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In mathematics, the Morrey–Campanato spaces L λ , p {\displaystyle L^{\lambda ,p}} 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 {\displaystyle L^{\lambda ,p}} are Hölder continuous functions over the domain Ω {\displaystyle \Omega }.
The seminorm of the Morrey spaces is given by
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}.