$W^{2,2}(\mathbb R^n)=\{u\in L^2(\mathbb R^n):D^\alpha u\in L^2(\mathbb R^n)\, \forall |\alpha|\le2\},$ where $n\in \mathbb N^+$.
I have shown that, $f\in W^{2,2}(\mathbb R^n)$ if and only if $f\in L^2(\mathbb R^n)$ and $\Delta f\in L^2(\mathbb R^n).$
Now consider the case $\mathbb R^2$.
I strongly feel that, the condition that $f\in L^2(\mathbb R^2)$ and $\partial^2 f/\partial x_1 \partial x_2\in L^2(\mathbb R^2)$ is not enough to conclude that $f\in W^{2,2}(\mathbb R^2)$. But I am having trouble constructing such a counter example.
My question is, can someone give me a function $g$ s.t. $g\in L^2(\mathbb R^2)$ and $\partial ^2g/\partial x_1 \partial x_2\in L^2(\mathbb R^2)$ but $g\notin W^{2,2}(\mathbb R^2).$
Thanks for help.
Sure, let $f(x,y)=\begin{cases}\sqrt{x}+\sqrt{y}&\text{ if }0<x<1,0<y<1\\0&\text{ else}\end{cases}$
Then $f\in L^2(\mathbb R^2)$ clearly. Also $\frac{\partial^2f}{\partial x\partial y}=0$ which is in $L^2(\mathbb R^2)$. But $\Delta f=-\frac14(\frac{1}{x^{3/2}}+\frac{1}{y^{3/2}})\not\in L^2(\mathbb R^2).$