Cfrgtkky

I'm taking a Nonlinear PDEs course this semester and the last time our professor introduced us to Morrey & Campanato Spaces. We have for $lambda gt 0$ that:

• The Morrey space $L^{2,lambda}(Omega)$ consists of all functions $fin L^2(Omega)$ for which the seminorm

$[f]_{L^{2,lambda}}=sup_{x in Omega} sup_{r lt diamOmega} frac{1}{r^lambda} int_{B(x,r)cap Omega} |f|^2 dy lt infty$

• The Campanato space $mathcal L^{2,lambda}(Omega)$ consists of all functions $fin L^2(Omega)$ for which the seminorm

$[f]_{mathcal L^{2,lambda}}=sup_{x in Omega} sup_{r lt diamOmega} frac{1}{r^lambda} int_{B(x,r)cap Omega} |f-f_{x,r}|^2 dy lt infty$

where $f_{x,r}:=frac{1}{mathcal L^n(B(x,r)capOmega)} int_{B(x,r)cap Omega} f dy$ and $Omega subset mathbb R^n$

The first examples that the professor gave are:

1. $fin L^{infty} Rightarrow f in L^{2,lambda} ;;forall lambda in (0,n]$


2. $f in W^{1,infty} Rightarrow f in mathcal L^{2,n+2}$


3. $f in C^{0,alpha} Rightarrow f in mathcal L^{2,n+2alpha}$

Although they seem to be quite trivial since there is no special proof of the above nowhere, I have trouble understanding them. I think this double $sup$ in the definition confuses me a lot because I don't know how to handle them. Why do these 3 examples hold?

I only have some thoughts about 2.:

If $f in W^{1,infty}(Omega)$ then $f$ is a Lipschitz function. So we write

$|f(x)-f_{x,r}(x)|=|frac{1}{mathcal L^n(B(x,r)cap Omega)} int_{B(x,r)cap Omega} f(x)-f(y) dy| le frac{1}{mathcal L^n(B(x,r)cap Omega)} int_{B(x,r)cap Omega} |f(x)-f(y)| dy le frac{1}{mathcal L^n(B(x,r)cap Omega)} int_{B(x,r)cap Omega} M|x-y| dy le frac{1}{mathcal L^n(B(x,r)cap Omega)} int_{B(x,r)cap Omega} Mr; dy=Mr$

Hence $sup_{x in Omega} sup_{r lt diamOmega} frac{1}{r^lambda} int_{B(x,r)cap Omega} |f-f_{x,r}|^2 dy le sup_{x in Omega} sup_{r lt diamOmega} frac{1}{r^lambda} int_{B(x,r)cap Omega} M^2 r^2 dy=sup_{x in Omega} sup_{r lt diamOmega} frac{1}{r^{lambda-2}} M^2 mathcal L^n(B(x,r)cap Omega)$

At this point I've been stuck. How do I proceed? Could somebody provide me some hints in order to prove the rest too?

Any help is much appreciated. Thanks in advance!