Reference for compact embedding for Holder space on $mathbb{R}^n$
$begingroup$
Suppose $0<alpha<beta$, and $Omega$ is a bounded subset of $mathbb{R}^n$. Then the Holder space $C^{beta}(Omega)$ is compactly embedded into $C^{alpha}(Omega)$. But if $Omega=mathbb{R}^n$, then the compact embedding is not true.
However, if we consider the weaker weighted Holder space $C^{alpha, -delta}(mathbb{R}^n)$ (for any $delta>0$) instead of $C^{alpha}(mathbb{R}^n)$. Then is $C^{beta}(mathbb{R}^n)$ compactly embedded to $C^{alpha, -delta}(mathbb{R}^n)$?
Here
$$
|f|_{C^{alpha, -delta}}=|(1+|cdot|^2)^{-frac{delta}{2}}f|_{C^{alpha}}.
$$
I could not find a precise reference from some books on functional analysis. Thanks in advance.
functional-analysis interpolation-theory
asked Jan 18 at 17:34
TomasTomas
651418
$endgroup$
add a comment |
$begingroup$
Suppose $0<alpha<beta$, and $Omega$ is a bounded subset of $mathbb{R}^n$. Then the Holder space $C^{beta}(Omega)$ is compactly embedded into $C^{alpha}(Omega)$. But if $Omega=mathbb{R}^n$, then the compact embedding is not true.
However, if we consider the weaker weighted Holder space $C^{alpha, -delta}(mathbb{R}^n)$ (for any $delta>0$) instead of $C^{alpha}(mathbb{R}^n)$. Then is $C^{beta}(mathbb{R}^n)$ compactly embedded to $C^{alpha, -delta}(mathbb{R}^n)$?
Here
$$
|f|_{C^{alpha, -delta}}=|(1+|cdot|^2)^{-frac{delta}{2}}f|_{C^{alpha}}.
$$
I could not find a precise reference from some books on functional analysis. Thanks in advance.
functional-analysis interpolation-theory
asked Jan 18 at 17:34
TomasTomas
651418
$endgroup$
add a comment |
$begingroup$
Suppose $0<alpha<beta$, and $Omega$ is a bounded subset of $mathbb{R}^n$. Then the Holder space $C^{beta}(Omega)$ is compactly embedded into $C^{alpha}(Omega)$. But if $Omega=mathbb{R}^n$, then the compact embedding is not true.
However, if we consider the weaker weighted Holder space $C^{alpha, -delta}(mathbb{R}^n)$ (for any $delta>0$) instead of $C^{alpha}(mathbb{R}^n)$. Then is $C^{beta}(mathbb{R}^n)$ compactly embedded to $C^{alpha, -delta}(mathbb{R}^n)$?
Here
$$
|f|_{C^{alpha, -delta}}=|(1+|cdot|^2)^{-frac{delta}{2}}f|_{C^{alpha}}.
$$
I could not find a precise reference from some books on functional analysis. Thanks in advance.
functional-analysis interpolation-theory
asked Jan 18 at 17:34
TomasTomas
651418
$endgroup$
Suppose $0<alpha<beta$, and $Omega$ is a bounded subset of $mathbb{R}^n$. Then the Holder space $C^{beta}(Omega)$ is compactly embedded into $C^{alpha}(Omega)$. But if $Omega=mathbb{R}^n$, then the compact embedding is not true.
However, if we consider the weaker weighted Holder space $C^{alpha, -delta}(mathbb{R}^n)$ (for any $delta>0$) instead of $C^{alpha}(mathbb{R}^n)$. Then is $C^{beta}(mathbb{R}^n)$ compactly embedded to $C^{alpha, -delta}(mathbb{R}^n)$?
Here
$$
|f|_{C^{alpha, -delta}}=|(1+|cdot|^2)^{-frac{delta}{2}}f|_{C^{alpha}}.
$$
I could not find a precise reference from some books on functional analysis. Thanks in advance.
functional-analysis interpolation-theory
functional-analysis interpolation-theory
asked Jan 18 at 17:34
TomasTomas
651418
asked Jan 18 at 17:34
TomasTomas
651418
asked Jan 18 at 17:34
TomasTomas
651418
asked Jan 18 at 17:34
TomasTomas
651418
asked Jan 18 at 17:34
TomasTomas
651418
651418
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
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%2f3078545%2freference-for-compact-embedding-for-holder-space-on-mathbbrn%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%2f3078545%2freference-for-compact-embedding-for-holder-space-on-mathbbrn%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
