Let $T:= \sum_{i=1}^{\infty} i^{\alpha}(\delta_{1/i} - \delta_{-1/i})$. I want to find the values of $\alpha$ such that this operator is, in fact, a distribution in $D'(\mathbb{R} \setminus \{0\})$.
Clearly it is linear but now I have two problems.
- The sum is infinite (but maybe I can only take the partial sums and then the limit, however I'm stuck).
- We are working in $\mathbb{R} \setminus \{0\}$. Could this fact rise up problems? (For example I can't take compact subsets like $[-a,a]$ in which I usally worked).
So, how to prove that for all compact $K \subset\mathbb{R} \setminus \{0\}$ there exists $p \in \mathbb{N}$ and $c > 0 $ such that $$ |\langle T,\phi\rangle | \leq c \Vert \phi\Vert_{C_K^p}? $$
Your second concern is actually the key to resolve your first.
Compact sets of $\mathbb{R} \setminus \{0\}$ must be disjoint from some open neighborhood of $0,$ so for test functions $\phi$ supported on compact $K \subset \mathbb{R}\setminus\{0\},$ the operator $T$ applied to $\phi$ becomes a finite sum (whose "size" is dependent only on $K$)