Let's consider the following PDE in $\mathbb R^d$ : $$\frac{\partial^d u}{\partial x_1...\partial x_d}=f$$ where $f$ is a tempered distribution with support in $\mathbb R^d_+$. There is a result by Hörmander that states that there exists at least one tempered distribution solution to this PDE (this has to do with the division of tempered distributions by polynomials). I want to know if (because of the particularly simple form of this particular equation) all solutions to this PDE are in fact tempered?

This is false. For example for $d=2$ you have solutions of the form $F(x,y)+g(x)+h(y)$ where $F$ is the standard tempered solution obtained by integration and $g$ and $h$ are ANY distributions, hence not necessarily tempered. 

