3
$\begingroup$

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.

$\endgroup$
2
+100
$\begingroup$

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 $0<x<1$.

$\endgroup$
4
  • $\begingroup$ It seems that the link doesn't work $\endgroup$ – 89085731 May 4 '19 at 19:30
  • $\begingroup$ I think I have fixed the link; somehow that site registers an unique ID for each download of the lecture notes. $\endgroup$ – Carlo Beenakker May 4 '19 at 19:41
  • $\begingroup$ 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. $\endgroup$ – 89085731 May 4 '19 at 19:49
  • $\begingroup$ are you sure of this embedding relation? theorem 0.0.12 (4) in these notes has a different inequality with a proof. $\endgroup$ – Carlo Beenakker May 4 '19 at 20:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.