Morrey–Campanato space explained

In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato)

L ^{λ,} (\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 , elements of the space L ^{λ,p} (\Omega) are Hölder continuous functions over the domain \Omega .

The seminorm of the Morrey spaces is given by

l ([u] _{λ,p} r ) ^{p} = \sup _{0} (\Omega), x _{0} \in \Omega} (1)/(r ^{λ}) \int _{B _{r(x} _{0)} \cap \Omega} | u(y) | ^{p} dy. 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 ^{infty} , due to the Lebesgue differentiation theorem. When λ > n , the space contains only the 0 function.

Note that this is a norm for

p \geq 1 .

The seminorm of the Campanato space is given by

l ([u] _{λ,p} r ) ^{p} = \sup _{0} (\Omega), x _{0} \in \Omega} (1)/(r ^{λ}) \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)\cap} \Omega|) \int _{B _{r(x} _{0)\cap} \Omega} u(y) dy. It is known that the Morrey spaces with 0 \leq λ < n are equivalent to the Campanato spaces with the same value of λ when \Omega is a sufficiently regular domain, that is to say, when there is a constant A such that |\Omega \cap B _{r(x} _{0)|} > A r ^{n} for every x _{0} \in \Omega and r < \operatorname{diam}(\Omega) .

When

n =λ , the Campanato space is the space of functions of bounded mean oscillation. When n < λ \leq n + p , the Campanato space is the space of Hölder continuous functions C ^{\alpha(\Omega)} with \alpha = (λ- n)/(p) . For λ > n + p , the space contains only constant functions