Defect measures on graded lie groups

In this article, we define a generalisation of microlocal defect measures (also known as H-measures) to the setting of graded nilpotent Lie groups. This requires to develop the notions of homogeneous symbols and classical pseudo-differential calculus adapted to this setting and defined via the representations of the groups. Our method relies on the study of the C *-algebra of 0-homogeneous symbols. Then, we compute microlocal defect measures for concentrating and oscillating sequences, which also requires to investigate the notion of oscillating sequences in graded Lie groups. Finally, we discuss compacity compactness approaches in the context of graded nilpotent Lie groups

Differential structure on the dual of a compact Lie group

In this paper we define difference operators and homogeneous Sobolev-type spaces on the dual of a compact Lie group. As an application and to show that this defines a relevant differential structure, we state and prove multiplier theorems of Hörmander, Mihlin and Marcinkiewicz types together with the sharpness in the Sobolev exponent for the result of Hörmander type.</p

Asymptotics and zeta functions on compact nilmanifolds

In this paper, we obtain asymptotic formulae on nilmanifolds $\Gamma \backslash G$, wher $G$ is any stratified (or even graded) nilpotent Lie group equipped with a co-compact discrete subgroup $\Gamma$. We study especially the asymptotics related to the sub-Laplacians naturally coming from the stratified structure of the group $G$ (and more generally any positive Rockland operators when $G$ is graded). We show that the short-time asymptotic on the diagonal of the kernels of spectral multipliers contains only a single non-trivial term. We also study the associated zeta functions.Comment: arXiv admin note: substantial text overlap with arXiv:2101.0702

QUANTUM EVOLUTION AND SUB-LAPLACIAN OPERATORS ON GROUPS OF HEISENBERG TYPE

In this paper we analyze the evolution of the time averaged energy densities associated with a family of solutions to a Schrödinger equation on a Lie group of Heisenberg type. We use a semi-classical approach adapted to the stratified structure of the group and describe the semi-classical measures (also called quantum limits) that are associated with this family. This allows us to prove an Egorov's type Theorem describing the quantum evolution of a pseudodifferential semi-classical operator through the semi-group generated by a sub-Laplacian

Some remarks on semi-classical analysis on two-step Nilmanifolds

In this paper, we present recent results about the developement of a semiclassical approach in the setting of nilpotent Lie groups and nilmanifolds. We focus on two-step nilmanifolds and exhibit some properties of the weak limits of sequence of densities associated with eigenfunctions of a sub-Laplacian. We emphasize the influence of the geometry on these properties

Quantization on Groups and Garding inequality

In this paper, we introduce Wick's quantization on groups and discuss its links with Kohn-Nirenberg's. By quantization, we mean an operation that associates an operator to a symbol. The notion of symbols for both quantizations is based on representation theory via the group Fourier transform and the Plancherel theorem. As an application, we give a simple proof of Garding inequalities for three globally symbolic pseudo-differential calculi on groups

Pre- and post-transition behavior of shear-thickening fluids in oscillating shear

The dynamic shear-thickening behavior of concentrated colloidal suspensions of fumed silica in polypropylene glycol has been investigated. Dynamic frequency sweeps showed that, for any given solids concentration, the complex viscosity at different imposed strain amplitudes followed a unique power-law-type behavior up to the onset of strain thickening. Moreover, similar behavior was also observed in the post-transition state, i.e., the viscosities again superimposed at frequencies beyond the transition frequency. In an attempt to develop a parametric description of this behavior, both the Delaware-Rutgers rule and the concept of a critical shear stress for the onset of shear thickening in steady-state experiments were considered. However, neither approach could account for the observed trends over the entire range of strains and frequency investigated. Plots of the critical shear strains for the onset and the end-point of the transition as a function of frequency were, therefore, used to describe the state of the suspensions for an arbitrary combination of strain and frequency. Finally, Fourier transform (FT) rheology was used to evaluate the extent of non-linearity in the response of the suspensions to dynamic shear, and it was shown that the observed behavior was not significantly influenced by wall slip at the tool-specimen interfac

Semi-classical analysis on H-type groups

In this paper, we develop a semi-classical analysis on H-type groups. We define semi-classical pseudodifferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consists of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account the finite-dimensioned representations of the group, even though they are negligible with respect to the Plancherel measure.Comment: 29 page