Summable function that is not infinitesimal This comes out from this exercise:
Let $u$ Harmonic on $\mathbb{R}^n$ and $\int_{_{\mathbb{R}^n}}u(x)^2 dx<+\infty$ prove that $u(x)=0$.
I want to use that $\mathbb{R}^n$ and $\int_{_{\mathbb{R}^n}}u(x)^2 dx<+\infty$ implies $\lim_{|x|\to+\infty}|u(x)|=0$. I am not sure of this property, is it real?
I mean, one can prove the exercise and than it shows that is real, but to prove the exercise one use Cauchy Shwartz on $\langle 1,u\rangle$ (i can give more datails).
So the question is when does $\mathbb{R}^n$ and $\int_{_{\mathbb{R}^n}}u(x)^2 dx<+\infty$ implies $\lim_{|x|\to+\infty}|u(x)|=0$ in general?
 A: I don't think you can have what what you ask. You really need to use specific properties of harmonic functions. Indeed, consider an arbitrary nonnegative function $\varphi \in C_c^{\infty} (\mathbb{R})$ such
that $\int_{\mathbb{R}} \varphi (x) d x = 1$. Define the infinitely differentiable function (no more with compact support)
$$ \Psi (x) := \sum_{n \in \mathbb{N}} \varphi \left( \frac{x - n}{r_n}
   \right) $$
where $(r_n)\to 0$ is a sequence of positive real numbers that shrinks the support
of $\varphi$. Further conditions on $(r_n)$ to be found later. We have
$$ \int_{\mathbb{R}} \Psi (x) d x = \sum_{n \in \mathbb{N}}
   \int_{\mathbb{R}} \varphi \left( \frac{x - n}{r_n} \right) d x =\sum_{n \in \mathbb{N}} r_n \int_{\mathbb{R}} \varphi
   (x) d x = \sum_{n \in \mathbb{N}} r_n . $$
Therefore if you choose $(r_n)\to 0$ so that the series converges to some $c > 0$, then
$$ \int_{\mathbb{R}} \Psi (x) d x = c. $$
However
$$\limsup_{x \rightarrow + \infty} \Psi (x)= \sup_{x \in \mathbb{R}} \varphi
(x) \, .$$
