Related papers: The Wodzicki residue for pseudo-differential opera…

The Wodzicki residue is the unique trace on the algebra of classical pseudodifferential operators on a closed manifold, and Connes in 1988 proved that it coincides with the Dixmier trace. A Carnot manifold is a manifold $M$ whose tangent…

Classical pseudo-differential operators of order zero on a graded nilpotent Lie group $G$ form a $^*$-subalgebra of the bounded operators on $L^2(G)$. We show that its $C^*$-closure is an extension of a noncommutative algebra of principal…

We provide a new type of proof for known or new Gohberg lemmas for pseudodifferential operators on Abelian locally compact groups $\mathrm{X}$. We use $C^*$-algebraic techniques, which also give spectral results to which the Gohberg lemma…

Given a compact Lie group $G$, in this paper we establish $L^p$-bounds for pseudo-differential operators in $L^p(G)$. The criteria here are given in terms of the concept of matrix symbols defined on the non-commutative analogue of the phase…

In this paper we use Riesz spectral Theory and Gershgorin Theory to obtain explicit information concerning the spectrum of pseudo-differential operators defined on the unit circle $\mathbb{T} := \mathbb{R}/ 2 \pi \mathbb{ Z}$. For symbols…

Inspired by Gilkey's invariance theory, Getzler's rescaling method and Scott's approach to the index via Wodzicki residues, we give a localisation formula for the $\mathbb Z_2$-graded Wodzicki residue of the logarithm of a class of…

We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$, invariant by a finite group of symmetry $G$, whose Weyl quantization $\hat{H}$ is a selfadjoint operator on $L^2(\mathbb{R}^d)$. If $\chi$ is an irreducible character of $G$, we…

Originally, the noncommutative residue was studied in the 80's by Wodzicki in his thesis and Guillemin. In this article we give a definition of the Wodzicki residue, using the langage of r-fibered distributions in the context of filtered…

The purpose of this paper is to introduce new definitions of H\"ormander classes for pseudo-differential operators over the compact group of $p$-adic integers. Our definitions possess a symbolic calculus, asymptotic expansions and…

For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical…

In this paper we present symbolic criteria for invariant operators on compact topological groups $G$ characterising the Schatten-von Neumann classes $S_{r}(L^{2}(G))$ for all $0<r\leq\infty$. Since it is known that for pseudo-differential…

Given a compact Lie group $G$ and its unitary dual $\widehat{G}$, we establish the weak (1,1) continuity for pseudo-differential operators in the global H\"ormander classes of order $-n(1-\rho)/2$ on $G\times \widehat{G}$. Our approach…

In this paper we develop the calculus of pseudo-differential operators on the lattice $\mathbb{Z}^n$, which we can call pseudo-difference operators. An interesting feature of this calculus is that the phase space is compact so the symbol…

We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion $a\sim \sum_{j=0}^\infty a_{m-j}, a_{m-j}(x,\xi)=\sum_{l=0}^k a_{m-j,l}(x,\xi) \log^l|\xi|,$ where…

In this work we establish a subelliptic sharp G\r{a}rding inequality on compact Lie groups for pseudo-differential operators with symbols belonging to global subelliptic H\"ormander classes. In order for the inequality to hold we require…

Given a smooth manifold $M$ (with or without boundary), in this paper we study the regularisation of traces for the global pseudo-differential calculus in the context of non-harmonic analysis. Indeed, using the global pseudo-differential…

Global quantization of pseudo-differential operators on compact Lie groups is introduced relying on the representation theory of the group rather than on expressions in local coordinates. Operators on the 3-dimensional sphere and on group…

In this work we obtain continuity results on H\"older spaces for operators belonging to a Weyl-H\"ormander calculus for metrics such that the class of the associated operators contains, in particular, certain hypoelliptic laplacians. With…

We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$, invariant by a Lie group of symmetry $G$, whose Weyl quantization $\hat{H}$ is a selfadjoint operator on $L^2(\mathbb{R}^d)$. If $\chi$ is an irreducible character of $G$, we…

In this paper. we study properties such as $L^r$-boundedness, compactness, belonging to Schatten classes and nuclearity, Riesz spectral theory, Fredholmness, ellipticity and Gohberg's lemma, among others, for pseudo-differential operators…