I'm trying to find a function $u:\mathbb{R}^2 \rightarrow \mathbb{R}$ with compact support which has a weak first derivative and satisfies the following conditions:
$$ \int \limits_{\mathbb{R}^2} |\partial_x u|^{3/2} < \infty, \qquad \int \limits_{\mathbb{R}^2} |\partial_y u|^{3/2} < \infty, \qquad \int \limits_{\mathbb{R}^2} |u|^2|\partial_x u| = \infty $$
Can anyone provide me an example of such a function, or a proof that no such function exists?
For context, I'm investigating embeddings of different function spaces into others. So far, I know that it won't work with a function of the form $u(x,y)=x^k$ for any $k \in \mathbb{R}$ (brought to zero away from the origin).
There does not exist any such function.
It is clear that $u$ belongs to the Sobolev space $W^{1,\frac{3}{2}}(\Omega)$ for some $\Omega \subset \mathbb{R}^2$ as defined in this paper, since the definition does not assume any integrability on $u$. The paper states that there is an embedding:
$$ W^{1,p}(\Omega) \hookrightarrow L^q(\Omega) $$
where $q=np/(n-p)$. In our case, we have $q = 2*1.5/(2-1.5) = 6$. Then we can use Hölder's inequality to get:
$$ \int \limits_{\Omega} |u|^2|\partial_x u| \leq \left\| |u|^2|\partial_x u| \right\|_1 \leq \left\| |u|^2 \right\|_3 \left\| |\partial_x u| \right\|_{3/2} = \left\| u \right\|_6 \left\| \partial_x u \right\|_{3/2} $$
Then, since we have established that $u \in W^{1, \frac{3}{2}}(\Omega) \subset L^6(\Omega)$, we conclude that the product must be finite.