Morrey–Campanato space

In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato) ${\displaystyle L^{\lambda ,p}(\Omega )}$ 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 ${\displaystyle \lambda }$, elements of the space ${\displaystyle L^{\lambda ,p}(\Omega )}$ are Hölder continuous functions over the domain ${\displaystyle \Omega }$.

The seminorm of the Morrey spaces is given by

${\displaystyle {\bigl (}[u]_{\lambda ,p}{\bigr )}^{p}=\sup _{0<r<\operatorname {diam} (\Omega ),x_{0}\in \Omega }{\frac {1}{r^{\lambda }}}\int _{B_{r}(x_{0})\cap \Omega }|u(y)|^{p}dy.}$

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

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

The seminorm of the Campanato space is given by

${\displaystyle {\bigl (}[u]_{\lambda ,p}{\bigr )}^{p}=\sup _{0<r<\operatorname {diam} (\Omega ),x_{0}\in \Omega }{\frac {1}{r^{\lambda }}}\int _{B_{r}(x_{0})\cap \Omega }|u(y)-u_{r,x_{0}}|^{p}dy}$

where

${\displaystyle u_{r,x_{0}}={\frac {1}{|B_{r}(x_{0})\cap \Omega |}}\int _{B_{r}(x_{0})\cap \Omega }u(y)dy.}$

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

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