Integrability of a given Laurent polynomial

Question

Does the following hold for all exponents $\alpha$ such that $|\alpha|<1$ ?

The following integral on $\mathbb{R}^n$ is finite: $$ \int_{\mathbb{R}^n} \frac{1}{|y|^{\alpha}}dy < \infty,$$ with $|\cdot|$ denoting the modulus of the evaluated vector.

Any advice would be appreciated.

Answer 1

This does not hold at all. Consider $n=1$ and $\alpha=1/2$ - $$\int_{\mathbb R}\frac{1}{\sqrt{|y|}}\mathrm dy \\ \text{diverges}$$