Morrey–Campanato space

In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato) $L^{\lambda, p}(\Omega)$ 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 $\lambda$ , elements of the space $L^{\lambda,p}(\Omega)$ are Hölder continuous functions over the domain $\Omega$ .

The seminorm of the Morrey spaces is given by

\left[u\right]_{\lambda,p}^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 $\lambda = 0$ , the Morrey space is the same as the usual $L^p$ space. When $\lambda = n$ , the spatial dimension, the Morrey space is equivalent to $L^\infty$ , due to the Lebesgue differentiation theorem. When $\lambda > n$ , the space contains only the 0 function.

The seminorm of the Campanato space is given by

\left[u\right]_{\lambda,p}^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} = \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 $0 \leq \lambda < n$ are equivalent to the Campanato spaces with the same value of $\lambda$ 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 $n=\lambda$ , the Campanato space is the space of functions of bounded mean oscillation. When $n < \lambda \leq n+p$ , the Campanato space is the space of Hölder continuous functions $C^\alpha(\Omega)$ with $\alpha = \frac{\lambda - n}{p}$ . For $\lambda > n+p$ , 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 105, Princeton University Press, ISBN 978-0-691-08330-8

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