---
tag: 18Q1BBK6C0
kind: definition
name: Morrey–Campanato space
msc: ["46E30", 42B35, "46E35"]
defines: NX1JQMFC4F
uses: [SASP14MA1Z, QJYTXFT8Y4, 012PBA0FMD]
sources:
  - url: https://en.wikipedia.org/wiki/Morrey%E2%80%93Campanato_space
---

## Definition

Let $\Omega \subseteq \R^n$ be a bounded open set with diameter $d$, let $1 \le p < \infty$ and let $\lambda \ge 0$.
For $x_0 \in \Omega$ and $r > 0$ put $\Omega(x_0, r) = \Omega \cap B(x_0, r)$, where $B(x_0, r)$ is the open ball of radius $r$ about $x_0$.
For $f \in L^p(\Omega)$ let $f_{x_0, r}$ be the mean value of $f$ over $\Omega(x_0, r)$.

1. The *Morrey space* $L^{p, \lambda}(\Omega)$ is the set of $f$ in the [[Lebesgue space|SASP14MA1Z]] $L^p(\Omega)$ with
$$\sup_{x_0 \in \Omega,\ 0 < r < d} \ r^{-\lambda} \int_{\Omega(x_0, r)} \abs{f(x)}^p \, dx < \infty.$$
2. The *Campanato space* $\mathcal{L}^{p, \lambda}(\Omega)$ is the set of $f \in L^p(\Omega)$ with
$$[f]_{p, \lambda}^p = \sup_{x_0 \in \Omega,\ 0 < r < d} \ r^{-\lambda} \int_{\Omega(x_0, r)} \abs{f(x) - f_{x_0, r}}^p \, dx < \infty.$$

The norms are $\norm{f}_{L^p}$ plus the $p$-th root of the supremum.

## Remarks

Example: let $\Omega$ be the unit ball of $\R^n$, let $0 < a < n/p$ and let $f(x) = \abs{x}^{-a}$.
The function $\abs{x}^{-ap}$ is radially decreasing, so its integral over a ball of radius $r$ is largest for the ball centred at $0$.
That integral equals $c \, r^{n - ap}$ for a constant $c$.
Hence $f \in L^{p, \lambda}(\Omega)$ with $\lambda = n - ap$.

Non-example: if $\lambda > n$, then $f = 1$ is not in $L^{p, \lambda}(\Omega)$.
Indeed $r^{-\lambda} \abs{\Omega(x_0, r)}$ grows like $r^{n - \lambda}$ as $r \to 0$, and this tends to infinity.
In fact $L^{p, \lambda}(\Omega) = \set{0}$ for $\lambda > n$, by the Lebesgue differentiation theorem.

Further facts:

- $L^{p, 0}(\Omega) = \mathcal{L}^{p, 0}(\Omega) = L^p(\Omega)$, and $L^{p, n}(\Omega) = L^\infty(\Omega)$.
- $\mathcal{L}^{p, n}(\Omega)$ is a space of functions of [[bounded mean oscillation|QJYTXFT8Y4]].
- If $\Omega$ has Lipschitz boundary and $n < \lambda \le n + p$, then $\mathcal{L}^{p, \lambda}(\Omega)$ equals the space of [[Hölder continuous|012PBA0FMD]] functions with exponent $\alpha = (\lambda - n)/p$ (Campanato's theorem).
- If $\lambda > n + p$ and $\Omega$ is connected, then $\mathcal{L}^{p, \lambda}(\Omega)$ contains only the constant functions.