Integration by Parts for multivariable functions Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$. Let also $f$ be twice continuously differentiable, $f \in C^2(\mathbb{R}^2)$,
and the function $\frac{\partial}{\partial x_1} \frac{\partial}{\partial x_2} f(x_1,x_2)$ be absolutely integrable,
$\frac{\partial}{\partial x_1} \frac{\partial}{\partial x_2} f(x_1,x_2)\in L_1(\mathbb{R}^2)$.
I am wondering what are sufficient conditions for the following identity to hold
\begin{align*}
  &\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \frac{\partial}{\partial x_1} \frac{\partial}{\partial x_2} f(x_1,x_2)  dx_1dx_2\\
  &\qquad\qquad=
  \left(f(x_1,x_2)\Big\vert_{x_1=-\infty}^{x_1=+\infty} \right)\Big\vert_{x_2=-\infty}^{x_2=+\infty}\\
   &\qquad\qquad= 
  f(+\infty,+\infty) - f(+\infty,-\infty) -  f(-\infty,+\infty) + f(-\infty,-\infty)?
\end{align*}   
I would appreciate any ideas, suggestions, counterexamples. Thanks!
 A: Let's first resolve the ambiguity of notation such as $f(+\infty, +\infty)$. This could mean the double limit $\lim_{x,y \to +\infty}f(x,y)$, either of the iterated limits $\lim_{x \to +\infty} \lim_{y \to +\infty} f(x,y)$, a symmetric limit $\lim_{x \to +\infty} f(x,x)$, etc.  For functions in general, some, all or none of these limits may exist.  If all exist then they may not be equal.
As will be shown in this case, all such limits will exist and we can without bias select an iterated limit as the definition.
You have already given sufficient conditions by stating that the mixed second partial derivative exists everywhere and is absolutely integrable.
Absolute integrability and Fubini's theorem stipulates the integral over any rectangle $[a,b] \times [c,d]$ can be taken as iterated integrals in any order and there exists an upper bound $M$ such that
$$\int_c^d \left(  \int_a^b\left|\frac{\partial}{\partial x_1}\frac{\partial f}{\partial x_2} \right| \, dx_1 \right)\, dx_2  \leqslant M. $$
By dominated convergence, the integral over $\mathbb{R}^2$ exists as a limit (in any order) 
$$\int_{-\infty}^{+\infty} \left(  \int_{-\infty}^{+\infty}\frac{\partial}{\partial x_1}\frac{\partial f}{\partial x_2}  \, dx_1 \right)\, dx_2  = \lim_{b,d \to +\infty,a,c \to -\infty} \int_c^d \left(  \int_a^b\frac{\partial}{\partial x_1}\frac{\partial f}{\partial x_2}  \, dx_1 \right)\, dx_2.$$
Since $f \in C^2$ the first partial derivatives are continuous and integrable, and since the mixed second partial derivative is integrable we can apply the fundamental theorem of calculus (FTC) to obtain
$$  \int_a^b\frac{\partial}{\partial x_1}\frac{\partial f}{\partial x_2}  \, dx_1 = \frac{\partial f}{\partial x_2}(b,x_2) - \frac{\partial f}{\partial x_2}(a,x_2).$$ 
Integrating and applying the FTC again over $[c,d]$ we obtain
$$  \int_c^d \left(\int_a^b\frac{\partial}{\partial x_1}\frac{\partial f}{\partial x_2}  \, dx_1 \right) \, dx_2= f(b,d) - f(b,c) - f(a,d) + f(a,c).$$
It is then valid to take the limit of both sides as $b,d \to +\infty$ and $a,c \to -\infty$ in any order to obtain the final result.  
