Kind of passage to the limit in the sense of distributions
up vote
0
down vote
favorite
Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$, and $f$ a locally integrable function. Suppose $F$ is a closed set with empty interior and $M>0$ such that the distribution defined by $f$ satisfies
begin{equation}
int_{Bsetminus F}f(t)phi(t)dtleq M
end{equation} for all test function $phi$. We know that if we had
$$f(x)leq M$$ for all $xin Bsetminus F$, we could get the same inequality by passing to the limit (supposing that $f$ is continuous); is it possible, in the same way, to circumvent around $F$ in (1) and prove that (1) holds for integrals over $B$?
real-analysis integration distribution-theory
asked Nov 19 at 6:29
M. Rahmat
289211
add a comment |
up vote
0
down vote
favorite
Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$, and $f$ a locally integrable function. Suppose $F$ is a closed set with empty interior and $M>0$ such that the distribution defined by $f$ satisfies
begin{equation}
int_{Bsetminus F}f(t)phi(t)dtleq M
end{equation} for all test function $phi$. We know that if we had
$$f(x)leq M$$ for all $xin Bsetminus F$, we could get the same inequality by passing to the limit (supposing that $f$ is continuous); is it possible, in the same way, to circumvent around $F$ in (1) and prove that (1) holds for integrals over $B$?
real-analysis integration distribution-theory
asked Nov 19 at 6:29
M. Rahmat
289211
1
Something is off in your formulation. The left-hand side of your inequality is linear with respect to $phi$, and the right-hand side is not. Then again, does $M$ depend on $x$?
– TZakrevskiy
Nov 21 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 at 13:34
Ok, how does $M$ depend on $varphi$? This information is important. Imagine that $M = |f|_{L^1(B)} |varphi|_infty$, in this case your inequality has no new information at all.
– TZakrevskiy
Nov 22 at 13:38
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$, and $f$ a locally integrable function. Suppose $F$ is a closed set with empty interior and $M>0$ such that the distribution defined by $f$ satisfies
begin{equation}
int_{Bsetminus F}f(t)phi(t)dtleq M
end{equation} for all test function $phi$. We know that if we had
$$f(x)leq M$$ for all $xin Bsetminus F$, we could get the same inequality by passing to the limit (supposing that $f$ is continuous); is it possible, in the same way, to circumvent around $F$ in (1) and prove that (1) holds for integrals over $B$?
real-analysis integration distribution-theory
asked Nov 19 at 6:29
M. Rahmat
289211
Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$, and $f$ a locally integrable function. Suppose $F$ is a closed set with empty interior and $M>0$ such that the distribution defined by $f$ satisfies
begin{equation}
int_{Bsetminus F}f(t)phi(t)dtleq M
end{equation} for all test function $phi$. We know that if we had
$$f(x)leq M$$ for all $xin Bsetminus F$, we could get the same inequality by passing to the limit (supposing that $f$ is continuous); is it possible, in the same way, to circumvent around $F$ in (1) and prove that (1) holds for integrals over $B$?
real-analysis integration distribution-theory
real-analysis integration distribution-theory
asked Nov 19 at 6:29
M. Rahmat
289211
asked Nov 19 at 6:29
M. Rahmat
289211
edited Nov 22 at 4:39
asked Nov 19 at 6:29
M. Rahmat
289211
asked Nov 19 at 6:29
M. Rahmat
289211
asked Nov 19 at 6:29
M. Rahmat
289211
289211
1
Something is off in your formulation. The left-hand side of your inequality is linear with respect to $phi$, and the right-hand side is not. Then again, does $M$ depend on $x$?
– TZakrevskiy
Nov 21 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 at 13:34
Ok, how does $M$ depend on $varphi$? This information is important. Imagine that $M = |f|_{L^1(B)} |varphi|_infty$, in this case your inequality has no new information at all.
– TZakrevskiy
Nov 22 at 13:38
add a comment |
1
Something is off in your formulation. The left-hand side of your inequality is linear with respect to $phi$, and the right-hand side is not. Then again, does $M$ depend on $x$?
– TZakrevskiy
Nov 21 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 at 13:34
Ok, how does $M$ depend on $varphi$? This information is important. Imagine that $M = |f|_{L^1(B)} |varphi|_infty$, in this case your inequality has no new information at all.
– TZakrevskiy
Nov 22 at 13:38
1
1
Something is off in your formulation. The left-hand side of your inequality is linear with respect to $phi$, and the right-hand side is not. Then again, does $M$ depend on $x$?
– TZakrevskiy
Nov 21 at 13:19
Something is off in your formulation. The left-hand side of your inequality is linear with respect to $phi$, and the right-hand side is not. Then again, does $M$ depend on $x$?
– TZakrevskiy
Nov 21 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 at 4:40
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 at 7:08
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 at 13:34
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 at 13:34
Ok, how does $M$ depend on $varphi$? This information is important. Imagine that $M = |f|_{L^1(B)} |varphi|_infty$, in this case your inequality has no new information at all.
– TZakrevskiy
Nov 22 at 13:38
Ok, how does $M$ depend on $varphi$? This information is important. Imagine that $M = |f|_{L^1(B)} |varphi|_infty$, in this case your inequality has no new information at all.
– TZakrevskiy
Nov 22 at 13:38
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3004590%2fkind-of-passage-to-the-limit-in-the-sense-of-distributions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
Something is off in your formulation. The left-hand side of your inequality is linear with respect to $phi$, and the right-hand side is not. Then again, does $M$ depend on $x$?
– TZakrevskiy
Nov 21 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 at 13:34
Ok, how does $M$ depend on $varphi$? This information is important. Imagine that $M = |f|_{L^1(B)} |varphi|_infty$, in this case your inequality has no new information at all.
– TZakrevskiy
Nov 22 at 13:38