Bounding a Function When We Only Know Its Dot Product Against Test Functions.

Question

Let $\phi : \mathbb{R}^d\to \mathbb{R}^d $ be in $ C^{\infty}_0( \mathbb{R}^d)$ that is its continuously differentiable and has compact support. Take some other vector field $v: \mathbb{R}^d\to \mathbb{R}^d$.

$\textbf{Question : }$ Is there any hope in bounding the (euclidean) norm $\|v(x)\|$ $\textbf{if}$ I have a bound on $ | v(x)\cdot \phi(x) |$ $\forall \phi \in C^{\infty}_0( \mathbb{R}^d)$ ?

Answer 1

Not with the current hypothesis. If $d=1$ and $v(x)=x$, then $v$ is bounded on all compact sets, so $|v(x)\cdot\phi(x)|$ is always bounded, but of course $v$ is unbounded.