# What Morrey and Campanato space characterize

Morrey and Campanato space is some subspace of $$L^p$$. We know that for a bounded domain $$\Omega$$, $$L^p$$ space characterize how the function blow up at some point. I want to know what Morrey and Campanato space characterize?

For bounded domain $$\Omega$$ and fixed $$p$$, if $$\lambda<\mu$$, we have $$L^{p,\mu}\subset L^{p,\lambda}$$ for Morrey space ,then since $$L^{p,n}\cong L^{\infty}$$, so it seems that Morrey space is another way to characterize blow-up. It is a bit different for Campanato space when $$\lambda\geq n$$, for $$\mathcal{L}^{p,\mu}\cong C^{0,\alpha}$$. Thus it seems it characterize kind of oscillation.

The lecture notes by Melanie Rupflin answer the question "What is a Morrey Space? What is a Campanato Space?"

The Morrey space $$L^{p,\lambda}$$ is a subset of $$L^p$$ containing functions $$f$$ on a domain $$\Omega\in\mathbb{R}^m$$ such that the integral of $$|f|^p$$ over a ball $$B$$ of radius $$r$$ centered at $$x_0$$ goes to zero at least as fast as $$r^\lambda$$ uniformly in $$x_0$$.

The purpose of the restriction of $$L^p$$ to the subspace $$L^{p,\lambda}$$ is that it allows for an alternative to the Sobolev-Embedding Theorem which can be used with exponent $$p$$ adapted to the equation (and not, for example, fixed by dimensionality). Choosing a smaller $$p$$ results in having to show a Morrey-Property with larger $$\lambda$$.

The Campanato space $${\cal L}^{p,\lambda}$$ is defined similarly as the corresponding Morrey space, but the integral of $$|f|^p$$ is replaced by the integral of $$|f-\bar{f}|^p$$, where $$\bar{f}$$ is the average of $$f$$ over the ball. This is less restrictive, so $$L^{p,\lambda}\subset {\cal L}^{p,\lambda}$$. The less restrictive definition of Campanato spaces allows for an integral characterization of Hölder continuity: $$C^{0,\alpha}={\cal L}^{p,m+p\alpha}$$.

Of particular interest is the case $$\lambda=m$$ where $$\lambda$$ equals the dimensionality of the domain $$\Omega$$. Then $${\cal L}^{p,m}={\cal L}^{1,m}$$ characterizes functions of bounded mean oscillations, meaning bounded $$|B|^{-1}\int_B|f(x)-\bar{f}|dx$$. It lies in between $$\cap_{p>1}L^p$$ and $$L^\infty$$.

A simple example of a function in $${\cal L}^{1,1}$$ is $$\log x$$ for $$0.

• It seems that the link doesn't work – 89085731 May 4 '19 at 19:30
• I think I have fixed the link; somehow that site registers an unique ID for each download of the lecture notes. – Carlo Beenakker May 4 '19 at 19:41
• Thanks a lot. I am thinking the embedding $L^{p,\lambda}\subset L^{q,\mu}$ as long as $\frac{n-p}{\lambda}\leq \frac{n-q}{\mu}$ and $p>q$. Since we are considering the blow up of the function at the point compared with $r^{\lambda}$. why we need the condition $p>q$, because I think $\frac{n-p}{\lambda}\leq \frac{n-q}{\mu}$ is enough. – 89085731 May 4 '19 at 19:49
• are you sure of this embedding relation? theorem 0.0.12 (4) in these notes has a different inequality with a proof. – Carlo Beenakker May 4 '19 at 20:31