Geometric BVPs, Hardy spaces, and the Cauchy integral and transform on regions with corners

Abstract

In this paper we give a new perspective on the Cauchy integral and transform and Hardy spaces for Diractype operators on manifolds with corners of codimension two. Instead of considering Banach or Hilbert spaces, we use polyhomogeneous functions on a geometrically "blown-up" version of the manifold called the total boundary blow-up introduced by Mazzeo and Melrose [R.R. Mazzeo, R.B. Melrose, Analytic surgery and the eta invariant, Geom. Funct. Anal. 5 (1) (1995) 14-75]. These polyhomogeneous functions are smooth everywhere on the original manifold except at the corners where they have a "Taylor series" (with possible log terms) in polar coordinates. The main application of our analysis is a complete Fredholm theory for boundary value problems of Dirac operators on manifolds with corners of codimension two.

Bergman and Calderón Projectors for Dirac Operators

“…We shall also call the operators of Lemma 28 log-free classical operators and denote this class by I ℓ lf (X × ∂X, ∆ ∂ ) and I ℓ lf (∂X × X, ∆ ∂ ). Notice that, since the restriction of a function in C ∞ (X × 0 X) to the right boundary gives a function in C ∞ (X × 0 ∂X), we deduce that an operator I −n−2 (X ×X, ∆ ∂ ) satisfying condition (29) induces naturally (by restriction to the boundary on the right variable) an operator in I −n−1 (X × ∂X, ∆ ∂ ) satisfying (32). This can also be seen by considering the oscillatory integrals restricted to x ′ = 0 but it is more complicated to prove.…”

“…We also refer to [13], [14] for an application of the Calderón projector of the Spin C Dirac operator, and a recent paper of Booss-Lesch-Zhu [6] for other generalizations of the work in [7]. Extensions of the Calderón projector for non-smooth boundaries were studied recently in [1,32].…”

Aspects of Dirac Operators in Analysis

“…Instead of trying to develop a function theory for the Dirac operator in the degree of generality described in the previous section and as is done in [9,23,27], we shall turn to look at the most basic example where our spin manifold is just a domain in R n .…”

“…Following [9,23,27] we shall now illustrate how some of the material we have looked at in the previous section generalizes to some spin manifolds. So in what follows we shall assume that M is a spin manifold with a spin bundle S and an Atiyah-Singer-Dirac operator D. We shall let D act on the space C ∞ 0 (S) of C ∞ sections in S having compact support.…”

“…As we have seen other examples of spin manifolds with invertible Dirac operator include C, with Dirac operator the CauchyRiemann operator d dz , and R n with euclidean Dirac operator. See [9,23,27] for further details. It should be noted that if M is a spin manifold with a bounded subdomain U with smooth compact boundary, then there is an argument via pseudo-differential projections, see [5], that allows one to invert the Dirac operator over C ∞ 0 (U ).…”