Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be **subspace** of $Y$ such that for any $\phi \in X$ there holds:
$$ \int\_{t^2>r^2} \phi'(t)^2\,dt \leq c\,\int\_{t^2>r^2-r^4} \phi(t)^2\,dt \quad \forall\, r\in (0,1).$$
Is it true that $X$ must be finite dimensional?



Finite dimensionality of a subspace
$\newcommand\de\delta$The answer is no. E.g., let $X$ be the linear span of the set $\{\de,\de',\de'',\dots\}$, where $\de$ is the Dirac delta distribution supported on $\{0\}$. Then $X$ satisfies your condition but is infinite dimensional.


