What Morrey and Campanato space characterize

Question

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?

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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.

Answer 1

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 $0<x<1$.