0

How can one proof the inclusion of the Schwarz Space $S(\mathbb{R}^n) \subseteq L^p(\mathbb{R}^n)$?

Here (Inclusion of Schwartz space on $L^p$) it is discussed for n=1, but i don't know how to extend this to $n>1$, since i don't know if $1/(1+|x|^2)$ is integrable on $\mathbb{R}^2$.

Mac Menders
  • 257
  • You can use an even sharper bound based on the definition of the Schwartz space. Think about what the spherical measure is and how you need to account for it in conjunction with the $L^p$ norm. What decay rate do you need? – Cameron L. Williams Jun 30 '23 at 16:31
  • 3
    $(1+|x|^2)^{-N}$ is integrable on $\Bbb{R}^N$, and $(1+|x|^2)^N$ is in the seminorms defining $S(\Bbb{R}^N)$. – Chad K Jun 30 '23 at 16:38
  • Ah thanks @JohnDoe, i got it! – Mac Menders Jun 30 '23 at 16:47
  • @CameronWilliams what is the spherical measure in $\mathbb{R}^n$ ? – Mac Menders Jun 30 '23 at 16:48
  • It's the $r^{n-1} ,dr,d\Omega$ bit, specifically the $r^{n-1}$ part is the important part because it is easy to deal with. The $d\Omega$ integral is just the surface area of the sphere, so it is finite. – Cameron L. Williams Jun 30 '23 at 16:53

0 Answers0