Morrey–Campanato space

In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato) $L^{\lambda ,p}(\mathrm{\Omega })$ 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 $L^{\lambda ,p}(\mathrm{\Omega })$ are Hölder continuous functions over the domain $\mathrm{\Omega }$. The seminorm of the Morrey spaces is given by

$([u{]}_{\lambda ,p}{)}^p=\underset{0<r<\mathrm{diam}(\mathrm{\Omega }),x_0\in \mathrm{\Omega }}{\sup }\frac{1}{r^{\lambda }}\underset{B_r(x_0)\cap \mathrm{\Omega }}{\int }|u(y){|}^{p}dy.$

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 $L^{\mathrm{\infty }}$, due to the Lebesgue differentiation theorem. When $\lambda >n$, the space contains only the 0 function. Note that this is a norm for $p\ge 1$. The seminorm of the Campanato space is given by

$([u{]}_{\lambda ,p}{)}^p=\underset{0<r<\mathrm{diam}(\mathrm{\Omega }),x_0\in \mathrm{\Omega }}{\sup }\frac{1}{r^{\lambda }}\underset{B_r(x_0)\cap \mathrm{\Omega }}{\int }|u(y)-u_{r,x_0}{|}^{p}dy$

where

$u_{r,x_0}=\frac{1}{|B_r(x_0)\cap \mathrm{\Omega }|}\underset{B_r(x_0)\cap \mathrm{\Omega }}{\int }u(y)dy.$

It is known that the Morrey spaces with $0\le \lambda <n$ are equivalent to the Campanato spaces with the same value of $\lambda$ when $\mathrm{\Omega }$ is a sufficiently regular domain, that is to say, when there is a constant A such that $|\mathrm{\Omega }\cap B_r(x_0)|>A{r}^n$ for every $x_0\in \mathrm{\Omega }$ and $r<\mathrm{diam}(\mathrm{\Omega })$. When $n=\lambda$, the Campanato space is the space of functions of bounded mean oscillation. When $n<\lambda \le n+p$, the Campanato space is the space of Hölder continuous functions $C^{\alpha }(\mathrm{\Omega })$ 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