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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set with diameter $d$, let $1 \\le p \u003c \\infty$ and let $\\lambda \\ge 0$.\nFor $x_0 \\in \\Omega$ and $r \u003e 0$ put $\\Omega(x_0, r) = \\Omega \\cap B(x_0, r)$, where $B(x_0, r)$ is the open ball of radius $r$ about $x_0$.\nFor $f \\in L^p(\\Omega)$ let $f_{x_0, r}$ be the mean value of $f$ over $\\Omega(x_0, r)$.\n\n1. The *Morrey space* $L^{p, \\lambda}(\\Omega)$ is the set of $f$ in the [[Lebesgue space|SASP14MA1Z]] $L^p(\\Omega)$ with\n$$\\sup_{x_0 \\in \\Omega,\\ 0 \u003c r \u003c d} \\ r^{-\\lambda} \\int_{\\Omega(x_0, r)} \\abs{f(x)}^p \\, dx \u003c \\infty.$$\n2. The *Campanato space* $\\mathcal{L}^{p, \\lambda}(\\Omega)$ is the set of $f \\in L^p(\\Omega)$ with\n$$[f]_{p, \\lambda}^p = \\sup_{x_0 \\in \\Omega,\\ 0 \u003c r \u003c d} \\ r^{-\\lambda} \\int_{\\Omega(x_0, r)} \\abs{f(x) - f_{x_0, r}}^p \\, dx \u003c \\infty.$$\n\nThe norms are $\\norm{f}_{L^p}$ plus the $p$-th root of the supremum.\n\n## Remarks\n\nExample: let $\\Omega$ be the unit ball of $\\R^n$, let $0 \u003c a \u003c n/p$ and let $f(x) = \\abs{x}^{-a}$.\nThe function $\\abs{x}^{-ap}$ is radially decreasing, so its integral over a ball of radius $r$ is largest for the ball centred at $0$.\nThat integral equals $c \\, r^{n - ap}$ for a constant $c$.\nHence $f \\in L^{p, \\lambda}(\\Omega)$ with $\\lambda = n - ap$.\n\nNon-example: if $\\lambda \u003e n$, then $f = 1$ is not in $L^{p, \\lambda}(\\Omega)$.\nIndeed $r^{-\\lambda} \\abs{\\Omega(x_0, r)}$ grows like $r^{n - \\lambda}$ as $r \\to 0$, and this tends to infinity.\nIn fact $L^{p, \\lambda}(\\Omega) = \\set{0}$ for $\\lambda \u003e n$, by the Lebesgue differentiation theorem.\n\nFurther facts:\n\n- $L^{p, 0}(\\Omega) = \\mathcal{L}^{p, 0}(\\Omega) = L^p(\\Omega)$, and $L^{p, n}(\\Omega) = L^\\infty(\\Omega)$.\n- $\\mathcal{L}^{p, n}(\\Omega)$ is a space of functions of [[bounded mean oscillation|QJYTXFT8Y4]].\n- If $\\Omega$ has Lipschitz boundary and $n \u003c \\lambda \\le n + p$, then $\\mathcal{L}^{p, \\lambda}(\\Omega)$ equals the space of [[Hölder continuous|012PBA0FMD]] functions with exponent $\\alpha = (\\lambda - n)/p$ (Campanato's theorem).\n- If $\\lambda \u003e n + p$ and $\\Omega$ is connected, then $\\mathcal{L}^{p, \\lambda}(\\Omega)$ contains only the constant functions.\n",
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