We have an $L^2$ function $u$ defined on $\mathbb{R^2}$ with compact support such that $u \in H^{2/3}$ (H stands for Sobolev spaces, as always), $\partial_y u \in L^2$, and $(x\partial_y - y\partial_x)u \in L^2$. I want to conclude that $u$ lies in some higher order $L^p$ space, that is, for some $p > 2$. Ideally, I would like to use all three pieces of information so that it leads to a high enough $p$. Any ideas?

Thanks a lot!

Edit: Is there some book that contains exercises specifically of this nature?

Another edit: I am sorry for not pointing this out explicitly: I am familiar of the conclusion that can be drawn from Sobolev Embedding type theorems. My main concern is how to make use of the facts $\partial_y u \in L^2$ and $(x\partial_y - y\partial_x)u \in L^2$, particularly the latter one.