I have the following exercise (with also a solution), which I can't understand.
Let $u(x,y)=(\sqrt{x^2+y^2})^{\alpha}$ and let $D$ be the unit disk. For which values of $\alpha$ is it true that $u \in H^{-1}(D)?$
The answer starts with:
We must find a constant $C$ satisfying
$$|\int_D uv | \leq D (\int_D |\nabla v|^2)^{\frac{1}{2}}$$ for every $v \in H_0^1(D)$
I cannot really understand why he's saying so. So, I know that $H^{-1}(D)$ is the space of bounded linear functionals from $H_0^1(D)$ to $\mathbb{R}$, while $u$ is just a function, not a functional.
So, I think he's using some kind of identification, like he's identifying $u$ with some functional, but I can't see which one!