Kind of passage to the limit in the sense of distributions
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 '18 at 6:29
M. Rahmat
332212
add a comment |
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 '18 at 6:29
M. Rahmat
332212
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 '18 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 '18 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 '18 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 '18 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 '18 at 13:38
add a comment |
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 '18 at 6:29
M. Rahmat
332212
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 '18 at 6:29
M. Rahmat
332212
asked Nov 19 '18 at 6:29
M. Rahmat
332212
edited Nov 22 '18 at 4:39
asked Nov 19 '18 at 6:29
M. Rahmat
332212
asked Nov 19 '18 at 6:29
M. Rahmat
332212
asked Nov 19 '18 at 6:29
M. Rahmat
332212
332212
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 '18 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 '18 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 '18 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 '18 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 '18 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 '18 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 '18 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 '18 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 '18 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 '18 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 '18 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 '18 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 '18 at 4:40
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 '18 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 '18 at 7:08
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 '18 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 '18 at 13:34
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 '18 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 '18 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 '18 at 13:38
add a comment |
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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
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 '18 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 '18 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 '18 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 '18 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 '18 at 13:38