Weighted Hölder spaces and their Dirichlet kernels are Banach

Let $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\beta }$ abbreviate the stated weighted norm for the fixed $k,\alpha ,\beta$. Sums and real multiples preserve local $C^{k,\alpha }$ regularity, and the termwise triangle and homogeneity inequalities preserve finiteness of this norm. If the norm is zero, the order-zero suprema on the core and annuli force the function to vanish everywhere, since these sets cover the exterior. Thus the space is a real normed vector space, as already justified in [F1]. Compactness makes $\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty },S}$ finite for every $v\in C^0(S)$; the absolute-value inequalities and pointwise separation give its norm axioms, including the specified empty-boundary case.

Let $(w_n)$ be Cauchy in $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\beta }$. Each coordinate derivative through order $k$ is uniformly Cauchy on every $K_i$ and every fixed $A_R$, since its fixed positive coefficient in the norm can be divided out. The same holds on any sufficiently small compact coordinate ball or boundary half-ball $B$: cover its part in the core by the finitely many $K_i$, cover the remaining part by finitely many compact end patches lying in annuli, and express all derivatives in the coordinates of $B$ using [F7]. The transition coefficients are bounded on the compact intersections by [F8], giving a bound ${\max }_{\mathrm{\mid }I\mathrm{\mid }\le k}{\sup }_B\mathrm{\mid }{\mathrm{\partial }}^I(w_n-w_m)\mathrm{\mid }\le C_B\mathrm{\parallel }{w}_n-w_m{\mathrm{\parallel }}_{\beta }$ with $C_B$ independent of $n,m$. By [F5] these derivative fields converge uniformly on $B$ to continuous fields $v_I$. The order-zero limits define a single function $w$ because they are limits of the same scalar values on overlaps.

On each closed coordinate segment strictly inside a chart, apply [F6] to ${\mathrm{\partial }}^I{w}_n$ and its derivative in a coordinate direction whenever $\mathrm{\mid }I\mathrm{\mid }<k$. The value at any fixed point converges by step 1.2, and the derivative converges uniformly there by the same step; the mixed-partial identity in [F6] identifies that derivative with the corresponding canonical multi-index derivative. Induction on the order gives $v_I={\mathrm{\partial }}^{I}w$ in the interior. Their continuous uniform limits on boundary half-balls supply the required extensions to the boundary. Taking limits in the coordinate-change identities shows that these derivatives agree on overlaps. For $k=0$ only the continuous order-zero limit is needed.

A Cauchy sequence is bounded in norm: fix a tail whose distances to one term are at most one and include the finitely many preceding norms. Passing this bound to the limit in each top-order Hölder quotient gives finite Hölder bounds for the extended derivatives of $w$ on each $K_i$ and each fixed annulus. These bounds also give local Hölder regularity in any sufficiently small compact convex coordinate half-ball $B$, even if the compact $K_i$ have not been chosen with interiors covering the core. Indeed their intersections with $B$, together with finitely many compact end patches inside annuli, give a finite closed cover $E_1,\dots ,E_N$ of $B$. In the coordinates of $B$, [F7] transfers the top-order Hölder bounds to each $E_i$: its smooth coefficients are bounded and Lipschitz on the relevant compact overlaps, while lower-order derivatives of $w$ are Lipschitz on $B$ by their bounded first derivatives and [F7]. For two points of $B$, restrict this cover to their joining segment. By connectedness in [F8], the sets containing the endpoints can be joined by a chain of intersecting cover sets, using at most $N$ sets; otherwise the finite unions of the two groups would disconnect that segment. Choose intersection points on the segment and telescope their value differences. Each difference is bounded by $H$ times the segment length to the power $\alpha$, where $H$ is the maximum of the finitely many bounds on the $E_i$, so the total is at most $NH$ times that power. Thus $w$ is locally $C^{k,\alpha }$ up to the boundary in the specified convention.

Given $\epsilon >0$, choose $J$ so that $\mathrm{\parallel }{w}_n-w_m{\mathrm{\parallel }}_{\beta }\le \epsilon$ for $n,m\ge J$. Fix $n\ge J$ and a radius $R$. Each derivative supremum and Hölder seminorm of $w_n-w$ is at most the lower limit of the corresponding quantity for $w_n-w_m$: every pointwise value or two-point quotient converges by steps 1.2 and 2.1, and one then takes the supremum. Finite nonnegative sums and maxima preserve this lower-limit inequality. Consequently $N_K(w_n-w)$ plus the entire weighted annular expression at this fixed $R$ is at most $\epsilon$. Taking the supremum over $R$ yields $\mathrm{\parallel }{w}_n-w{\mathrm{\parallel }}_{\beta }\le \epsilon$. Step 3.1 supplies the local regularity, and $\mathrm{\parallel }w{\mathrm{\parallel }}_{\beta }\le \mathrm{\parallel }{w}_n{\mathrm{\parallel }}_{\beta }+\epsilon <\mathrm{\infty }$ supplies membership in the weighted space. This proves completeness and hence the Banach property.

For every $w$ in the space, its boundary extension is continuous and $\mathrm{\parallel }w{\mathrm{\mid }}_S{\mathrm{\parallel }}_{\mathrm{\infty },S}\le N_K(w)\le \mathrm{\parallel }w{\mathrm{\parallel }}_{\beta }$, since the $K_i$ cover $S$ and include the order-zero suprema. Thus restriction is a continuous linear map into the normed space $C^0(S)$ defined in [F4]. Its zero-trace kernel is a linear subspace. If $v$ is outside that kernel, some $p\in S$ has $v(p)\mathrm{\ne }0$; every $u$ with $\mathrm{\parallel }u-v{\mathrm{\parallel }}_{\beta }<\mathrm{\mid }v(p)\mathrm{\mid }\mathrm{/}2$ still has $u(p)\mathrm{\ne }0$. The complement is therefore open and the kernel is closed, including the case $S=\mathrm{\varnothing }$, when the kernel is the whole space. Applying [F9] to the Banach space established in step 4.1 makes this kernel complete for its restricted norm. Taking $k=4$ gives $X_{\beta }$ and taking $k=2$ with weight $\beta -2$ gives $Y_{\beta }$ as real Banach spaces.