Suppose $S=\{x\in\mathbb R^2:\|x\|_2\leq1\textrm{ and }x\neq0\}.$ What is the simplest way to characterize the set of smooth, square integrable functions on $S$?
The only way our integral can diverge is as $x\to0$, so I’m assuming there is some condition to check that involves a limit of $\|x\|_2^p \cdot f(x)$ as $x\to0$ for some $p$ but I got a bit stuck articulating the details. I’d like to work with smooth rather than holomorphic functions. Thanks!