In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato) L λ , p ( Ω ) 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 λ , elements of the space L λ , p ( Ω ) are Hölder continuous functions over the domain Ω .
The seminorm of the Morrey spaces is given by
[ u ] λ , p p = sup 0 < r < diam ( Ω ) , x 0 ∈ Ω 1 r λ ∫ B r ( x 0 ) ∩ Ω | u ( y ) | p d y . When λ = 0 , the Morrey space is the same as the usual L p space. When λ = n , the spatial dimension, the Morrey space is equivalent to L ∞ , due to the Lebesgue differentiation theorem. When λ > n , the space contains only the 0 function.
The seminorm of the Campanato space is given by
[ u ] λ , p p = sup 0 < r < diam ( Ω ) , x 0 ∈ Ω 1 r λ ∫ B r ( x 0 ) ∩ Ω | u ( y ) − u r , x 0 | p d y where
u r , x 0 = 1 | B r ( x 0 ) ∩ Ω | ∫ B r ( x 0 ) ∩ Ω u ( y ) d y . It is known that the Morrey spaces with 0 ≤ λ < n are equivalent to the Campanato spaces with the same value of λ when Ω is a sufficiently regular domain, that is to say, when there is a constant A such that | Ω ∩ B r ( x 0 ) | > A r n for every x 0 ∈ Ω and r < diam ( Ω ) .
When n = λ , the Campanato space is the space of functions of bounded mean oscillation. When n < λ ≤ n + p , the Campanato space is the space of Hölder continuous functions C α ( Ω ) with α = λ − n p . For λ > n + p , the space contains only constant functions.
