# Example of a function in the weighted Morrey space

Let $$1 and $$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 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.

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.