On Some Microlocal Properties of the Range of a Pseudo-Differential Operator of Principal Type

The purpose of this paper is to obtain microlocal analogues of results by L. H \"ormander about inclusion relations between the ranges of first order differential operators with coefficients in $C^\infty$ which fail to be locally solvable. Using similar techniques, we shall study the properties of the range of classical pseudo-differential operators of principal type which fail to satisfy condition $(\Psi)$.Comment: With 1 figure. Corrected typos. Rewrote parts of Section 2 to account for new language. Corrected an erroneous comment following Theorem 2.1

Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions

We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.Comment: 43 page

The classical limit of the time dependent Hartree-Fock equation. I. The Weyl symbol of the solution

We study the time evolution of the Weyl symbol of a solution of the time dependent Hartree Fock equation, assuming that for t=0, it has a Weyl symbol which is integrable in the phase space, such as all its derivatives. We prove that the solution has the same property for all t, and we give an asymptotic expansion, in L1 sense, of this Weyl symbol

Every P-convex subset of R2\R^2 is already strongly P-convex

A classical result of Malgrange says that for a polynomial P and an open subset $\Omega$ of $\R^d$ the differential operator $P(D)$ is surjective on $C^\infty(\Omega)$ if and only if $\Omega$ is P-convex. H\"ormander showed that $P(D)$ is surjective as an operator on $\mathscr{D}'(\Omega)$ if and only if $\Omega$ is strongly P-convex. It is well known that the natural question whether these two notions coincide has to be answered in the negative in general. However, Tr\`eves conjectured that in the case of d=2 P-convexity and strong P-convexity are equivalent. A proof of this conjecture is given in this note

On the Quillen determinant

We explain the bundle structures of the {\it Determinant line bundle} and the {\it Quillen determinant line bundle} considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line bundles and trivial on some subspaces. Also we remark a relation of the {\it Quillen determinant line bundle} and the {\it Maslov line bundle}.Comment: 11 pages, to appear in the Journal of Geometry and Physic

Global wave-front sets of Banach, Fr{\'e}chet and Modulation space types, and pseudo-differential operators

We introduce global wave-front sets $\operatorname{WF}_{{\mathcal B}} (f)$, $f\in {\mathscr S}^\prime(\textbf{R}^d)$, with respect to suitable Banach or Fr\'echet spaces ${\mathcal B}$. An important special case is given by the modulation spaces ${\mathcal B}=M(\omega,\mathscr B)$, where $\omega$ is an appropriate weight function and $\mathscr B$ is a translation invariant Banach function space. We show that the standard properties for known notions of wave-front set extend to $\operatorname{WF}_{{\mathcal B}} (f)$. In particular, we prove that micro locality and microellipticity hold for a class of globally defined pseudo-differential operators $\operatorname{Op}_t(a)$, acting continuously on the involved spaces.Comment: 51 pages, mistakes and typos correction, reorganized material

The relationship between the Wigner-Weyl kinetic formalism and the complex geometrical optics method

The relationship between two different asymptotic techniques developed in order to describe the propagation of waves beyond the standard geometrical optics approximation, namely, the Wigner-Weyl kinetic formalism and the complex geometrical optics method, is addressed. More specifically, a solution of the wave kinetic equation, relevant to the Wigner-Weyl formalism, is obtained which yields the same wavefield intensity as the complex geometrical optics method. Such a relationship is also discussed on the basis of the analytical solution of the wave kinetic equation specific to Gaussian beams of electromagnetic waves propagating in a ``lens-like'' medium for which the complex geometrical optics solution is already available.Comment: Extended version comprising two new section

Sharp polynomial bounds on the number of Pollicott-Ruelle resonances

We give a sharp polynomial bound on the number of Pollicott-Ruelle resonances. These resonances, which are complex numbers in the lower half-plane, appear in expansions of correlations for Anosov contact flows. The bounds follow the tradition of upper bounds on the number of scattering resonances and improve a recent bound of Faure-Sj\"ostrand. The complex scaling method used in scattering theory is replaced by an approach using exponentially weighted spaces introduced by Helffer-Sj\"ostrand in scattering theory and by Faure-Sj\"ostrand in the theory of Anosov flows.Comment: 18 pages; minor revision based on referee's comments. To appear in Erg. Theory Dyn. Sys

On the microlocal properties of the range of systems of principal type

The purpose of this paper is to study microlocal conditions for inclusion relations between the ranges of square systems of pseudodifferential operators which fail to be locally solvable. The work is an extension of earlier results for the scalar case in this direction, where analogues of results by L. H\"ormander about inclusion relations between the ranges of first order differential operators with coefficients in $C^\infty$ which fail to be locally solvable were obtained. We shall study the properties of the range of systems of principal type with constant characteristics for which condition (\Psi) is known to be equivalent to microlocal solvability.Comment: Added Theorem 4.7, Corollary 4.8 and Lemma A.4, corrected misprints. The paper has 40 page

The role of Fourier modes in extension theorems of Hartogs-Chirka type

We generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of graph(F)\cup(\partial D\times D) -- where D is the open unit disc and graph(F) denotes the graph of a continuous D-valued function F -- to the bidisc. We extend holomorphic functions by applying the Kontinuitaetssatz to certain continuous families of analytic annuli, which is a procedure suited to configurations not covered by Chirka's theorem.Comment: 17 page