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Endpoint multiplier bounds extended to negative spectral orders

Statement

Assume the Axiom of Choice. Let $Ω = R^{n} ∖ Θ$ , $n \geq 2$ , where $Θ$ is compact with smooth boundary and $Ω$ is open. Let $A = I - Δ_{D} \geq I$ be the shifted Dirichlet Laplacian and let $H_{D}^{s}$ be the associated Hilbert scale, with pivot $L^{2} (Ω)$ and norm $∥ A^{s / 2} u ∥_{2}$ on its spectral core. For every complex-valued $a \in W^{2, \infty} (Ω)$ and every $- 2 \leq s \leq 2$ , multiplication by $a$ extends to a bounded complex-linear operator on $H_{D}^{s}$ , with norm at most $C_{s, Ω} ∥ a ∥_{W^{2, \infty}}$ . Here $W^{k, p} (Ω; C)$ is the space of almost-everywhere classes with weak derivatives through order $k$ in $L^{p}$ , and $H^{k} (Ω) = W^{k, 2} (Ω; C)$ for $k = 0, 1, 2$ , with $∥ u ∥_{H^{k}}^{2} = \sum_{∣ α ∣ \leq k} ∥ D^{α} u ∥_{2}^{2}$ . In particular $W^{2, \infty}$ has norm $∥ a ∥_{W^{2, \infty}} = \max_{∣ α ∣ \leq 2} ∥ D^{α} a ∥_{\infty}$ .

For $s > 0$ , the negative-order extension is defined by $⟨ a f, u ⟩_{- s, s} = ⟨ f, \overline{a} u ⟩_{- s, s}$ for $f \in H_{D}^{- s}$ and $u \in H_{D}^{s}$ , where the pivot pairing is linear in its first argument. It agrees with ordinary multiplication on $L^{2}$ . Thus the result applies in particular to smooth cutoffs with bounded derivatives through order two and to any potentials belonging to $W^{2, \infty} (Ω)$ . Complex conjugation is an isometry on each of these scale spaces.

Facts & Assumptions

[F1]

The exterior Dirichlet realization gives $D (A) = H^{2} (Ω) \cap H_{0}^{1} (Ω)$ with equivalent graph and $H^{2}$ norms, and $D (A^{1 / 2}) = H_{0}^{1} (Ω)$ with the energy norm Exterior Dirichlet realization, H² domain and form domain. Here $H_{0}^{1} (Ω)$ is the $H^{1}$ closure of $C_{c}^{\infty} (Ω)$ .

[F2]

The positive spectral-scale contract identifies positive levels with domains of powers, identifies negative levels with pivot antiduals, gives bounded-operator interpolation on the scale, and makes conjugation an isometry for the real Dirichlet form Positive spectral Hilbert scale: duality and interpolation.

[F3]

The weak Leibniz rule gives the product formula through any finite order when the multiplier is smooth and the indicated weak derivatives exist Weak Leibniz rule with a smooth factor.

[F4]

The Axiom of Choice is assumed for the spectral and realization suppliers and implies the Countable Choice hypotheses of the Sobolev, product, mollification and extension suppliers. It also permits the approximating sequence used below The Axiom of Choice.

[F5]

Integer-order Sobolev spaces are spaces of almost-everywhere classes characterized by the weak test identities; their norms are derivative sums for finite $p$ and the maximum of essential derivative bounds for $p = \infty$ Integer-order Sobolev spaces and their norms. We use complex scalars, $H^{k} = W^{k, 2}$ and the norm convention in the Statement.

[F6]

For an open set $G$ , $b \in W^{k, p} (G; C)$ , $1 \leq p \leq \infty$ , and a nonnegative smooth unit-mass kernel $ρ$ supported in the unit ball, the interior mollification $b_{ε} = ρ_{ε} * E_{0} b$ is smooth on $G_{ε} = {x \in G : dist (x, R^{n} ∖ G) > ε}$ and $D^{α} b_{ε} = ρ_{ε} * E_{0} (D^{α} b)$ there for $∣ α ∣ \leq k$ Interior mollification commutes with weak derivatives. Here $ρ_{ε} (x) = ε^{- n} ρ (x / ε)$ , $E_{0}$ denotes extension by zero, and $G_{ε} = R^{n}$ if $G = R^{n}$ .

[F7]

For the same mollification, if $b \in W^{k, p} (G; C)$ and $1 \leq p < \infty$ , then $b_{ε} \to b$ in $W^{k, p} (U; C)$ for every open $U$ compactly contained in $G$ Local smooth approximation in integer-order Sobolev spaces. We apply this with $p = 2$ , on bounded interior sets for the coefficient and on $R^{n}$ for compactly supported $H^{1}$ functions; no $W^{k, \infty}$ norm convergence is used.

[F8]

If $v \in W^{k, p} (G; C)$ vanishes outside a compact subset of the open set $G$ , then $E_{0} v \in W^{k, p} (R^{n}; C)$ , $D^{α} E_{0} v = E_{0} D^{α} v$ for $∣ α ∣ \leq k$ , and its Sobolev norm is preserved Compactly supported Sobolev functions extend by zero in every integer order.

[F9]

Complex Holder bounds $∣ \int f g ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ , and complex Minkowski bounds finite sums in $L^{2}$ Complex Holder, Minkowski, and the quotient norm. These apply with Lebesgue measure on every open set used below.

[F10]

A compact subset of an open Euclidean set has a smooth compactly supported cutoff valued in $[0, 1]$ and equal to one near that subset Test function cutoffs and euclidean localization. Applied to ${0} \subset B_{1} (0)$ and normalized by its positive finite integral, this supplies a nonnegative $ρ \in C_{c}^{\infty} (B_{1} (0))$ with integral one.

Proof

Given: The domain, scale, choice assumption and multiplier in the Statement.

Goal: Prove the multiplier bounds at all orders from $- 2$ to $2$ and identify the negative-order action.

1.1F4F5F6F7F10given

Fix $k \in {1, 2}$ , $u \in H^{k} (Ω)$ and $φ \in C_{c}^{\infty} (Ω)$ . Choose bounded open sets $U, V$ with $supp φ \subset U$ , $\overline{U} \subset V$ and $\overline{V} \subset Ω$ , both closures compact, and choose the kernel $ρ$ of [F10]. Let $a_{ε} = ρ_{ε} * E_{0} a$ . For small $ε$ , [F6] gives smoothness on $U$ and $∣ D^{β} a_{ε} (x) ∣ \leq ∥ D^{β} a ∥_{\infty}$ there for $∣ β ∣ \leq k$ , since the convolution is an average with a nonnegative unit-mass kernel. Because $V$ is bounded, $a ∣_{V} \in W^{k, 2} (V)$ : each derivative has finite $L^{2} (V)$ norm bounded by $∣ V ∣^{1 / 2} ∥ D^{β} a ∥_{\infty}$ . For $ε < dist (\overline{U}, R^{n} ∖ V)$ its mollification agrees on $U$ with $a_{ε}$ . Thus [F7] gives $D^{β} a_{ε} \to D^{β} a$ in $L^{2} (U)$ for every $∣ β ∣ \leq k$ , including the highest order.

2.1F3F5F9step 1.1given

Apply [F3] on $U$ to $a_{ε}$ and $u ∣_{U}$ . In the weak test identity for $∣ α ∣ \leq k$ , pass to the limit against $φ$ : the left-hand error is bounded by $∥ a_{ε} - a ∥_{L^{2} (U)} ∥ u D^{α} φ ∥_{L^{2} (U)}$ , and the error in each right-hand product term is bounded by $∥ D^{β} a_{ε} - D^{β} a ∥_{L^{2} (U)} ∥ D^{α - β} u φ ∥_{L^{2} (U)}$ , both tending to zero by step 1.1 and [F9]. Since the test function was arbitrary, the weak formula is $D^{α} (a u) = \sum_{β \leq α} (\binom{α}{β}) D^{β} a D^{α - β} u$ on $Ω$ . Every summand is globally in $L^{2}$ , with norm at most $∥ D^{β} a ∥_{\infty} ∥ D^{α - β} u ∥_{2}$ . Finite-sum estimates and [F5] therefore give $a u \in H^{k} (Ω)$ and $∥ a u ∥_{H^{k}} \leq C_{n, k} ∥ a ∥_{W^{k, \infty}} ∥ u ∥_{H^{k}}$ . For $k = 0$ the bound is the pointwise $L^{\infty}$ estimate.

3.1F1F4F5F6F7F8step 2.1given

If $u \in H_{0}^{1} (Ω)$ , choose $u_{j} \in C_{c}^{\infty} (Ω)$ converging to $u$ in $H^{1}$ . Step 2.1 gives $a u_{j} \to a u$ in $H^{1}$ . For fixed $j$ , $v = a u_{j}$ belongs to $H^{1}$ and vanishes outside $K_{j} = supp u_{j}$ , a compact subset of $Ω$ . By [F8], $E_{0} v \in H^{1} (R^{n})$ . Its mollification is smooth by [F6] and vanishes when $dist (x, K_{j}) > ε$ , directly from the support of $ρ_{ε}$ in its defining integral. Hence for all sufficiently small $ε$ it has compact support in $Ω$ , with all these supports in one bounded ball $B$ . By [F7] on $R^{n}$ it converges to $E_{0} v$ in $H^{1} (B)$ , and therefore globally in $H^{1} (R^{n})$ , since the differences and their weak derivatives vanish outside $B$ . Restriction to $Ω$ proves $a u_{j} \in H_{0}^{1} (Ω)$ ; closure then gives $a u \in H_{0}^{1} (Ω)$ . The case $v = 0$ uses the zero approximants. The same proof applies to $\overline{a}$ . Combined with step 2.1 this preserves $H^{2} \cap H_{0}^{1}$ as well.

4.1F1F2step 2.1step 3.1

By the domain and norm identifications, steps 2.1 and 3.1 give multiplier bounds on $H_{D}^{0} = L^{2}$ , $H_{D}^{1} = H_{0}^{1}$ and $H_{D}^{2} = H^{2} \cap H_{0}^{1}$ , each bounded by a constant times $∥ a ∥_{W^{2, \infty}}$ . The endpoint operators are consistent because they are all ordinary pointwise multiplication. Interpolation between orders $0$ and $1$ , and between orders $1$ and $2$ , yields the asserted bound for every $0 \leq s \leq 2$ . The same bounds hold for multiplication by $\overline{a}$ .

5.1F2step 4.1given∎

For $0 < s \leq 2$ , the antidual identification makes $u \mapsto ⟨ f, \overline{a} u ⟩_{- s, s}$ a bounded conjugate-linear functional on $H_{D}^{s}$ , with norm at most $C_{s, Ω} ∥ a ∥_{W^{2, \infty}} ∥ f ∥_{H_{D}^{- s}}$ . It therefore determines a unique $a f \in H_{D}^{- s}$ with that bound. If $f \in L^{2}$ , then $⟨ f, \overline{a} u ⟩_{L^{2}} = ⟨ a f, u ⟩_{L^{2}}$ for $u \in H_{D}^{s}$ , so this definition agrees with ordinary multiplication. The real Dirichlet form also gives the stated conjugation isometry on the scale. The cutoff and potential cases follow by substituting the stated regularity into the bound.

Source notes

The proof uses local coefficient mollification and the test-function definition of the zero-boundary space; it does not require a trace product theorem or a higher-power Dirichlet boundary condition. No assertion is made here for orders beyond two.

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