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## Über dieses Buch

This volume, like its predecessors, is based on the special session on pseudo-differential operators, one of the many special sessions at the 11th ISAAC Congress, held at Linnaeus University in Sweden on August 14-18, 2017. It includes research papers presented at the session and invited papers by experts in fields that involve pseudo-differential operators.

The first four chapters focus on the functional analysis of pseudo-differential operators on a spectrum of settings from Z to Rn to compact groups. Chapters 5 and 6 discuss operators on Lie groups and manifolds with edge, while the following two chapters cover topics related to probabilities. The final chapters then address topics in differential equations.

## Inhaltsverzeichnis

### Discrete Analogs of Wigner Transforms and Weyl Transforms

Abstract
We first introduce the discrete Fourier–Wigner transform and the discrete Wigner transform acting on functions in $$L^2({\mathbb Z})$$. We prove that properties of the standard Wigner transform of functions in $$L^2({\mathbb R}^n)$$ such as the Moyal identity, the inversion formula, time-frequency marginal conditions, and the resolution formula hold for the Wigner transforms of functions in $$L^2({\mathbb Z})$$. Using the discrete Wigner transform, we define the discrete Weyl transform corresponding to a suitable symbol on $${\mathbb Z}\times {\mathbb S}^1$$. We give a necessary and sufficient condition for the self-adjointness of the discrete Weyl transform. Moreover, we give a necessary and sufficient condition for a discrete Weyl transform to be a Hilbert–Schmidt operator. Then we show how we can reconstruct the symbol from its corresponding Weyl transform. We prove that the product of two Weyl transforms is again a Weyl transform and an explicit formula for the symbol of the product of two Weyl transforms is given. This result gives a necessary and sufficient condition for the Weyl transform to be in the trace class.
Shahla Molahajloo, M. W. Wong

### Characterization and Spectral Invariance of Non-Smooth Pseudodifferential Operators with Hölder Continuous Coefficients

Abstract
Smooth pseudodifferential operators on $$\mathbb {R}^n$$ can be characterized by their mapping properties between L p −Sobolev spaces due to Beals and Ueberberg. In applications such a characterization would also be useful in the non-smooth case, for example to show the regularity of solutions of a partial differential equation. Therefore, we will show that every linear operator P, which satisfies some specific continuity assumptions, is a non-smooth pseudodifferential operator of the symbol-class $$C^{\tau } S^m_{1,0}(\mathbb {R}^n \times \mathbb {R}^n)$$. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols.
Helmut Abels, Christine Pfeuffer

### Fredholmness and Ellipticity of ΨDOs on and

Abstract
We give a condition under which a pseudodifferential operator with symbol in $$S^{m}\left (\mathbb {R}^{n}\times \mathbb {R}^{n}\right )$$ cannot be a Fredholm operator when acting on suitable Besov and Triebel-Lizorkin spaces. As a corollary, we show that, if a classical pseudodifferential operator on $$\mathbb {R}^{n}$$ is Fredholm in one of these spaces, then this operator must be elliptic.
Pedro T. P. Lopes

### Characterizations of Self-Adjointness, Normality, Invertibility, and Unitarity of Pseudo-Differential Operators on Compact and Hausdorff Groups

Abstract
We give explicit formulas for the adjoint, product and inverse of a bounded pseudo-differential operator in terms of its symbol on a compact and Hausdorff group. As applications we give necessary and sufficient conditions to insure that a bounded pseudo-differential operator on a compact and Hausdorff group G is self-adjoint, normal, and unitary on L 2(G), and invertible on L p(G) for 1 ≤ p < .
Majid Jamalpourbirgani, M. W. Wong

### Multilinear Commutators in Variable Lebesgue Spaces on Stratified Groups

Abstract
In this paper, we study the multilinear fractional integrals and Calderón–Zygmund singular integrals on stratified groups. We obtain the boundedness of the commutators of the multilinear fractional integrals and Calderón–Zygmund singular integrals in variable Lebesgue spaces.
Dongli Liu, Jian Tan, Jiman Zhao

### Volterra Operators with Asymptotics on Manifolds with Edge

Abstract
We study Volterra property and parabolicity of a class of anisotropic pseudo-differential operators on a manifold with edge. This exposition belongs to a more comprehensive approach. In the present consideration we focus on asymptotic aspects of parametrices or inverses in the subalgebra of anisotropic operators of Mellin plus Green type. In the zero order case we also add the identity map. The resulting space constitutes a necessary step for constructing Volterra parametrices in general.
M. Hedayat Mahmoudi, B.-W. Schulze

### Bismut’s Way of the Malliavin Calculus for Non-Markovian Semi-groups: An Introduction

Abstract
We give a review of our recent works related to the Malliavin calculus of Bismut type for non-Markovian generators. Part IV is new and relates the Malliavin calculus and the general theory of elliptic pseudo-differential operators.
Rémi Léandre

### Operator Transformation of Probability Densities

Abstract
We describe an operator relation that relates two arbitrary probability densities. The relation may be thought of as a generalization of the Edgeworth and Gram–Charlier series of probability theory. We apply the relation to a number of issues. We generalize to relate a probability distribution with itself but at different times and show that it can be used to obtain approximate solutions. We apply the scale operator to the case of the product of two independent random variables, and generalize the concept of cumulants for that case. An operator relation between the energy density of a signal and the energy density of the spectrum is obtained. In addition, we show that the spectral moments of a signal may be expressed in terms of the Bell polynomials.
Leon Cohen

### The Time-Frequency Interference Terms of the Green’s Function for the Harmonic Oscillator

Abstract
The harmonic oscillator is a fundamental prototype for all types of resonances, and hence plays a key role in the study of physical systems governed by differential equations. The time-frequency representation of its Green’s function, obtained through the Wigner distribution, reveals the time-varying frequency structure of resonances. Unfortunately, the Wigner distribution of the Green’s function is affected by strong interference terms with a highly oscillatory structure. We characterize these interference terms by evaluating the ambiguity function of the Green’s function. The obtained result shows that, in the ambiguity domain, the interference terms are localized and separate from the resonance component, and hence they can be reduced by a proper filtering.
Lorenzo Galleani

### On the Solvability in the Sense of Sequences for Some Non-Fredholm Operators Related to the Anomalous Diffusion

Abstract
We study solvability of some linear nonhomogeneous elliptic problems and prove that under reasonable technical conditions the convergence in $$L^{2}({\mathbb R}^{d})$$ of their right sides implies the existence and the convergence in $$H^{2s}({\mathbb R}^{d})$$ of the solutions. The equations involve the second order non-Fredholm differential operators raised to certain fractional powers s and we use the methods of spectral and scattering theory for Schrödinger type operators developed in our preceding work (Volpert and Vougalter, Electron J Differ Equ 160:16 pp, 2013).
Vitali Vougalter, Vitaly Volpert
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