Mixing
Locally Sobolev space and Sobolev spaces
$begingroup$
In Reed, Simon, Methods of Mathematical Physics, it defines the Sobolev space $W_a, ain mathbb{R}$ as the set of tempered distributions $muin S'(mathbb{R}^n)$ such that its Fourier transform $hat{mu}$ is measurable and
$$
|| mu ||_a^2 = int (1+p^2)^a |hat{mu}|^2 dp <infty
$$
Let $Omega subseteq mathbb{R}^n$ be open. Then the local Sobolev spaces on $W_a (Omega)$ is the set of distributions $mu in D'$ such that $phi mu in W_a$ for every $phi in D(Omega) = C_c^infty(Omega)$.
It seems that if $Omega = mathbb{R}^n$, then the local Sobolev space $W_a (Omega)$ should equal the Sobolev space $W_a$. However, I can't seem to prove it. Any help would be appreciated.
sobolev-spaces distribution-theory
asked Jan 7 at 6:32
Andrew YuanAndrew Yuan
1508
$endgroup$
add a comment |
$begingroup$
In Reed, Simon, Methods of Mathematical Physics, it defines the Sobolev space $W_a, ain mathbb{R}$ as the set of tempered distributions $muin S'(mathbb{R}^n)$ such that its Fourier transform $hat{mu}$ is measurable and
$$
|| mu ||_a^2 = int (1+p^2)^a |hat{mu}|^2 dp <infty
$$
Let $Omega subseteq mathbb{R}^n$ be open. Then the local Sobolev spaces on $W_a (Omega)$ is the set of distributions $mu in D'$ such that $phi mu in W_a$ for every $phi in D(Omega) = C_c^infty(Omega)$.
It seems that if $Omega = mathbb{R}^n$, then the local Sobolev space $W_a (Omega)$ should equal the Sobolev space $W_a$. However, I can't seem to prove it. Any help would be appreciated.
sobolev-spaces distribution-theory
asked Jan 7 at 6:32
Andrew YuanAndrew Yuan
1508
$endgroup$
add a comment |
$begingroup$
In Reed, Simon, Methods of Mathematical Physics, it defines the Sobolev space $W_a, ain mathbb{R}$ as the set of tempered distributions $muin S'(mathbb{R}^n)$ such that its Fourier transform $hat{mu}$ is measurable and
$$
|| mu ||_a^2 = int (1+p^2)^a |hat{mu}|^2 dp <infty
$$
Let $Omega subseteq mathbb{R}^n$ be open. Then the local Sobolev spaces on $W_a (Omega)$ is the set of distributions $mu in D'$ such that $phi mu in W_a$ for every $phi in D(Omega) = C_c^infty(Omega)$.
It seems that if $Omega = mathbb{R}^n$, then the local Sobolev space $W_a (Omega)$ should equal the Sobolev space $W_a$. However, I can't seem to prove it. Any help would be appreciated.
sobolev-spaces distribution-theory
asked Jan 7 at 6:32
Andrew YuanAndrew Yuan
1508
$endgroup$
In Reed, Simon, Methods of Mathematical Physics, it defines the Sobolev space $W_a, ain mathbb{R}$ as the set of tempered distributions $muin S'(mathbb{R}^n)$ such that its Fourier transform $hat{mu}$ is measurable and
$$
|| mu ||_a^2 = int (1+p^2)^a |hat{mu}|^2 dp <infty
$$
Let $Omega subseteq mathbb{R}^n$ be open. Then the local Sobolev spaces on $W_a (Omega)$ is the set of distributions $mu in D'$ such that $phi mu in W_a$ for every $phi in D(Omega) = C_c^infty(Omega)$.
It seems that if $Omega = mathbb{R}^n$, then the local Sobolev space $W_a (Omega)$ should equal the Sobolev space $W_a$. However, I can't seem to prove it. Any help would be appreciated.
sobolev-spaces distribution-theory
sobolev-spaces distribution-theory
asked Jan 7 at 6:32
Andrew YuanAndrew Yuan
1508
asked Jan 7 at 6:32
Andrew YuanAndrew Yuan
1508
asked Jan 7 at 6:32
Andrew YuanAndrew Yuan
1508
asked Jan 7 at 6:32
Andrew YuanAndrew Yuan
1508
asked Jan 7 at 6:32
Andrew YuanAndrew Yuan
1508
1508
add a comment |
add a comment |
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