Let $1<p<\infty$ and $w\in A_p$ which is the well-known class of Muckenhoupt weights and let $\Omega$ be an open bounded smooth domain in $\mathbb{R}^N$ ($N\geq 2$). For $t>0$ we say that $u$ belong to the weighted Morrey space $L^{p,t}(\Omega,w)$, if $u\in L^p(\Omega,w)$ and $$ ||u||_{L^{p,t}(\Omega,w)}:=\big(\sup_{x\in\Omega, 0<r<d_{0}}\frac{r^t}{\mu(\Omega\cap B(x,r))}\int_{\Omega\cap B(x,r)}|u(y)|^pw(y)\,dy\big)^\frac{1}{p}<\infty, $$ where $d_0:=\text{diam}(\Omega)$, $\mu(\Omega\cap B(x,r))=\int_{\Omega\cap B(x,r)}w(y)\,dy$ and $$L^{p}(\Omega,w):=\{u:\Omega\to\mathbb{R}\text{ measurable }:\int_{\Omega}|u|^pw(y)\,dy<\infty\}.$$

Now the function $w(x)=|x|^\alpha\in A_p$ if and only if $-N<\alpha<N(p-1)$.

I want to find a function $w\in A_p$ such that $\frac{1}{w}\in L^{q,t}(\Omega,w)$ for some $q>N$ and $t>0$.

This space is defined on page 6 of the following reference: http://imar.ro/journals/Mathematical_Reports/Pdfs/2017/3/3.pdf

And my question is to find an example of $w$ mentioned in Theorem 1.3, page 4 of the above reference.

Please help me with an example. Thank you very much.


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