Morrey–Campanato space

In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato) $$L^{\lambda, p}(\Omega)$$ are Banach spaces which extend the notion of functions of bounded mean oscillation, describing situations where the oscillation of the function in a ball is proportional to some power of the radius other than the dimension. They are used in the theory of elliptic partial differential equations, since for certain values of $$\lambda$$, elements of the space $$L^{\lambda,p}(\Omega)$$ are Hölder continuous functions over the domain $$\Omega$$.

The seminorm of the Morrey spaces is given by
 * $$\bigl([u]_{\lambda,p}\bigr)^p = \sup_{0 < r< \operatorname{diam} (\Omega), x_0 \in \Omega} \frac{1}{r^\lambda} \int_{B_r(x_0) \cap \Omega} | u(y) |^p dy. $$

When $$\lambda = 0$$, the Morrey space is the same as the usual $$L^p$$ space. When $$\lambda = n$$, the spatial dimension, the Morrey space is equivalent to $$L^\infty$$, due to the Lebesgue differentiation theorem. When $$\lambda > n$$, the space contains only the 0 function.

Note that this is a norm for $$ p \geq 1 $$.

The seminorm of the Campanato space is given by
 * $$\bigl([u]_{\lambda,p}\bigr)^p = \sup_{0 < r< \operatorname{diam} (\Omega), x_0 \in \Omega} \frac{1}{r^\lambda} \int_{B_r(x_0) \cap \Omega} | u(y) - u_{r,x_0} |^p dy $$

where
 * $$u_{r,x_0} = \frac{1}{|B_r(x_0)\cap \Omega|} \int_{B_r(x_0)\cap \Omega} u(y) dy.$$

It is known that the Morrey spaces with $$0 \leq \lambda < n$$ are equivalent to the Campanato spaces with the same value of $$\lambda$$ when $$\Omega$$ is a sufficiently regular domain, that is to say, when there is a constant A such that $$|\Omega \cap B_r(x_0)| > A r^n $$ for every $$x_0 \in \Omega$$ and $$r < \operatorname{diam}(\Omega)$$.

When $$n=\lambda$$, the Campanato space is the space of functions of bounded mean oscillation. When $$n < \lambda \leq n+p$$, the Campanato space is the space of Hölder continuous functions $$C^\alpha(\Omega)$$ with $$\alpha = \frac{\lambda - n}{p}$$. For $$\lambda > n+p$$, the space contains only constant functions.