Say for all continuously differentiable vector fields $w \in C^\infty_c (\mathbb{R}^d \times \mathbb{R}^d) $ with compact support. (Note $ w : \mathbb{R}^d \to \mathbb{R}^d$), I have the
$$ \big\langle v (y) ,w(y) \big\rangle = L(w(y)), ~ \text{where}~ L:C^\infty_c (\mathbb{R}^d \times \mathbb{R}^d)\to \mathbb{R} $$
Can I use this to get a bound of the form
$$ \|v(y)\| \leq f(y,w(y),L(w(y)) $$
for some other functional $f$ ?