# Morrey–Campanato space

In mathematics, the **Morrey–Campanato spaces** (named after Charles B. Morrey, Jr. and Sergio Campanato) 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 , elements of the space are Hölder continuous functions over the domain .

The seminorm of the Morrey spaces is given by

When , the Morrey space is the same as the usual space. When , the spatial dimension, the Morrey space is equivalent to , due to the Lebesgue differentiation theorem. When , the space contains only the 0 function.

The seminorm of the Campanato space is given by

where

It is known that the Morrey spaces with are equivalent to the Campanato spaces with the same value of when is a sufficiently regular domain, that is to say, when there is a constant *A* such that for every and .

When , the Campanato space is the space of functions of bounded mean oscillation. When , the Campanato space is the space of Hölder continuous functions with . For , 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