Morrey–Campanato space

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

[math]\displaystyle{ \bigl([u]_{\lambda,p}\bigr)^p = \sup_{0 \lt r\lt \operatorname{diam} (\Omega), x_0 \in \Omega} \frac{1}{r^\lambda} \int_{B_r(x_0) \cap \Omega} | u(y) |^p dy. }[/math]

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

Note that this is a norm for [math]\displaystyle{ p \geq 1 }[/math].

The seminorm of the Campanato space is given by

[math]\displaystyle{ \bigl([u]_{\lambda,p}\bigr)^p = \sup_{0 \lt r\lt \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 }[/math]

where

[math]\displaystyle{ u_{r,x_0} = \frac{1}{|B_r(x_0)\cap \Omega|} \int_{B_r(x_0)\cap \Omega} u(y) dy. }[/math]

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

When [math]\displaystyle{ n=\lambda }[/math], the Campanato space is the space of functions of bounded mean oscillation. When [math]\displaystyle{ n \lt \lambda \leq n+p }[/math], the Campanato space is the space of Hölder continuous functions [math]\displaystyle{ C^\alpha(\Omega) }[/math] with [math]\displaystyle{ \alpha = \frac{\lambda - n}{p} }[/math]. For [math]\displaystyle{ \lambda \gt n+p }[/math], 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, 105, Princeton University Press, ISBN 978-0-691-08330-8