Morrey–Campanato space

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 } . 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 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 pgeq 1} .