Gerd Grubb

Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems

This paper introduces a class of pseudodifferential operators depending on a parameter in a particular way. The main application is a complete expansion of the trace of the resolvent of a Dirac-type operator with nonlocal boundary conditions of the kind introduced by Atiyah, Patodi, and Singer [APS]. This extends the partial expansion in [G2] to a complete one, and extends the complete expansion in [GS 1 ] to the case where the Dirac operator does not have a product structure near the boundary. A secondary application is to obtain a complete expansion of the resolvent of a ~bdo on a compact manifold, essentially reproving a result of Agranovich [Agr]. The resolvent expansion yields immediately an expansion of the trace of the heat kernel, and determines the singularities of the zeta function; moreover, a pseudodifferential factor can be allowed. A major motive for these expansions is to obtain index formulas for elliptic operators; there are many such applications in the physics and geometry literature. The index formula comes from one particular term in the expansion, but each term is a spectral invariant, and they have been used for other purposes as well as for the index. In particular, Branson and Gilkey have a number of papers (e.g. [BG] and [Gi]) analyzing these invariants, and drawing geometric consequences. Interest in the asymptotic behavior of the resolvent goes back to Carleman [C]. More recently, Agmon [Agm] developed it extensively for analytic applications; he introduced the fundamental idea of treating the resolvent parameter essentially as another cotangent variable. This idea was developed in [S1] to analyze the singularities of the zeta function of an elliptic Odo on a compact manifold, and in [

Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators

3] to analyze the resolvent of a differential operator with differential boundary conditions. The technique works smoothly for differential operators, producing so-called local invariants, integrals over the underlying

Zeta and eta functions for Atiyah-Patodi-Singer operators

Abstract Let P be a classical pseudodifferential operator of order m ∈ C on an n-dimensional C ∞ manifold Ω 1 . For the truncation P Ω to a smooth subset Ω there is a well-known theory of boundary value problems when P Ω has the transmission property (preserves C ∞ ( Ω ¯ ) ) and is of integer order; the calculus of Boutet de Monvel. Many interesting operators, such as for example complex powers of the Laplacian ( − Δ ) μ with μ ∉ Z , are not covered. They have instead the μ-transmission property defined in Hormanders books, mapping x n μ C ∞ ( Ω ¯ ) into C ∞ ( Ω ¯ ) . In an unpublished lecture note from 1965, Hormander described an L 2 -solvability theory for μ-transmission operators, departing from Vishik and Eskins results. We here develop the theory in L p Sobolev spaces ( 1 p ∞ ) in a modern setting. It leads to not only Fredholm solvability statements but also regularity results in full scales of Sobolev spaces ( s → ∞ ). The solution spaces have a singularity at the boundary that we describe in detail. We moreover obtain results in Holder spaces, which radically improve recent regularity results for fractional Laplacians.

Trace expansions for pseudodifferential boundary problems for Dirac-type operators and more general systems

AbstractThis paper concerns Dirac-type operatorsP on manifoldsX with boundary which are “product-type” near the boundary. That is,

On coerciveness and semiboundedness of general boundary problems

P = \sigma \left( {\frac{\partial }{{\partial x_n }} + A} \right)

Spectral Boundary Conditions for Generalizations of Laplace and Dirac Operators

for a unitary morphism σ and a self-adjoint first-order operatorA onbdry(X);xn denotes the normal coordinate. For a realizationPB defined by a boundary operatorB of Atiyah-Patodi-Singer type, the paper gives a complete description of the singularities of the traces of the meromorphic continuations of Γ(s)D(Δi)−s and Γ(s)DP(Δi)−s where Δ1 =PB*PB, Δ2 =PBPB*, andD is any differential operator onX which is tangential and independent of 4xn nearbdry(X). This implies expansions for the associated heat kernels and resolvents, containing the usual powers (with both “local” and “global” coefficients) together with logarithmic terms.

Heat operator trace expansions and index for general Atiyah-Patodi-Singer boundary problems

for general realizations Du of first-order differential operators D (e.g. Dirac-type operators) on a manifold X with pseudodifferential boundary conditions: B(ulx,)=O at the boundary OX=X ~. In (1.1), ~ denotes a compactly supported morphism. The coefficients without primes are locally determined, the primed coefficients global. Such realizations were considered first by Atiyah, Patodi and Singer in [APS] who showed an interesting index formula in the so-called product case, when X is compact. We say that D is of Dirac-type when D=o-(Ox,~+A1) on a collar neighborhood of X ~, with a unitary morphism cr and a first-order differential operator A1 such that AI=A+x~PI+Po with A selfadjoint on X ~ and constant in xn and the Pj of order j; the p~vduct case is where P1 =P0 =0. The operator B was in [APS] taken equal to the orthogonal projection II> onto the eigenspace for A associated with eigenvalues >_0. For Dirac-type operators on compact manifolds, finite expansions (1.1) (up to k=0, with ~ =1 and ai,0=0) were shown in [G4], implying the index formula

Solution of Parabolic Pseudo-differential initial-Boundary Value Problems

These works are almost exclusively concerned with the L 2 theory, whereas little has been done in an L p framework. The purpose of this paper is to extend the calculus to L p generalizations of Sobolev spaces, mainly the Bessel-potential spaces H ps and the Besov spaces B p s (and B p,q s ) for 1