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The maximal distributional domain of a geometric Dirac operator is H1

Statement

Assume full AC. Let $M$ be closed, Riemannian, and of even dimension $n \geq 2$ . Let $D_{V}$ be the formally symmetric geometric Dirac operator of Graded Hermitian Clifford modules and geometric Dirac operators, and let $P = D_{V}^{2}$ have the Lichnerowicz form of Lichnerowicz formula for a geometric Dirac operator. Then $dom ({\overline{D}}_{V}) = H^{1} (M, V) = {u \in L^{2} : D_{V} u \in L^{2} distributionally},$ its graph norm is equivalent to the $H^{1}$ norm, smooth sections are a core, and ${\overline{D}}_{V}$ is selfadjoint.

Facts & Assumptions

Given: Full AC, closed compact $M$ , the module and operator above, and the compact $L^{2}$ and Sobolev spaces.

[F1]

The Lichnerowicz formula says $P = \nabla^{*} \nabla + R_{0}$ , where $R_{0} = scal / 4 + c (F^{V / S})$ is a smooth bounded endomorphism (Lichnerowicz formula for a geometric Dirac operator).

[F2]

For a formally symmetric Laplace-type bundle operator on compact boundaryless $M$ , $(P + λ) : H^{2} \to L^{2}$ is an isomorphism for sufficiently large $λ > 0$ , its inverse $B$ maps $L^{2} \to H^{2}$ and is selfadjoint, and smooth data are preserved (Compact Laplace-type bundle operators: regularity and shifted parametrices).

[F3]

$P$ has a smooth finite-dimensional kernel, closed range, and a Green inverse; the exact $H^{2}$ domain and higher regularity are as in Compact Laplace-type operators: smooth kernel, closed range and Green inverse.

[F4]

The geometric Dirac operator is globally defined and formally symmetric on closed $M$ under Countable Choice (The geometric Dirac operator is globally well defined and formally symmetric).

[A1]

Full AC supplies the choice assumptions of [F2]–[F4] (The Axiom of Choice).

Proof

Given: The hypotheses in the Statement.

1.1F1F3F4given

For smooth $u$ , formal symmetry [F4] and integration of [F1] against $u$ give $∥ D u ∥_{2}^{2} = ∥ \nabla u ∥_{2}^{2} + ⟨ R_{0} u, u ⟩$ . Since $R_{0}$ is bounded and the connection Sobolev norm is equivalent to a finite-chart $H^{1}$ norm, this gives $∥ u ∥_{H^{1}} \leq C (∥ D u ∥_{2} + ∥ u ∥_{2})$ . Smooth density extends the estimate to $H^{1}$ , while the first-order local formula gives $∥ D u ∥_{2} \leq C ∥ u ∥_{H^{1}}$ . Thus the graph norm is equivalent to $H^{1}$ , and smooth sections are dense in that graph norm.

1.2F2F3F4givenalgebra

Let $B = (P + λ)^{- 1}$ from [F2]. It maps $L^{2} \to H^{2}$ , so $D B : L^{2} \to H^{1}$ is bounded. On smooth $h$ , $D (B h) = B (D h)$ : both solve $(P + λ) z = D h$ because $D$ commutes with $D^{2}$ on smooth sections and the solution is unique. Density extends the symmetry identity $⟨ D B f, h ⟩ = ⟨ f, D B h ⟩$ to $f, h \in L^{2}$ .

2.1F2F4step 1.2

Suppose $u \in L^{2}$ and $D u = f \in L^{2}$ distributionally. Then $(P + λ) u = D f + λ u$ distributionally. Test against $B h$ for smooth $h$ : formal symmetry gives $⟨ u, h ⟩ = ⟨ f, D B h ⟩ + λ ⟨ u, B h ⟩ = ⟨ D B f + λ B u, h ⟩$ , using step 1.2 and selfadjointness of $B$ . Hence $u = D B f + λ B u$ as distributions. The first term is in $H^{1}$ by the bounded map $D B$ , and the second is in $H^{2}$ by [F2]; therefore $u \in H^{1}$ . Conversely every $H^{1}$ section has $D u \in L^{2}$ by the first-order local formula, so the maximal distributional domain equals $H^{1}$ .

3.1A1F2F3F4step 1.1step 2.1algebra∎

The closure of smooth $D$ has domain $H^{1}$ by step 1.1, so its closure is contained in the maximal distributional operator. If $u \in dom (D^{*})$ , then the distribution $D u$ equals the $L^{2}$ vector $D^{*} u$ by the definition of adjoint testing against smooth sections. Step 2.1 gives $u \in H^{1}$ , and formal symmetry plus density shows $D^{*} u = D u$ . Thus $D^{*} = \overline{D}$ ; the closure is selfadjoint. No compact-resolvent theorem for a general first-order operator is assumed. Full AC supplies the choice assumptions of [F2]–[F4] as recorded in [A1].

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Sources

Liviu I. Nicolaescu, Notes on the Atiyah–Singer Index Theorem, revision 15 November 2013 (standard reference, not scraped)
Konstantin Wernli, Lecture Notes on Spin Geometry (2019) (standard reference, not scraped)