given $M=\{x \in \mathbb{R}^n | \|x\|_2 \ge 1 \}$, $\alpha >0$
For which values of p is $$ f_{\alpha}(x)=\left\{\begin{array}{ll} \frac{1}{\|x\|_2^{\alpha}}, & x\in M \\ 0, & x\not\in M\end{array}\right. . $$ a member of $L_p(\mathbb{R}^n)?$
I thought if I transform this problem into spherical coordiantes, the problem boils down to show that $\int_1^{\infty} (f_{\alpha}(r)r^n)^p dr$ is bounded. But I think thats not the case for any p. Is this correct or am I mistaken?
Greetings.