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 $|\Omega \cap B_{r}(x_{0})|>Ar^{n}$ 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.

References

• Campanato, Sergio (1963), "Proprietà di hölderianità di alcune classi di funzioni", Ann. Scuola Norm. Sup. Pisa (3), 17: 175–188
• Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, ISBN 978-0-691-08330-8