decay estimate for a elliptic operator

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I have a function $f : mathbbR² rightarrow mathbbC$ such that $|- Delta f+ f | < 1/(1+r)^1+sigma$, where $r$ is the distance to the origin and $0<sigma <1$, and I want to show that $|f| < K/(1+r)^1+sigma$ and $|nabla f| < K/(1+r)^2+sigma$ for some constant $K$.



I can show the first inequality using Bessel functions, but I can't find how to do it for the derivatives. If anyone has an idea or a reference, I would be interested :)










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    I have a function $f : mathbbR² rightarrow mathbbC$ such that $|- Delta f+ f | < 1/(1+r)^1+sigma$, where $r$ is the distance to the origin and $0<sigma <1$, and I want to show that $|f| < K/(1+r)^1+sigma$ and $|nabla f| < K/(1+r)^2+sigma$ for some constant $K$.



    I can show the first inequality using Bessel functions, but I can't find how to do it for the derivatives. If anyone has an idea or a reference, I would be interested :)










    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I have a function $f : mathbbR² rightarrow mathbbC$ such that $|- Delta f+ f | < 1/(1+r)^1+sigma$, where $r$ is the distance to the origin and $0<sigma <1$, and I want to show that $|f| < K/(1+r)^1+sigma$ and $|nabla f| < K/(1+r)^2+sigma$ for some constant $K$.



      I can show the first inequality using Bessel functions, but I can't find how to do it for the derivatives. If anyone has an idea or a reference, I would be interested :)










      share|cite|improve this question













      I have a function $f : mathbbR² rightarrow mathbbC$ such that $|- Delta f+ f | < 1/(1+r)^1+sigma$, where $r$ is the distance to the origin and $0<sigma <1$, and I want to show that $|f| < K/(1+r)^1+sigma$ and $|nabla f| < K/(1+r)^2+sigma$ for some constant $K$.



      I can show the first inequality using Bessel functions, but I can't find how to do it for the derivatives. If anyone has an idea or a reference, I would be interested :)







      pde estimation bessel-functions elliptic-operators






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      asked Sep 10 at 9:26









      Eliot

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