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.

Note that this is a norm for .

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

Function spaces