Weighted Hölder norms on a three-dimensional exterior

PDF p. 13: §3.1, opening norm display

Let ${\mathrm{\Sigma }}_1$ be a smooth three-dimensional exterior with compact smooth boundary and a fixed end chart $x$ identifying the end with $\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>r_0\}$ for some $r_0>0$. Write $r=\mathrm{\mid }x\mathrm{\mid }$, $K={\mathrm{\Sigma }}_1\setminus \{r>2r_0\}$ and $A_R=\{R<r<2R\}$ for $R\ge r_0$. Fix the finite interior and boundary chart subsets covering $K$ and the compact-core Hölder norm $N_K=\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{C^{k,\alpha }(K)}$ specified by the exterior-data definition: it is the sum of the coordinate derivative suprema through order $k$ and the order-$k$ Hölder seminorms on those subsets.

Let $k\ge 0$ be an integer, $0<\alpha <1$ and $\beta \in \mathbb{R}$. For distinct points of a set $E$ in Euclidean coordinates, put $[v{]}_{\alpha ;E}={\sup }_{x\mathrm{\ne }y\in E}\mathrm{\mid }v(x)-v(y)\mathrm{\mid }\mathrm{/}\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }$. A multi-index $I=(i_1,i_2,i_3)$ has nonnegative integer entries, order $\mathrm{\mid }I\mathrm{\mid }=i_1+i_2+i_3$, and ${\mathrm{\partial }}_x^I={\mathrm{\partial }}_{x^1}^{i_1}{\mathrm{\partial }}_{x^2}^{i_2}{\mathrm{\partial }}_{x^3}^{i_3}$. For a real scalar function $w$ that is locally $C^{k,\alpha }$ up to the boundary, define the extended-valued quantity

$\mathrm{\parallel }w{\mathrm{\parallel }}_{C_{\beta }^{k,\alpha }}=N_K(w)+{\sup }_{R\ge r_0}{R}^{-\beta }({\sum }_{j=0}^k{R}^j{\max }_{\mathrm{\mid }I\mathrm{\mid }=j}{\sup }_{A_R}\mathrm{\mid }{\mathrm{\partial }}_x^{I}w\mathrm{\mid }+R^{k+\alpha }{\max }_{\mathrm{\mid }I\mathrm{\mid }=k}[{\mathrm{\partial }}_x^{I}w{]}_{\alpha ;A_R})\mathrm{.}$

Here locally $C^{k,\alpha }$ up to the boundary means that, in each interior or boundary chart, derivatives through order $k$ are continuous up to the boundary and the order-$k$ derivatives have finite Hölder seminorm on each compact chart subset. The space $C_{\beta }^{k,\alpha }({\mathrm{\Sigma }}_1)$ consists of these functions for which the displayed quantity is finite. For a finite array of components, apply the definition componentwise and sum the component norms; in particular, coordinate components of a tensor on the end can be measured this way, with fixed local components on the core.

Weighted exterior elliptic domains and admissible weights

Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains

The exterior-data definition supplies the fixed chart subsets and the compact-core norm, and the local Hölder definition supplies the coordinate derivative and seminorm conventions. Each displayed term is nonnegative and homogeneous, and the triangle inequality follows from the absolute-value triangle inequality before taking suprema. The core and annuli cover ${\mathrm{\Sigma }}_1$, so a zero quantity forces $w=0$ everywhere. Thus the finite-valued class is a vector space and the displayed quantity is a norm on it. The fixed choices are part of the definition; no equivalence of different choices, completeness, or elliptic mapping property is asserted here.

The article's notation ${\mathrm{\partial }}_x^{j}w$ is made explicit by the maximum over coordinate multi-indices of order $j$. The article reference records attribution, while the definition and its justification use the stated local suppliers.