1
$\begingroup$

I'm working with Morrey spaces, which are the spaces $$L^{p,\lambda}(\Omega):= \left\{ u \in L^1_{loc}(\Omega): \sup_{x \in \Omega, r > 0} r^{-\lambda}\int_{B(x,r)\cap \Omega}|u(y)|^pdy< \infty\right\},$$ equipped with the norm $$\left\|u\right\|_{p,\lambda} = \left(r^{-\lambda}\int_{B(x,r)\cap\Omega}|u(y)|^p\right)^{1/p}.$$ Some notes (for example http://w3.ualg.pt/~ssamko/dpapers/files/251_Morrey_overview.pdf) said that they are Banach spaces with above norms. However, I cannot find a reference for proof of the case $\Omega$ is unbounded.

I want to ask for a book or a publish, where I can find the proof. Thanks for your help.

Improve this question
$\endgroup$
2
  • $\begingroup$ If you are going to find any such result, it's bound to be in Morrey's book Multiple integrals in the calculus of variations. Have you checked there? $\endgroup$
    – Chris Gerig
    Commented Sep 25, 2017 at 6:43
  • $\begingroup$ I checked it, but there was no proof for the Theorem I need. $\endgroup$
    – Marry Mag
    Commented Sep 25, 2017 at 9:58

0

Reset to default

You must log in to answer this question.