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  "body": "\n## Summary\n\nThe Hölder space $C^{k,\\alpha}$ consists of the functions whose derivatives up to order $k$ are bounded.\nIn addition, the derivatives of order $k$ must be [[Hölder continuous|012PBA0FMD]] with exponent $\\alpha$, where $0 \u003c \\alpha \\le 1$.\nWith the natural norm it is a [[Banach space|06GZ956657]].\nHölder spaces are the setting of the classical regularity theory of elliptic and parabolic equations, as in the [[Schauder estimates|HSEKZMAH1V]].\nThey fill the gaps between the spaces $C^k$ of $k$ times continuously differentiable functions.\n\n## Explanations\n\n### L1 human\n\nA Hölder continuous function cannot change faster than a fixed power of the distance: $\\abs{u(x) - u(y)} \\le C \\abs{x - y}^\\alpha$.\nThe exponent $\\alpha$ measures roughness: $\\alpha = 1$ means Lipschitz, and small $\\alpha$ allows sharp spikes like $\\sqrt{\\abs{x}}$.\nThe space $C^{k,\\alpha}$ asks this of the $k$-th derivatives.\nFor example, $x \\abs{x}$ is in $C^{1,1}$ on $(-1, 1)$, because its derivative $2 \\abs{x}$ is Lipschitz, but it is not twice differentiable at $0$.\n",
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