Integral in termsof supremum
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I am facing difficulty to prove the following fact:
If $w$ is a locally integrable positive function in $Omega$, then
$$
sup_{B(x,R)}lvert vrvert=lim_{ptoinfty}lVert vrVert_{L^p(B(x,R),w))},
$$
where $B(x,R)subset Omega$ and $Omega$ is a bounded domain, and $lVert vrVert_{L^p(B(x,R),w))}$ is defined to be
$$
left(int_{B(x,R)}lvert vrvert^pw(x),dxright)^{1/p}.
$$
For $w=1$, I know this is true. But for non-constant $w$, even for $A_p$ weights does the same hold true? I have sen this fact is applied in the paper attached in the link: see page 15 or 16.
http://imar.ro/journals/Mathematical_Reports/Pdfs/2017/3/3.pdf
real-analysis functional-analysis pde sobolev-spaces
edited 11 hours ago
user10354138
6,124523
asked 13 hours ago
Mathlover
937
add a comment |
up vote
0
down vote
favorite
I am facing difficulty to prove the following fact:
If $w$ is a locally integrable positive function in $Omega$, then
$$
sup_{B(x,R)}lvert vrvert=lim_{ptoinfty}lVert vrVert_{L^p(B(x,R),w))},
$$
where $B(x,R)subset Omega$ and $Omega$ is a bounded domain, and $lVert vrVert_{L^p(B(x,R),w))}$ is defined to be
$$
left(int_{B(x,R)}lvert vrvert^pw(x),dxright)^{1/p}.
$$
For $w=1$, I know this is true. But for non-constant $w$, even for $A_p$ weights does the same hold true? I have sen this fact is applied in the paper attached in the link: see page 15 or 16.
http://imar.ro/journals/Mathematical_Reports/Pdfs/2017/3/3.pdf
real-analysis functional-analysis pde sobolev-spaces
edited 11 hours ago
user10354138
6,124523
asked 13 hours ago
Mathlover
937
1
If $w=0$ this cannot be true.
– daw
12 hours ago
1
For $A_p$ weights $w$ of course $w$ is nonzero (a.e.), so the same proof works.
– user10354138
11 hours ago
Thank you very much.
– Mathlover
7 hours ago
Can you kindly have a look at the foolwing question in the link : math.stackexchange.com/questions/2996709/…
– Mathlover
6 hours ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am facing difficulty to prove the following fact:
If $w$ is a locally integrable positive function in $Omega$, then
$$
sup_{B(x,R)}lvert vrvert=lim_{ptoinfty}lVert vrVert_{L^p(B(x,R),w))},
$$
where $B(x,R)subset Omega$ and $Omega$ is a bounded domain, and $lVert vrVert_{L^p(B(x,R),w))}$ is defined to be
$$
left(int_{B(x,R)}lvert vrvert^pw(x),dxright)^{1/p}.
$$
For $w=1$, I know this is true. But for non-constant $w$, even for $A_p$ weights does the same hold true? I have sen this fact is applied in the paper attached in the link: see page 15 or 16.
http://imar.ro/journals/Mathematical_Reports/Pdfs/2017/3/3.pdf
real-analysis functional-analysis pde sobolev-spaces
edited 11 hours ago
user10354138
6,124523
asked 13 hours ago
Mathlover
937
I am facing difficulty to prove the following fact:
If $w$ is a locally integrable positive function in $Omega$, then
$$
sup_{B(x,R)}lvert vrvert=lim_{ptoinfty}lVert vrVert_{L^p(B(x,R),w))},
$$
where $B(x,R)subset Omega$ and $Omega$ is a bounded domain, and $lVert vrVert_{L^p(B(x,R),w))}$ is defined to be
$$
left(int_{B(x,R)}lvert vrvert^pw(x),dxright)^{1/p}.
$$
For $w=1$, I know this is true. But for non-constant $w$, even for $A_p$ weights does the same hold true? I have sen this fact is applied in the paper attached in the link: see page 15 or 16.
http://imar.ro/journals/Mathematical_Reports/Pdfs/2017/3/3.pdf
real-analysis functional-analysis pde sobolev-spaces
real-analysis functional-analysis pde sobolev-spaces
edited 11 hours ago
user10354138
6,124523
asked 13 hours ago
Mathlover
937
edited 11 hours ago
user10354138
6,124523
asked 13 hours ago
Mathlover
937
edited 11 hours ago
user10354138
6,124523
edited 11 hours ago
user10354138
6,124523
edited 11 hours ago
user10354138
6,124523
6,124523
asked 13 hours ago
Mathlover
937
asked 13 hours ago
Mathlover
937
asked 13 hours ago
Mathlover
937
937
1
If $w=0$ this cannot be true.
– daw
12 hours ago
1
For $A_p$ weights $w$ of course $w$ is nonzero (a.e.), so the same proof works.
– user10354138
11 hours ago
Thank you very much.
– Mathlover
7 hours ago
Can you kindly have a look at the foolwing question in the link : math.stackexchange.com/questions/2996709/…
– Mathlover
6 hours ago
add a comment |
1
If $w=0$ this cannot be true.
– daw
12 hours ago
1
For $A_p$ weights $w$ of course $w$ is nonzero (a.e.), so the same proof works.
– user10354138
11 hours ago
Thank you very much.
– Mathlover
7 hours ago
Can you kindly have a look at the foolwing question in the link : math.stackexchange.com/questions/2996709/…
– Mathlover
6 hours ago
1
1
If $w=0$ this cannot be true.
– daw
12 hours ago
If $w=0$ this cannot be true.
– daw
12 hours ago
1
1
For $A_p$ weights $w$ of course $w$ is nonzero (a.e.), so the same proof works.
– user10354138
11 hours ago
For $A_p$ weights $w$ of course $w$ is nonzero (a.e.), so the same proof works.
– user10354138
11 hours ago
Thank you very much.
– Mathlover
7 hours ago
Thank you very much.
– Mathlover
7 hours ago
Can you kindly have a look at the foolwing question in the link : math.stackexchange.com/questions/2996709/…
– Mathlover
6 hours ago
Can you kindly have a look at the foolwing question in the link : math.stackexchange.com/questions/2996709/…
– Mathlover
6 hours ago
add a comment |
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1
If $w=0$ this cannot be true.
– daw
12 hours ago
1
For $A_p$ weights $w$ of course $w$ is nonzero (a.e.), so the same proof works.
– user10354138
11 hours ago
Thank you very much.
– Mathlover
7 hours ago
Can you kindly have a look at the foolwing question in the link : math.stackexchange.com/questions/2996709/…
– Mathlover
6 hours ago