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Annular scaling and fixed interior covers

Statement

Let $n \geq 1$ and $k \geq 0$ be integers and $0 < α < 1$ . In $R^{n}$ set $A = {y : 1 / 2 < ∣ y ∣ < 4}$ , $A^{'} = {y : 1 < ∣ y ∣ < 2}$ and $K = \overline{A^{'}} = {y : 1 \leq ∣ y ∣ \leq 2}$ . For $R \geq 1$ and $u \in C^{k} (R A; R)$ , where $R S = {R y : y \in S}$ , define $v (y) = u (R y)$ and $w (y) = R^{- 1} u (R y)$ on $A$ . Multi-index derivatives are taken in their canonical order. No boundary extension of $u$ to $\partial (R A)$ is assumed.

For every multi-index $β$ with $∣ β ∣ \leq k$ and every nonempty open $S \subseteq A$ , the exact scaling identities are $D_{y}^{β} v (y) = R^{∣ β ∣} (D_{x}^{β} u) (R y), D_{y}^{β} w (y) = R^{∣ β ∣ - 1} (D_{x}^{β} u) (R y),$ $[D_{y}^{β} v]_{α; S} = R^{∣ β ∣ + α} [D_{x}^{β} u]_{α; R S}, [D_{y}^{β} w]_{α; S} = R^{∣ β ∣ - 1 + α} [D_{x}^{β} u]_{α; R S} .$ Here $[f]_{α; S} = \sup_{y, z \in S, y \neq z} ∣ f (y) - f (z) ∣ / ∣ y - z ∣^{α}$ ; seminorms may equal $+ \infty$ .

For $h \in C^{k} (S; R)$ use the ordinary norm convention $∥ h ∥_{C^{k, α} (S)} = \sum_{j = 0}^{k} \max_{∣ β ∣ = j} \sup_{S} ∣ D^{β} h ∣ + \sum_{∣ β ∣ = k} [D^{β} h]_{α; S} .$ Consequently the annular rescaling, with the same convention for any such $S$ , satisfies $∥ v ∥_{C^{k, α} (S)} = \sum_{j = 0}^{k} R^{j} \max_{∣ β ∣ = j} \sup_{R S} ∣ D_{x}^{β} u ∣ + R^{k + α} \sum_{∣ β ∣ = k} [D_{x}^{β} u]_{α; R S}, ∥ w ∥_{C^{k, α} (S)} = R^{- 1} ∥ v ∥_{C^{k, α} (S)} .$

There is a fixed finite family of centers $p_{1}, \dots, p_{N} \in K$ , depending only on $n$ , such that the balls $B_{r} (p_{i})$ cover $K$ , where $r = 1 / 16$ , and every enlarged ball $B_{4 r} (p_{i})$ lies in $A$ . In particular the smaller concentric balls $B_{2 r} (p_{i})$ also cover $K$ . For every $h \in C^{k} (A; R)$ with $M : = \max_{i} ∥ h ∥_{C^{k, α} (B_{2 r} (p_{i}))} < \infty$ , $∥ h ∥_{C^{k, α} (A^{'})} \leq C_{n, k, α} M, C_{n, k, α} : = (k + 1) + N_{n, k} \max {1, 2 r^{- α}}, N_{n, k} : = # {β \in N^{n} : ∣ β ∣ = k} .$ Thus ball estimates on the smaller balls, obtained using these enlarged balls, imply a full annular estimate, including the global Hölder seminorm, with a constant independent of $R$ .

The same conclusion applies to families indexed by $R$ and any further parameter whenever the local estimates have common constants on the specified parameter set; uniformity at a parameter endpoint is not inferred from separate pointwise estimates. In the three-dimensional exterior-chart setting $E = {x : ∣ x ∣ > R_{0}}$ with $R_{0} > 0$ , if $u$ is given only on $E$ , require $R \geq 1$ and $R / 2 > R_{0}$ so that $R A \subset E$ . Uniform end bounds retain their common-chart, common-radius and common-constant meaning.

Facts & Assumptions

[F1]

The class $C^{k}$ requires continuous iterated coordinate derivatives through order $k$ ; $D^{β}$ uses the fixed canonical order. Cᵏ maps and multi-index derivative notation in Euclidean space

[F2]

Ordinary Hölder seminorms and norms use Euclidean distances, derivative suprema and top-order difference quotients; scaled norms attach the indicated powers of radius. Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains

[F3]

Exterior differentiated bounds and parameter uniformity require a specified Cartesian chart and common radii and constants on the specified parameter set. Differentiated asymptotic bounds and parameter uniformity

[F4]

A closed bounded subset of $R^{n}$ , $n \geq 1$ , is compact in its Euclidean metric: every open cover has a finite subcover. Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line

Proof

Given: The hypotheses and norm conventions in the Statement.

Goal: Prove the exact scaling identities and the full fixed-cover annular estimate.

1.1givenF1algebra

If $k \geq 1$ , for any coordinate unit vector $e_{i}$ and any iterated coordinate derivative $q$ of $u$ of order less than $k$ , the difference quotient after dilation is $\frac{q (R (y + t e_{i})) - q (R y)}{t} = R \frac{q (R y + R t e_{i}) - q (R y)}{R t} ⟶ R (\partial_{i} q) (R y) (t \to 0) .$ For $y \in A$ the points involved lie in $R A$ for sufficiently small $∣ t ∣$ , since $R A$ is open. Apply this calculation successively to every coordinate-derivative word through order $k$ ; all required derivatives exist by [F1]. In particular, the canonical order gives the asserted identities for $D^{β} v$ , and multiplication by the constant $R^{- 1}$ gives those for $D^{β} w$ . Each resulting derivative is continuous, since its derivative of $u$ is continuous and $∣ R y - R z ∣ = R ∣ y - z ∣$ makes dilation continuous. Thus $v, w \in C^{k} (A)$ . Order zero is the defining identity, including when $k = 0$ .

1.2givenF4constructalgebra

The reverse triangle inequality $∣ ∣ y ∣ - ∣ z ∣ ∣ \leq ∣ y - z ∣$ shows directly that the complement of $K$ is open: at a point with $∣ y ∣ < 1$ take a ball of radius $(1 - ∣ y ∣) / 2$ , and at one with $∣ y ∣ > 2$ take radius $(∣ y ∣ - 2) / 2$ . Also $K \subset B_{3} (0)$ , so [F4] makes $K$ compact. The balls $B_{r} (p)$ , for $p \in K$ and $r = 1 / 16$ , form an open cover. Take a finite subcover with centers $p_{1}, \dots, p_{N}$ . For $y \in B_{4 r} (p_{i})$ , the same triangle inequalities give $3 / 4 < ∣ y ∣ < 9 / 4$ ; thus $B_{4 r} (p_{i}) \subset A$ . All these balls are fixed in $y$ coordinates, independently of $R$ , $u$ or any further parameter.

2.1givenF2step 1.1algebra

Dilation maps $S$ bijectively onto $R S$ , so the derivative identities give the corresponding supremum factors $R^{∣ β ∣}$ and $R^{∣ β ∣ - 1}$ . For distinct $y, z \in S$ , substitute $x = R y$ , $x^{'} = R z$ ; then $∣ x - x^{'} ∣ = R ∣ y - z ∣$ . Each difference quotient for $D_{y}^{β} v$ is therefore $R^{∣ β ∣ + α}$ times the corresponding quotient for $D_{x}^{β} u$ , and the factor for $w$ is $R^{∣ β ∣ - 1 + α}$ . The pair correspondence is bijective, so taking suprema proves both seminorm equalities, also if they are infinite. Summation in the stated norm convention proves the norm identities.

2.2givenF1F2step 1.2algebra

Put $M = \max_{i} ∥ h ∥_{C^{k, α} (B_{2 r} (p_{i}))}$ . Every $y \in A^{'}$ belongs to some $B_{r} (p_{i}) \subset B_{2 r} (p_{i})$ , so each derivative supremum in the norm on $A^{'}$ is at most $M$ . For distinct $y, z \in A^{'}$ with $∣ y - z ∣ < r$ , choose $i$ with $∣ y - p_{i} ∣ < r$ . Then $∣ z - p_{i} ∣ < 2 r$ , and the $α$ -Hölder quotient of each $D^{β} h$ with $∣ β ∣ = k$ is bounded by its seminorm on $B_{2 r} (p_{i})$ , hence by $M$ . Thus the fixed margin $r$ controls every nearby pair.

3.1givenF2step 2.2algebra

For the remaining pairs, $∣ y - z ∣ \geq r$ , each numerator $∣ D^{β} h (y) - D^{β} h (z) ∣$ is at most $2 M$ , so its Hölder quotient is at most $2 r^{- α} M$ . Together with step 2.2 this bounds each top-order seminorm by $\max {1, 2 r^{- α}} M$ . There are $N_{n, k}$ such multi-indices, and $k + 1$ derivative-supremum terms. Summing gives exactly $∥ h ∥_{C^{k, α} (A^{'})} \leq C_{n, k, α} M$ as stated.

4.1givenF3step 2.1step 1.2step 3.1algebra∎

Apply the fixed-cover estimate to $h = v$ or $h = w$ and use the exact rescaling identities of step 2.1. If the local estimates are bounded by a common local constant times a given right-hand side, their finite maximum has that same bound, and step 3.1 multiplies it only by $C_{n, k, α}$ . Neither the balls nor this factor depend on $R$ or a further parameter. For an exterior chart in dimension three, $y \in A$ gives $∣ R y ∣ > R / 2 > R_{0}$ , which verifies the domain condition; [F3] specifies precisely when end estimates and their constants are uniform. This proves all the claimed scaling and uniform annular conclusions.

Depends on

Used by · 1

Uniform annular Schauder bootstrap for spacelike graph equations

Dependency tree · 2 levels

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