Reference for compact embedding for Holder space on $mathbb{R}^n$












1












$begingroup$


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.










share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    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.










    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      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.










      share|cite|improve this question









      $endgroup$




      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






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 18 at 17:34









      TomasTomas

      651418




      651418






















          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
          });


          }
          });














          draft saved

          draft discarded


















          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
















          draft saved

          draft discarded




















































          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.




          draft saved


          draft discarded














          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





















































          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







          Popular posts from this blog

          Human spaceflight

          Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

          張江高科駅