Condition on a function to be in $L^{p}(R^{d})$ space

Question

Consider $L^{p}(R^{d})$ with Lebesgue measure. Let

$f_{0}(x)=|x|^{- \alpha}$ , $|x|<1$ and zero other wise

$f_{1}(x)=|x|^{- \alpha}$ , $|x|\geq1$ and zero other wise

Show $f_{0}\in L^{p}$ iff $p\alpha<d$ and $f_{1}\in L^{p}$ iff $p\alpha>d$ ?

Is there any way to show that with out using polar coordinates?!

Thanks.

Answer 1

Use the equivalence of the Euclidean norm $||x||_2=\sqrt{\sum_{i=1}^d x_i^2}$ with the $L^{\infty}$ norm $||x||_{\infty}=\max_{1\le i\le d}|x_i|$ and solve the same question with $||\cdot||_{\infty}$ instead. For the latter, one can do it by some explicit calculations only using Fubini's Theorem.