Let $f\in C^\infty(\mathbb{R}^n \backslash \{0\} )$ be homogeneous of degree $m\in \mathbb{R}$: $f(tx)=t^mf(x)$ for $t>0$ and $x\neq 0$. When can we say $f \in L^p(B_r(0))$?
Here is the motivation: we know $|x|^m \in W^{1,p} (B_r(0))$ iff $m+\frac{n}{p}>1$. Can we conclude the same result for general homogeneous functions?