Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of the metric

Abstract

We prove that the Atiyah-Singer Dirac operator / D g in L 2 depends Riesz continuously on L ∞ perturbations of complete metrics g on a smooth manifold. The Lipschitz bound for the map g → / D g (1 + / D

Evolution of Time-Harmonic Electromagnetic and Acoustic Waves Along Waveguides

“…Now we prove that T has a bounded H ∞ (Σ) functional calculus . The main idea is contained in [2], [7]. We estimate each summand separately.…”

Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of local boundary conditions

“…This paper is as mentioned a sequel to the authors' joint paper [6] with Alan McIntosh. During our work on this project, McIntosh untimely passed away, leaving us in great sorrow.…”

“…To estimate, a Weitzenböck-type inequality for D is needed. Turning to a manifold with boundary, one sees that (1.2) follows as in [6], mutatis mutandis. Instead, the presence of boundary forces (1.3) to be a nonstandard estimate, since new boundary terms appear in the absence of boundary conditions for the multiplier A 2 .…”

“…where D andD are self-adjoint elliptic first-order partial differential operators, acting on sections of a vector bundle V over a smooth manifold M. The symbol f (ζ) = ζ(1 + ζ 2 ) − 1 2 is a motivating example, yielding continuity results in the Riesz sense, but our methods apply equally well to more general holomorphic symbols around R, which may be discontinuous at ∞. In [6], together with Alan McIntosh, we obtained results on complete manifolds (M, g) without boundary. In that case, the main example of operators D andD was the Atiyah-Singer Dirac operators on M with respect to two different metrics g andg.…”

“…Like in [6], we use methods from real harmonic analysis to obtain (1.1). Initial operator theoretic reductions of the problem shows that two estimateŝ…”