$\frac{f}{|x|^4+1}$ lies in $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ for $f \in L^2(\mathbb{R}^n)$ and $n > 6$

Question

I'm trying to solve the following problem:

Show that $\frac{f}{|x|^4+1}$ lies in $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ for $f \in L^2(\mathbb{R}^n)$ and $n \le 7$, where $|x|$ denotes the euclidian Norm in $\mathbb{R}^n$

My first approach was to prove that the function $\frac{1}{|x|^4+1}$ is in $L^2$, but unfortunately, this is not the case.

Could you please provide a hint on how to solve this problem?

Answer 1

For $L^1$ try Hölder's inequality.

$$\int \frac{f}{1+|x|^4} \le \Vert f\Vert_{L^2} \left(\int \frac{1}{(1+|x|^4)^2} \right)^{1/2}.$$

For $L^2$, this is obvious because $1/(1+|x|^4)<1$.