Properties of function on $L_p$ spaces

Question

Given $L_p$ space with the lebesgue measure on $\mathbb{R}^n$ and the function

$f(x) = |x|^{-\alpha}$ if $|x| < 1$
$f(x) = 0$ if $|x| \geq 1$

I need to show that $f \in L_p$ if and only if $p\alpha < n$.
Any help would be appreciated.
Thank you

Answer 1

Hint
Look at $$\| f_\alpha (x) \|_p^p = \int_{B_1(0)} |x|^{-p\alpha} \mathrm dx$$ By a dimensional argument you can go down to $$\int_0^1 |x|^{-n} \mathrm dx < \infty \Leftrightarrow n < 1$$ (polar coordinates with a small adjustment essentially give you a factor of $|x|^n$)