For what $p \in [1, + \infty]$ and $a \in R$ the function $u(x)=(1+|x|)^{-a}$ defined on $R^n$ verify $||u||_{L_p}< \infty$?

Question

Has been 7 years from my last $L_p$ spaces experiences, now I have an exam about this. I have difficulties with the very first exercise:

For what $p \in [1, + \infty]$ and $a \in R$ the function $u(x)=(1+|x|)^{-a}$ defined on $R^n$ verify $||u||_{L_p}< \infty$?

Can someone explain me the methods for handle this kind of exercise?

Answer 1

Split the cases where $a$ is positive and negative. The second one is trivial, for the first one the standard thing to do is to use spherical coordinates in $\mathbb{R}^n$. I could do the exercise for you, but from here it should be quite simple, as the integral becomes practially one dimensional (the only thing that could explode is the radial part, once you have changed coordinate system).

Answer 2

$\int (1+|x|)^{-ap}dx=C \int_0^{\infty} (1+r)^{-ap} r^{n-1}\, dr$ where $C$ is the measure of $S^{n-1}$. Can you finish now? [Answer: $ap >n$].