Morrey–Campanato space

Remarks

Example: let $\Omega$ be the unit ball of $\R^n$, let $0 < a < n/p$ and let $f(x) = \abs{x}^{-a}$. The function $\abs{x}^{-ap}$ is radially decreasing, so its integral over a ball of radius $r$ is largest for the ball centred at $0$. That integral equals $c \, r^{n - ap}$ for a constant $c$. Hence $f \in L^{p, \lambda}(\Omega)$ with $\lambda = n - ap$.

Non-example: if $\lambda > n$, then $f = 1$ is not in $L^{p, \lambda}(\Omega)$. Indeed $r^{-\lambda} \abs{\Omega(x_0, r)}$ grows like $r^{n - \lambda}$ as $r \to 0$, and this tends to infinity. In fact $L^{p, \lambda}(\Omega) = \set{0}$ for $\lambda > n$, by the Lebesgue differentiation theorem.

Further facts:

• $L^{p, 0}(\Omega) = \mathcal{L}^{p, 0}(\Omega) = L^p(\Omega)$, and $L^{p, n}(\Omega) = L^\infty(\Omega)$.
• $\mathcal{L}^{p, n}(\Omega)$ is a space of functions of bounded mean oscillation.
• If $\Omega$ has Lipschitz boundary and $n < \lambda \le n + p$, then $\mathcal{L}^{p, \lambda}(\Omega)$ equals the space of Hölder continuous functions with exponent $\alpha = (\lambda - n)/p$ (Campanato's theorem).
• If $\lambda > n + p$ and $\Omega$ is connected, then $\mathcal{L}^{p, \lambda}(\Omega)$ contains only the constant functions.