{1/{pk}}&\leq N R \left(\frac{1}{w(B_R)} \int_{B_R} |f|^{p} w dx \right)^{1/{p}}. \end{align} Then the desired result follows from Jensen's inequality.