Hölder spaces $C^{k,\alpha }$, closure and interior scaled norms, and $C^{k,\alpha }$ domains

Let $n\ge 1$ be an integer, let $k\ge 0$ be an integer, let $0<\alpha <1$ and let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. Fix an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ and a function $u\in C^k(\mathrm{\Omega };\mathbb{K})$, where $C^k$ and the canonical-order derivatives $D^{\beta }u$ are those of $C^k$ maps and multi-index derivative notation in Euclidean space. Put $[u{]}_{k,\alpha ;\mathrm{\Omega }}:={\sum }_{\mathrm{\mid }\beta \mathrm{\mid }=k}{\sup }_{\begin{array}{c}x,y\in \mathrm{\Omega }\\ x\mathrm{\ne }y\end{array}}\frac{\mathrm{\mid }{D}^{\beta }u(x)-D^{\beta }u(y)\mathrm{\mid }}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }},\mathrm{\parallel }u{\mathrm{\parallel }}_{C^{k,\alpha }(\mathrm{\Omega })}:={\sum }_{j=0}^k{\sup }_{x\in \mathrm{\Omega }}{\max }_{\mathrm{\mid }\beta \mathrm{\mid }=j}\mathrm{\mid }{D}^{\beta }u(x)\mathrm{\mid }+[u{]}_{k,\alpha ;\mathrm{\Omega }},$ where the sums and maxima run over the finitely many multi-indices of the stated order. The quantity $[u{]}_{k,\alpha ;\mathrm{\Omega }}$ is the $k$-th order $\alpha$-Hölder seminorm of $u$ and the quantity $\mathrm{\parallel }u{\mathrm{\parallel }}_{C^{k,\alpha }(\mathrm{\Omega })}$ the $C^{k,\alpha }$ norm of $u$; both are taken in $[0,+\mathrm{\infty }]$, so the "norm" may be $+\mathrm{\infty }$ and only the class below carries a genuine normed-space structure.

The local Hölder class $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k,\alpha }(\mathrm{\Omega };\mathbb{K})$ consists of the $u\in C^k(\mathrm{\Omega };\mathbb{K})$ with $[u{]}_{k,\alpha ;{\mathrm{\Omega }}^{\mathrm{\prime }}}<+\mathrm{\infty }$ for every ${\mathrm{\Omega }}^{\mathrm{\prime }}⋐\mathrm{\Omega }$. The bounded class $C^{k,\alpha }(\mathrm{\Omega };\mathbb{K})=C_b^{k,\alpha }(\mathrm{\Omega };\mathbb{K})$ consists of the $u\in C^k(\mathrm{\Omega };\mathbb{K})$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{C^{k,\alpha }(\mathrm{\Omega })}<+\mathrm{\infty }$. Thus $C^{k,\alpha }(\mathrm{\Omega };\mathbb{K})\subseteq C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k,\alpha }(\mathrm{\Omega };\mathbb{K})$, and the inclusion may be strict: on $\mathrm{\Omega }={\mathbb{R}}^n$ the function $u(x)=\sin (\mathrm{\mid }x{\mathrm{\mid }}^2)$ lies in the local class (it is $C^1$, hence locally $\alpha$-Hölder on every compact set), while $[u{]}_{0,\alpha ;{\mathbb{R}}^n}=+\mathrm{\infty }$, because at the points $x_j=\sqrt{2\pi j}{e}_1$ and $y_j=x_j+(2\sqrt{2\pi j}{)}^{-1}{e}_1$ the difference quotients are $\asymp (2\sqrt{2\pi j}{)}^{\alpha }\to \mathrm{\infty }$. On the bounded class the displayed formula is a norm, and the assignment $u\mapsto \mathrm{\parallel }u{\mathrm{\parallel }}_{C^{k,\alpha }(\mathrm{\Omega })}$ is that norm.

Scaled interior norm. For a ball $B_R(x_0)\subseteq \mathrm{\Omega }$ of radius $R>0$ and centre $x_0$, write $[v{]}_{0,\alpha ;B}:=\sup \{\mathrm{\mid }v(x)-v(y)\mathrm{\mid }\mathrm{/}\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }:x,y\in B,x\mathrm{\ne }y\}$ for the plain Hölder seminorm on a set $B$, and put $\mathrm{\parallel }u{\mathrm{\parallel }}_{k,\alpha ;B_R(x_0)}^{\ast }:={\sum }_{j=0}^k{R}^j{\max }_{\mathrm{\mid }\beta \mathrm{\mid }=j}{\sup }_{B_R(x_0)}\mathrm{\mid }{D}^{\beta }u\mathrm{\mid }+R^{k+\alpha }{\max }_{\mathrm{\mid }\beta \mathrm{\mid }=k}[D^{\beta }u{]}_{0,\alpha ;B_R(x_0)}\mathrm{.}$ On every ball, $C^{2,\alpha }(B_R(x_0))$ therefore agrees with the finite-scaled-norm class of that dependency. This is the scaled interior norm of $u$ on the ball; for $k=2$ it is exactly the scaled quantity of Local Hölder and scaled C-two-alpha norms on balls, and it takes values in $[0,+\mathrm{\infty }]$ as well. It is read off the open ball alone.

Bounded $C^{k,\alpha }$ domains. A bounded $C^{k,\alpha }$ domain in ${\mathbb{R}}^n$ is a bounded nonempty open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ with the following local graph property: for every $x\in \mathrm{\partial }\mathrm{\Omega }$ there are an open neighbourhood $W$ of $x$, a rigid motion $R(p)=Qp+b$ with orthogonal $Q$ and $b\in {\mathbb{R}}^n$, an open ball $B\subseteq {\mathbb{R}}^{n-1}$ and a function $\phi \in C^{k,\alpha }(B;\mathbb{R})$ such that, after shrinking $W$ so that $R(W)\subseteq B\times \mathbb{R}$, $R(\mathrm{\Omega }\cap W)=R(W)\cap \{y=(y^{\mathrm{\prime }},y_n)\in B\times \mathbb{R}:y_n<\phi (y^{\mathrm{\prime }})\}\mathrm{.}$ This generalises the integer-order notion of Bounded C^k domains and boundary charts to the Hölder scale, with the same one-sided graph convention; the regularity $\phi \in C^{k,\alpha }(B)$ is the only strengthening, connectedness is not required, and no boundary seminorm is attached to the interior norms above. In dimension $n=1$ the corresponding sets are finite disjoint unions of bounded open intervals.

The boundary-extension class. Let $\mathrm{\Omega }$ be bounded and let $u\in C_b^{k,\alpha }(\mathrm{\Omega };\mathbb{K})$. Say that $u$ lies in $C^{k,\alpha }(\bar{\mathrm{\Omega }};\mathbb{K})$ when each derivative field $D^{\beta }u$ with $\mathrm{\mid }\beta \mathrm{\mid }\le k$ extends continuously to $\bar{\mathrm{\Omega }}$. The extension of each field is then unique, since $\mathrm{\Omega }$ is dense in $\bar{\mathrm{\Omega }}$, and the sup and Hölder quantities formed with the extended fields and suprema over $\bar{\mathrm{\Omega }}$ agree with those displayed above, formed over $\mathrm{\Omega }$: a supremum over the dense subset $\mathrm{\Omega }$ already computes the supremum of the extension, and for the seminorm the inequality $[\tilde{v}{]}_{0,\alpha ;\bar{\mathrm{\Omega }}}\le [v{]}_{0,\alpha ;\mathrm{\Omega }}$ follows by approximating a pair in $\bar{\mathrm{\Omega }}$ by pairs in $\mathrm{\Omega }$, so the two seminorms are equal. We equip $C^{k,\alpha }(\bar{\mathrm{\Omega }};\mathbb{K})$ with this common norm. This is the Hölder-scale analogue of the interior-up-to-boundary convention for integer order $C^m(\bar{\mathrm{\Omega }})$ fixed in Bounded C1 domains and their outward normals.

• Scaling. If $R>0$, $x_0\in {\mathbb{R}}^n$ and $v(z):=u(x_0+Rz)$ on $B_1(0)$, then $D^{\beta }v(z)=R^{\mathrm{\mid }\beta \mathrm{\mid }}{D}^{\beta }u(x_0+Rz)$, and consequently $\mathrm{\parallel }v{\mathrm{\parallel }}_{k,\alpha ;B_1(0)}^{\ast }=\mathrm{\parallel }u{\mathrm{\parallel }}_{k,\alpha ;B_R(x_0)}^{\ast }$. The powers $R^j$ and $R^{k+\alpha }$ are exactly what makes this identity hold; the same computation with $k=2$ is the one recorded in Local Hölder and scaled C-two-alpha norms on balls.
• Boundary extension. For $k=0$, every element of $C_b^{0,\alpha }(\mathrm{\Omega })$ is uniformly continuous and extends uniquely to $\bar{\mathrm{\Omega }}$: for any boundary point choose an interior sequence converging to it, use the Hölder bound to make its values Cauchy, and compare two sequences by the same bound. For $k\ge 1$, boundedness of the lower-order fields need not give their boundary limits on an arbitrary open set. For example, let $\mathrm{\Omega }=((0,1)\times (0,2))\cup ((1,2)\times (0,2))$ and let $u$ equal $0$ on the first component and $1$ on the second. All positive-order derivatives vanish, so $\mathrm{\parallel }u{\mathrm{\parallel }}_{C^{k,\alpha }(\mathrm{\Omega })}=1$ for $k\ge 1$, but $u$ has no limit at $(1,1)$. This set fails the one-sided boundary graph condition at the removed interface. The definitions assert no trace theorem, completeness or compactness.
• Local versus bounded. The local class tests compactly contained subsets; the bounded class additionally controls all derivative suprema and the global top-order seminorm. On arbitrary bounded open sets, lower-order suprema can also fail: on the disjoint intervals $I_j=(2^{-j},2^{-j}+2^{-j-2})$, the function equal to $j$ on $I_j$ is locally smooth with every positive-order derivative zero, but is unbounded. Thus boundedness of the domain alone does not identify the two classes.
• Choice. The definition itself uses no choice principle; later completeness, approximation and embedding statements on this page state their own Countable Choice hypotheses.