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?



Different definitions of Morrey and Campanato Spaces
Ok I got an answer to the question meanwhile:



> 
> 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-$A$.
> 
> 
> 


