1
$\begingroup$

Let $f \in L^p(\mathbb{R}^n,\mathbb{R}^m)$. If $m=1$ then the essential support of $f$ is a mainstream definition; see here for example. However, when $m>1$ is the following definition used?

$$ \operatorname{ess-supp}(f) := \bigcap \left\{ K \subseteq \mathbb{R}^n : \, K \mbox{ closed and } [f](x)=0 \mbox{ $\mu$-a.e. } x \not\in K \right\}, $$ where here $0$ denotes the zero-vector in $\mathbb{R}^D$?

Improve this question
$\endgroup$
2
  • $\begingroup$ If I'm not mistaken this definition is equivalent to $\operatorname{ess-sup}(f)\otimes 1_{\mathbb{R}^m}$ when identifying $f$ with an element of $L^p(\mathbb{R}^n,\mathbb{R})\otimes \mathbb{R}^m$? But maybe I'm wrong. $\endgroup$
    – ABIM
    Commented Mar 18, 2020 at 20:41
  • 2
    $\begingroup$ I think so, yes... But still this doesn't answer the question if the notation/terminology is used or not... $\endgroup$
    – ABIM
    Commented Mar 18, 2020 at 20:45

0

Reset to default

You must log in to answer this question.