Let $c>0$ and let $X$ be a subspace of $C^{\infty}_c((0,1);\mathbb R)$ such that for any $\phi \in X$ there holds: $$ \int_{r}^1 \phi'(t)^2\,dt \leq c\,\int_{r-r^2}^1 \phi(t)^2\,dt \quad \forall\, r\in (0,1).$$ Is it true that $X$ must be finite dimensional?

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?