In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato) L λ , p ( Ω ) {\displaystyle L^{\lambda ,p}(\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 λ {\displaystyle \lambda } , elements of the space L λ , p ( Ω ) {\displaystyle L^{\lambda ,p}(\Omega )} are Hölder continuous functions over the domain Ω {\displaystyle \Omega } . 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 . {\displaystyle {\bigl (}[u]_{\lambda ,p}{\bigr )}^{p}=\sup _{0<r<\operatorname {diam} (\Omega ),x_{0}\in \Omega }{\frac {1}{r^{\lambda }}}\int _{B_{r}(x_{0})\cap \Omega }|u(y)|^{p}dy.}
When λ = 0 {\displaystyle \lambda =0} , the Morrey space is the same as the usual L p {\displaystyle L^{p}} space. When λ = n {\displaystyle \lambda =n} , the spatial dimension, the Morrey space is equivalent to L ∞ {\displaystyle L^{\infty }} , due to the Lebesgue differentiation theorem. When λ > n {\displaystyle \lambda >n} , the space contains only the 0 function. Note that this is a norm for p ≥ 1 {\displaystyle p\geq 1} . 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 {\displaystyle {\bigl (}[u]_{\lambda ,p}{\bigr )}^{p}=\sup _{0<r<\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}
where
u r , x 0 = 1 | B r ( x 0 ) ∩ Ω | ∫ B r ( x 0 ) ∩ Ω u ( y ) d y . {\displaystyle u_{r,x_{0}}={\frac {1}{|B_{r}(x_{0})\cap \Omega |}}\int _{B_{r}(x_{0})\cap \Omega }u(y)dy.}
It is known that the Morrey spaces with 0 ≤ λ < n {\displaystyle 0\leq \lambda <n} are equivalent to the Campanato spaces with the same value of λ {\displaystyle \lambda } when Ω {\displaystyle \Omega } 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 {\displaystyle |\Omega \cap B_{r}(x_{0})|>Ar^{n}} for every x 0 ∈ Ω {\displaystyle x_{0}\in \Omega } and r < diam ( Ω ) {\displaystyle r<\operatorname {diam} (\Omega )} . When n = λ {\displaystyle n=\lambda } , the Campanato space is the space of functions of bounded mean oscillation. When n < λ ≤ n + p {\displaystyle n<\lambda \leq n+p} , the Campanato space is the space of Hölder continuous functions C α ( Ω ) {\displaystyle C^{\alpha }(\Omega )} with α = λ − n p {\displaystyle \alpha ={\frac {\lambda -n}{p}}} . For λ > n + p {\displaystyle \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