Riesz measure associated with a superharmonic function
$begingroup$
Suppose $B$ is a ball in $mathbb{R}^{n}$ ($ ngeq3 $), and $v_{t}$ a family of superharmonic functions on a neighborhood of the closure of $B$, and the index $t$ varies in an interval $(a,b)$ of $ mathbb{R} $. We know that if $mu_{t} $ is the Riesz measure associated with $v_{t}$, then we have
begin{align}
int_{B}phi(x)dmu_{t}(x)=a_{n}int_{B}v_{t}(x)Deltaphi(x)dx
end{align}
for all $phiin C_{c}^{infty}(B)$ ( $a_{n}$ is a positive constant). We know also that the potential given in the Riesz decomposition theorem is
$$p_{t}(x)=int_{B}dfrac{dmu_{t}(y)}{|x-y|^{n-2}}.$$
My question is: suppose $F(t)$ is the function on the right of the top =, and suppose I can prove that $tmapsto F(t)$ is continuous on $(a,b)$ and bounded:
$$ |f(x)|leq M$$
($M$ depends on $phi$). Can I conclude that $tmapsto p_{t}(x)$ is continuous for $x$ fixed?
real-analysis measure-theory potential-theory
asked Dec 29 '18 at 4:21
M. RahmatM. Rahmat
291212
$endgroup$
add a comment |
$begingroup$
Suppose $B$ is a ball in $mathbb{R}^{n}$ ($ ngeq3 $), and $v_{t}$ a family of superharmonic functions on a neighborhood of the closure of $B$, and the index $t$ varies in an interval $(a,b)$ of $ mathbb{R} $. We know that if $mu_{t} $ is the Riesz measure associated with $v_{t}$, then we have
begin{align}
int_{B}phi(x)dmu_{t}(x)=a_{n}int_{B}v_{t}(x)Deltaphi(x)dx
end{align}
for all $phiin C_{c}^{infty}(B)$ ( $a_{n}$ is a positive constant). We know also that the potential given in the Riesz decomposition theorem is
$$p_{t}(x)=int_{B}dfrac{dmu_{t}(y)}{|x-y|^{n-2}}.$$
My question is: suppose $F(t)$ is the function on the right of the top =, and suppose I can prove that $tmapsto F(t)$ is continuous on $(a,b)$ and bounded:
$$ |f(x)|leq M$$
($M$ depends on $phi$). Can I conclude that $tmapsto p_{t}(x)$ is continuous for $x$ fixed?
real-analysis measure-theory potential-theory
asked Dec 29 '18 at 4:21
M. RahmatM. Rahmat
291212
$endgroup$
add a comment |
$begingroup$
Suppose $B$ is a ball in $mathbb{R}^{n}$ ($ ngeq3 $), and $v_{t}$ a family of superharmonic functions on a neighborhood of the closure of $B$, and the index $t$ varies in an interval $(a,b)$ of $ mathbb{R} $. We know that if $mu_{t} $ is the Riesz measure associated with $v_{t}$, then we have
begin{align}
int_{B}phi(x)dmu_{t}(x)=a_{n}int_{B}v_{t}(x)Deltaphi(x)dx
end{align}
for all $phiin C_{c}^{infty}(B)$ ( $a_{n}$ is a positive constant). We know also that the potential given in the Riesz decomposition theorem is
$$p_{t}(x)=int_{B}dfrac{dmu_{t}(y)}{|x-y|^{n-2}}.$$
My question is: suppose $F(t)$ is the function on the right of the top =, and suppose I can prove that $tmapsto F(t)$ is continuous on $(a,b)$ and bounded:
$$ |f(x)|leq M$$
($M$ depends on $phi$). Can I conclude that $tmapsto p_{t}(x)$ is continuous for $x$ fixed?
real-analysis measure-theory potential-theory
asked Dec 29 '18 at 4:21
M. RahmatM. Rahmat
291212
$endgroup$
Suppose $B$ is a ball in $mathbb{R}^{n}$ ($ ngeq3 $), and $v_{t}$ a family of superharmonic functions on a neighborhood of the closure of $B$, and the index $t$ varies in an interval $(a,b)$ of $ mathbb{R} $. We know that if $mu_{t} $ is the Riesz measure associated with $v_{t}$, then we have
begin{align}
int_{B}phi(x)dmu_{t}(x)=a_{n}int_{B}v_{t}(x)Deltaphi(x)dx
end{align}
for all $phiin C_{c}^{infty}(B)$ ( $a_{n}$ is a positive constant). We know also that the potential given in the Riesz decomposition theorem is
$$p_{t}(x)=int_{B}dfrac{dmu_{t}(y)}{|x-y|^{n-2}}.$$
My question is: suppose $F(t)$ is the function on the right of the top =, and suppose I can prove that $tmapsto F(t)$ is continuous on $(a,b)$ and bounded:
$$ |f(x)|leq M$$
($M$ depends on $phi$). Can I conclude that $tmapsto p_{t}(x)$ is continuous for $x$ fixed?
real-analysis measure-theory potential-theory
real-analysis measure-theory potential-theory
asked Dec 29 '18 at 4:21
M. RahmatM. Rahmat
291212
asked Dec 29 '18 at 4:21
M. RahmatM. Rahmat
291212
asked Dec 29 '18 at 4:21
M. RahmatM. Rahmat
291212
asked Dec 29 '18 at 4:21
M. RahmatM. Rahmat
291212
asked Dec 29 '18 at 4:21
M. RahmatM. Rahmat
291212
291212
add a comment |
add a comment |
0
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%2f3055526%2friesz-measure-associated-with-a-superharmonic-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
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.
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%2f3055526%2friesz-measure-associated-with-a-superharmonic-function%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
