BoundedH_∞-Calculus for Differential Operators on Conic Manifolds with Boundary

Abstract

Abstract. We derive conditions that ensure the existence of a bounded H∞-calculus in weighted Lp-Sobolev spaces for closed extensions A T of a differential operator A on a conic manifold with boundary, subject to differential boundary conditions T . In general, these conditions ask for a particular pseudodifferential structure of the resolvent (λ − A T ) −1 in a sector Λ ⊂ C. In case of the minimal extension they reduce to parameter-ellipticity of the boundary value problem A T . Examples concern the Dirichlet and Neumann Laplacians.

Boundary value problems and layer potentials on manifolds with cylindrical ends

“…See also [39]. Also, see [47] for an extension of the above results to L p -spaces, and [10] for some applications to non-linear evolution equations.…”

Bounded $H_\infty $-calculus for pseudodifferential operators and applications to the Dirichlet-Neumann operator

“…This shows the closedness of A, 3 as well as the desired norm estimate for the resolvent of A (with c possibly enlarged).…”

“…The paper [4] extends the results of [2] to domains with boundary. In [3] differential operators on conic manifolds with boundary are investigated.…”

Maximal $$L_{p}$$ L p -regularity of non-local boundary value problems

“…Boutet de Monvel's calculus can also be exploited to demonstrate existence of bounded imaginary powers and even of a bounded H ∞ -calculus, cf. Duong [7], Abels [1] and Coriasco, Schrohe, Seiler [4] for example; as it turns out, the strategy of proof we use in the present work is closely related to that of Abels [1]. On one hand, bounded H ∞ -calculus is stronger than R-boundedness, on the other hand the concept of R-boundedness applies to operator-families more general than the resolvent of a fixed operator.…”