Sandro Coriasco

Global wave-front sets of Banach, Fréchet and Modulation spacetypes, and pseudo-differential operators

We introduce global wave-front sets with respect to suitable Banach or Frechet spaces. An important special case appears when choosing these spaces as modulation spaces. We show that the standard p ...

Bounded imaginary powers of differential operators on manifolds with conical singularities

Abstract. We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted Lp-spaces over B, 1<p<∞. Under suitable ellipticity assumptions we can define a family of complex powers Az, zℂ. We also obtain sufficient information on the resolvent of A to show the boundedness of the purely imaginary powers. Examples concern unique solvability and maximal regularity for the solution of the Cauchy problem for the Laplacian on conical manifolds as well as certain quasilinear diffusion equations.

Global L^p continuity of Fourier Integral Operators

In this paper we establish global L-p(R-n)-regularity properties of Fourier integral operators. The orders of decay of the amplitude are determined for operators to be bounded on L-p(R-n), 1 < p < infinity, as well as to be bounded from Hardy space H-1(R-n) to L-1(R-n). This extends local L-p-regularity properties of Fourier integral operators, as well as results of global L-2(R-n) boundedness, to the global setting of L-p(R-n). Global boundedness in weighted Sobolev spaces W-s(sigma,p) (R-n) is also established, and applications to hyperbolic partial differential equations are given.

Bounded H ∞-Calculus for Differential Operators on Conic Manifolds with Boundary

We prove the existence of a bounded H ∞-calculus in weighted L p -Sobolev spaces for a closed extension A T of a differential operator A on a conic manifold with boundary, subject to a differential boundary condition T, provided the resolvent (λ − A T )−1 exists in a sector Λ ⊂ ℂ and has a certain pseudodifferential structure that we describe. In case A T is the minimal extension of A, this condition reduces to parameter-ellipticity of the boundary value problem . Examples concern the Dirichlet and Neumann Laplacians.

Realizations of Differential Operators on Conic Manifolds with Boundary

We study the closed extensions (realizations) of differential operators subject to homogeneous boundary conditions on weighted Lp-Sobolev spaces over a manifold with boundary and conical singularities. Under natural ellipticity conditions we determine the domains of the minimal and the maximal extension. We show that both are Fredholm operators and give a formula for the relative index.

On the Spectral Asymptotics of Operators on Manifolds with Ends

We deal with the asymptotic behaviour, for , of the counting function of certain positive self-adjoint operators P with double order , > , defined on a manifold with ends M. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier integral operators associated with weighted symbols globally defined on . By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae for and show how their behaviour depends on the ratio and the dimension of M.

Global wave-front sets of intersection and union type

We show that a temperate distribution belongs to an ordered intersection or union of admissible Banach or Frechet spaces if and only if the corresponding global wave-front set of union or intersection type is empty. We also discuss the situation where intersections and unions of sequences of spaces with two indices are involved. A main situation where the present theory applies is given by sequences of weighted, general modulation spaces.

A Class of Fourier Integral Operators on Manifolds with Boundary

We study a class of Fourier integral operators on compact manifolds with boundary X and Y , associated with a natural class of symplectomorphisms \(\mathcal{X} :\;T^{*}Y \; \backslash \;0\rightarrow\;T^{*}X\; \backslash \;0\), namely, those which preserve the boundary. A calculus of Boutet de Monvel’s type can be defined for such Fourier integral operators, and appropriate continuity properties established. One of the key features of this calculus is that the local representations of these operators are given by operator-valued symbols acting on Schwartz functions or temperate distributions. Here we focus on properties of the corresponding local phase functions, which allow to prove this result in a rather straightforward way.

Global Wave-Front Properties for Fourier Integral Operators and Hyperbolic Problems

We continue our analysis of the global wave-front sets we introduced on modulation spaces, here in relation with the corresponding class of Fourier integral operators. We obtain propagation results in terms of canonical transformations of the phase space, without requiring that the involved phase functions and amplitude are classical.

A note on the Einstein–Hilbert action and Dirac operators on {\mathbb{R}^n}

We prove an extension to