Different definitions of Morrey and Campanato Spaces

Question

The book by Giaquinta defines Campanato spaces using the seminorm:

$$[u]_{p,\lambda} = \left(\sup_{\substack{{x_0\in\Omega \\ 0<r<\text{diam}(\Omega)}}}r^{-\lambda}\int_{B_r(x_0)\cap\Omega}|u(x) - u_{x_0,r}|^p \right)^{1/p}$$

our lecture on the other hand uses:

$$[u]_{p,\lambda} = \left(\sup_{\substack{{x_0\in\Omega\\ 0<r<1}}}r^{-\lambda}\int_{B_r(x_0)\cap\Omega}|u(x) - u_{x_0,r}|^p \right)^{1/p}$$

and I have also seen the following definition used:

$$[u]_{p,\lambda} = \left(\sup_{\substack{{x_0\in\Omega \\ 0<r<\min(1,\text{diam}(\Omega))}}}r^{-\lambda}\int_{B_r(x_0)\cap\Omega}|u(x) - u_{x_0,r}|^p \right)^{1/p}$$

and similar for the definition of the Morrey spaces and for the definition "of type A".

Are those definitions equivalent? Or when are they equivalent?

Answer 1

Ok I got an answer to the question meanwhile (answered myself here https://mathoverflow.net/questions/173634/different-definitions-of-morrey-and-campanato-spaces):

The difference between the choice of upper bound $r$ is mainly related to the assumptions one requires about $\Omega$. If $\Omega$ is bounded, one wishes to have $A$ scaling-invariant and therefore chooses $r = diam(\Omega)$. But any other finite number would work as well. Therefore the choice of $1$ in the other definition. If $\Omega$ is not bounded on the other hand, one takes just $1$ by convention. The upper bound does therefore change $A$ but not the fact whether a $A$ exists or not therefore the set is still of Typ-$A$.