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...



Partial $L^2$ control on (part of) the Hessian of a harmonic function.
Okay so I thought about this some more and I believe it is just a (really) straightforward application of the Poincare/Wirtinger inequality and the obvious way you would solve $u\_{xy}=0$ classically. 


Lets work on the square $S=(0,2\pi)\_x\times(0,2\pi)\_y$ as it is simpler to work there and doesn't change much.  

We assume $u$ is $C^\infty\_0(S)$ (i.e. smooth and of compact support) for the time being but do not need it to be harmonic. 


Set $f(s)=\frac{1}{2\pi}\int\_0^{2\pi} u\_x(s, t)dt$ so $\int\_0^{2\pi} u\_{x}(s, t)-f(s) dt=0$ for all $s$.


By Wirtinger's inequality we have
$\int\_0^{2\pi} (u\_x(s,t)-f(s))^2 dt \leq \int\_0^{2\pi} u\_{xy}^2(s,t) dt$.


Now pick a $F$ so that $F'(x)=f(x)$.  

Set $G(t)=\frac{1}{2\pi}\int\_0^{2\pi} u(s,t)-F(s) ds$ so $\int\_0^{2\pi} u(s,t)-F(s)-G(t) ds=0$ for all $t$. Since $\frac{d}{ds}\left( u(s,t)-F(s)-G(t)\right)=u\_x(s,t)-f(s)$ we have by Wirtinger's inequality that:


$\int\_0^{2\pi} (u(s,t)-F(s)-G(t))^2 ds \leq \int\_0^{2\pi} (u\_x(s,t)-f(s))^2 $ for all $t$.
 Then by Fubini we have that


$\int\_{S} (u(x,y)-F(x)-G(y))^2 \leq \int\_S u^2\_{xy}$


Everything is valid if $u$ is in $H^2(S)$. Notice the $F,G$ are given explicitly in terms of $u$.
If $u$ is harmonic with the normalizing properties at $0$ then the $F=ax^2, G=-ay^2$.


At least I think this works...


