I have a simple little analysis question that I'm hoping is well known.

Suppose $D=\lbrace(x,y): x^2+y^2<1\rbrace$ is the unit disk and that $u$ is a harmonic function on $D$. Suppose in addition that $u(0)=0$ and $\nabla u(0)=0$. Lets also assume $u$ has finite $L^2$ norm -- i.e. $||u||_2<\infty$

If $u_{xy}=\partial_{xy}u=0$ on $D$ then it is clear that $u=a(x^2-y^2)$ for some $a$.

Suppose instead that we only know that $|| u_{xy}||_2<<1$. I'm pretty sure a compactness argument implies that there is an $a$ so that $||u-a(x^2-y^2)||_2<<1$ (or at least this is true on any smaller disk).

What I'm most interested in is whether there is a constant $C>0$ so that

$ \inf_{a\in \mathbb{R}} ||u-a(x^2-y^2)||_2\leq C ||u_{xy}||_2$.

(I'm also happy if the control was only on a smaller disk).

On a related note, does anyone know if $u_{xy}=0$ in a distributional sense then $u=F(x)+G(y)$. I can't remember now if this is true or not...