Let $u \in L^2(\mathbb{R}^n)$ and $$\frac{\partial}{\partial x_1}\frac{\partial}{\partial x_2} \dots \frac{\partial}{\partial x_n}u \in (1 + |x|)^{-n-1}L^2(\mathbb{R}^n)$$ where the derivatives are computed in the sense of distributions. The problem: use repeated integration to show that $u \in L^\infty(\mathbb{R}^n) \cap C^0(\mathbb{R}^n)$.
I was tempted to simply integrate $$ \frac{\partial}{\partial x_1}\frac{\partial}{\partial x_2} \dots \frac{\partial}{\partial x_n}u \in L^\infty(\mathbb{R}^n) \cap C^0(\mathbb{R}^n)$$ and get $u(x)$ (and hence a uniform bound on $u(x)$), but since it is not the actual derivative I don't think I can do it. Any help appreciated.