Microlocal properties of Shubin pseudodifferential and localization operators

The Quantization of Edge Symbols.- On Rays of Minimal Growth for Elliptic Cone Operators.- Symbolic Calculus of Pseudo-differential Operators and Curvature of Manifolds.- Weyl Transforms, Heat Kernels, Green Functions and Riemann Zeta Functions on Compact Lie Groups.- On the Fourier Analysis of Operators on the Torus.- Wave Kernels of the Twisted Laplacian.- Super-exponential Decay of Solutions to Differential Equations in ?d.- Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order.- A New Aspect of the L p-extension Problem for Inhomogeneous Differential Equations.- Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols.- Continuity and Schatten Properties for Pseudo-differential Operators on Modulation Spaces.- Algebras of Pseudo-differential Operators with Discontinuous Symbols.- A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform.- A Characterization of Stockwell Spectra.- Exact and Numerical Inversion of Pseudo-differential Operators and Applications to Signal Processing.- On the Product of Localization Operators.- Gelfand-Shilov Spaces, Pseudo-differential Operators and Localization Operators.- Continuity and Schatten Properties for Toeplitz Operators on Modulation Spaces.- Microlocalization within Some Classes of Fourier Hyperfunctions.

Pseudo-differential operators, Wigner transform and Weyl transform on the affine Poincaré group

This paper is the second part of a two-paper series whose aim is to give a detailed description of Connes’ pseudodifferential calculus on noncommutative [Formula: see text]-tori, [Formula: see text]. We make use of the tools introduced in the 1st part to deal with the main properties of pseudodifferential operators on noncommutative tori of any dimension [Formula: see text]. This includes the main results mentioned in [2, 5, 11]. We also obtain further results regarding action on Sobolev spaces, spectral theory of elliptic operators, and Schatten-class properties of pseudodifferential operators of negative order, including a trace-formula for pseudodifferential operators of order [Formula: see text].

In this article we characterize certain ultradifferential operators by the condition of being local. First, we examine the continuity properties of local linear operators on spaces of ultradifferentiable functions in the sense of Beurling and of Roumieu. Next, a structure theorem for vector-valued ultradistributions with support at the origin is proved. This result leads to a representation theorem for continuous local operators from spaces of ultradifferentiable functions into various spaces of ultradistributions. In combination with the continuity results we thus obtain in many cases the desired characterization.

Mapping properties of operator-valued pseudo-differential operators

Spectral properties of zeroth-order pseudodifferential operators

In this paper, using the [Formula: see text]-Dunkl harmonic analysis introduced in Bettaibi and Bettaieb [q-Analog of the Dunkl transform on the real line, Tamsui Oxf. J. Math. Sci. 25(2) (2007) 117–205] and motivated by Wong’s approach [M. W. Wong, Wavelet Transforms and Localization Operators, Vol. 136 (Springer Science & Business Media, Berlin, 2002)], we first define and study the [Formula: see text]-wavelets, the continuous [Formula: see text]-wavelet transforms. Next, we introduce the two-wavelet localization operators, the Schatten–von Neumann properties of these localization operators are established, and for trace class localization operators, the traces and the trace class norm inequalities are presented.

We review the proof of the existence of a fundamental solution for a pseudo-differential operator with polynomial symbol based on the existence of a meromorphic continuation for the local zeta function attached to the symbol. We compute fundamental solutions for quasielliptic and Schrodinger-type pseudo-differential operators. As an application we solve certain initial value problems for Schrodinger-type pseudo-differential equations. We pose several questions and problems about the connection between local zeta functions and pseudo-differential operators.

Properties of boundary operators are studied for a system of equations of elasticity theory in a full scale of Sobolev-type spaces.

Pseudo-differential operators and Localization operators on $S^{\mu}_{\nu}(\mathbb R)$ space involving fractional Fourier transform

We study the surface plasmon modes of an arbitrarily shaped nanoparticle in the electrostatic limit. We first deduce an eigenvalue equation for these modes, expressed in terms of the Dirichlet–Neumann operators. We then use the properties of these pseudo-differential operators for deriving the limit of the high-order modes.

We consider the spaces of ultradifferentiable functions $$\mathcal {S}_\omega $$ as introduced by Bjorck (and its dual $$\mathcal {S}'_\omega $$ ) and we use time-frequency analysis to define a suitable wave front set in this setting and obtain several applications: global regularity properties of pseudodifferential operators of infinite order and the micro-pseudolocal behaviour of partial differential operators with polynomial coefficients and of localization operators with symbols of exponential growth. Moreover, we prove that the new wave front set, defined in terms of the Gabor transform, can be described using only Gabor frames. Finally, some examples show the convenience of the use of weight functions to describe more precisely the global regularity of (ultra)distributions.

Following Shastry and Sutherland I construct the super Lax operators for the Calogero model in the oscillator potential. These operators can be used for the derivation of the eigenfunctions and integrals of motion of the Calogero model and its supersymmetric version. They allow to infer several relations involving the Lax matrices for this model in a fast way. It is shown that the super Lax operators for the Calogero and Sutherland models can be expressed in terms of the supercharges and so called local Dunkl operators constructed in our recent paper with M. Ioffe. Several important relations involving Lax matrices and Hamiltonians of the Calogero and Sutherland models are easily derived from the properties of Dunkl operators.

In this thesis, a pseudodifferential calculus for a degenerate hyperbolic Cauchy problem is developed. The model for this problem originates from a certain observation in fluid mechanics, and is then extended to a more general class of hyperbolic Cauchy problems where the coefficients degenerate like a power of $t + |x|^2$ as $(t,x) \to (0,0)$. Symbol classes and pseudodifferential operators are introduced. In this process, it becomes apparent that exactly in the origin, these operators are of type (1,1). Although these operators are not $L^2$-continuous in general, a proof of continuity in $\mathscr C([0,T],L^2(\mathbb R^d))$ is given for a suitable subclass. An adapted scale of function spaces is defined, where at $t = 0$ these spaces coincide with 2-microlocal Sobolev spaces with respect to the Lagrangian $\dot T^*_0\mathbb R^d$. In these spaces, energy estimates are derived, so that a symbolic approach can be applied to prove wellposedness of the Cauchy problem.

For homogeneous space G/H equipped with strongly quasi invariant measure \(\nu \), we introduce the two-wavelet constant for square integrable representations and the orthogonal subspaces of \(L^{2}(G/H)\), which are related to continuous wavelet transform (C.W.T). For admissible wavelet \(\eta \) and \(\Theta \in L^{p}(G/H);\) \(1\leqslant p \leqslant \infty \), the localization operator \(\Upsilon _{\Theta ,\eta }\) is introduced. The boundedness properties of localization operator is studied and it is shown that \(\Upsilon _{\Theta ,\eta }\) is in Schatten p-class.