Prove that $L^p\left ( \mathbb R^n, e^{\frac{-1}2 |x|^2} \right)\cap \theta(\mathbb R^n)$ is infinite dimensional

Question

Prove that the space $$L^p\left ( \mathbb R^n, e^{\frac{-1}2 |x|^2} \right)\cap \theta(\mathbb R^n)$$ is infinite dimensional for $1\le p\le \infty$.

Here, $\theta(\mathbb R^n)$ is the set of all harmonic functions on $\mathbb R^n$. I have proved in a previous exercise that the space $$L^p(\mathbb R^n)\cap \theta(\mathbb R^n)$$ only contains constant functions ($0$ function if $p<\infty$).

But, I do not have any idea about proving this exercise.

Answer 1

We need to assume $n \geq 2$, else you only have functions of the form $$ f(x)=ax+b. $$ You know that with $z=x_1+ix_2$ $$ f(z)=z^m $$ is holomorphic for any $m>0$ and hence $$ \Re(f(z)) $$ is a (real-vaued) harmonic polynomial in the variables $x_1,x_2$.

Furthermore, any polynomial is in $L^p(\mathbb{R}^n,e^{-\frac{1}{2}|x|^2}dx)$.

But now there are infinitely many: The real part of $(x_1+ix_2)^m$ is also a polynomial of degree $m$ and hence these polynomials are linearly independent for different choices of $m$. Therefore, the subspace $$ \{ \Re((x_1+ix_2)^m) \mid m \in \mathbb{N} \} \subset L^p(\mathbb{R}^n,e^{-\frac{1}{2}|x|^2}dx) \cap \theta(\mathbb{R}^n) $$ is infinite-dimensional.