Related papers: Pseudodifferential operators on manifolds with a L…

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show existence and uniqueness of $m$…

For an element $\Psi$ in the graded vector space $\Omega^*(M, TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $\Psi$-submanifold is defined as a submanifold $N$ of $M$ such that $\Psi_{|N} \in \Omega^*(N, TN)$. The class of…

Motivated by the study of layer potentials on manifolds with straight conical or cylindrical ends, we introduce and study two classes (or calculi) of pseudodifferential operators defined on manifolds with cylindrical ends: the class of…

We deal with the asymptotic behaviour for $\lambda\to+\infty$ of the counting function $N_P(\lambda)$ of certain positive selfadjoint operators $P$ with double order $(m,\mu)$, $m,\mu>0$, $m\not=\mu$, defined on a manifold with ends $M$.…

We build a longitudinally smooth differentiable groupoid associated to any manifold with corners. The pseudodifferential calculus on this groupoid coincides with the pseudodifferential calculus of Melrose (also called b-calculus). We also…

On a suitable class of non-compact manifolds, we study (pseudo)differential operators which feature an asymptotic translation-invariance along one axis and an asymptotic dilation-invariance, or asymptotic homogeneity with respect to…

We present in this paper the construction of a pseudodifferential calculus on smooth non-compact manifolds associated to a globally defined and coordinate independant complete symbol calculus, that generalizes the standard…

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie…

The importance of the theory of pseudo-differential operators in the study of non linear integrable systems is point out. Principally, the algebra $\Xi $ of nonlinear (local and nonlocal) differential operators, acting on the ring of…

In a previous paper ([1]), we associated a holonomy groupoid and a C*-algebra to any singular foliation (M,F). Using these, we construct the associated pseudodifferential calculus. This calculus gives meaning to a Laplace operator of any…

We extend the method of layer potentials to manifolds with boundary and cylindrical ends. To obtain this extension along the classical lines, we have to deal with several technical difficulties due to the non-compactness of the boundary,…

Let $\mathbb K$ be an algebraically closed field of characteristic zero. Let $V$ be a module over the polynomial ring $\mathbb K[x,y]$. The actions of $x$ and $y$ determine linear operators $P$ and $Q$ on $V$ as a vector space over $\mathbb…

We investigate properties of pseudodifferential operators on $L^2$ space on manifold with ends including asymptotically conical or hyperbolic ends. Our pseudodifferential operators are a generalization of the canonical quantization which…

We study the geometry and topology of (filtered) algebra-bundles ${\bf\Psi}^{\mathbb Z}$ over a smooth manifold $X$ with typical fibre $\Psi^{\mathbb Z}(Z; V)$, the algebra of classical pseudodifferential operators of integral order on the…

We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators on the line and the circle (with scalar or matrix coefficients). This defines a Poisson--Lie structure on the dual group of…

Fix a manifold M, and let V be an infinite dimensional Lie algebra of vector fields on M. Assume that V contains a finite dimensional semisimple maximal subalgebra A, the projective or conformal subalgebra. A projective or conformal…

We show that finitely subgraded Lie algebras of compact operators have invariant subspaces when conditions of quasinilpotence are imposed on certain components of the subgrading. This allows us to obtain some useful information about the…

The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related:…

In this article, we introduce and study $M$-elliptic pseudo-differential operators in the framework of non-harmonic analysis of boundary value problems on a manifold $\Omega$ with boundary $\partial \Omega$, introduced by Ruzhansky and…

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold…