1
$\begingroup$

Consider the set of all compactly supported distributions $v\in\mathcal{\mathcal{E}^{\prime}}(\mathbb{R}^{n})=\left(C^{\infty}\right)^{*}$ with compact support in a fixed compact set $\Omega$ . Denote this set by $E:=\left\{ v\in\mathcal{\mathcal{E}^{\prime}}(\mathbb{R}^{n}):\mbox{supp}v\subset\Omega\right\}$.

Given any fixed $f\in C^{\infty}$ , can one then find a uniform bound $\left|v(f)\right|\leq C$ that holds for all $v\in E$ ?

Obviously, we have a bound for each individual $v$ , but can we somehow utilize that their support is in the same compact set to get a uniform bounded.

I'd be very grateful for any thoughts.

$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

Let $\Omega=[-1,1]$, $f$ - a test function such that $f(0)=1$. Take now $v_n = n\delta_0$. Obviously, $v_n\in E$, yet $(v_n,f)=n$, which can not be boudned.

To be more general, the space $\mathcal E'$ is a linear space, hence for a given $f$ you can not find a uniform bound on $(v,f)$ for all $v\in\mathcal E'$, because we can always multiply $v$ by a constant.

$\endgroup$

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.