An algebra of pseudo‐differential operators

1. Integral transforms on a sphere.- 1. The generalized kernels (xy)?+-, (+-xy + iO)?.- 2. The operator E(?), its relation with the Fourier and Mellin transform.- 3. The action of E(?) on spherical functions.- 4. Operators related with the transform E(?).- 5. The spaces Hs(?, Sn-1). The operator E(?) on the spaces Hs(?, Sn-1).- 6. An analog of the Paley-Wiener theorem for the operator E(?).- 2. The Fourier transform and convolution operators on spaces with weighted norms.- 1. The spaces H?s(?n).- 2. The Fourier transform on the spaces Hs?(?n).- 3. Convolution operator on the spaces Hs?(?n).- 4. The spaces Hs?(?m,?m-n).- 5. Transversal operators and special representations.- 6. Estimates for the convolution operator on the spaces H?s(?m, ?m-n).- 3. Meromorphic pseudodifferential operators.- 1. Canonical meromorphic pseudodifferential operators.- 2. Operations on canonical meromorphic pseudodifferential operators.- 3. General meromorphic pseudodifferential operators.- 4. Traces of meromorphic pseudodifferential operators.- 5. Meromorphic pseudodifferential operators on strongly oscillating functions.- 6. Estimates for meromorphic pseudodifferential operators.- 7. Periodic meromorphic pseudodifferential operators.- 8. Change of variables in meromorphic pseudodifferential operators.- 4. Pseudodifferential operators with discontinuous symbols on manifolds with conical singularities.- 1. Pseudodifferential operators on ?n.- 2. Pseudodifferential operators on a conic manifold.- 3. Pseudodifferential operators on manifolds with conical points.- 4. Algebras generated by pseudodifferential operators of order zero.- 5. The spectrum of a C* -algebra of pseudodifferential operators with discontinuous symbols on a closed manifold.- 1. Results from the theory* of C* -algebras.- 2. The spectrum of a C* -algebra of pseudodifferential operators with discontinuities of the first kind in the symbols on a smooth closed manifold (statement of the main theorem).- 3. Representations of the algebra $$ \mathfrak{G} $$(?) generated by the operators E(?)-1F(o, ?)E(?).- 4. Representations of an algebra $$ \mathfrak{G} $$(lx).- 5. Proof of theorem 2.1.- 6. Ideals in the algebra of pseudodifferential operators with discontinuous symbols.- 7. Spectra of C* -algebras of pseudodifferential operators on a manifold with conical points.- 8. The spectrum of a C* -algebra of pseudodifferential operators with oscillating symbols.- 6. The spectrum of a C* -algebra of pseudodifferential operators on a manifold with boundary.- 1. The algebras $$ \mathfrak{G} $$c(?).- 2. The algebras $$ \mathfrak{G} $$(?).- 3. The algebras $$ \mathfrak{G} $$c(l?).- 4. The algebras $$ \mathfrak{G} $$(l?).- 5. The spectrum of an algebra of pseudodifferential operators on a manifold with boundary.- Bibliographical sketch.- References.

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.

Point symmetries are obtained for all equations in the KP hierarchy. The Lie algebra for each equation is infinite dimensional and involves several arbitrary functions of the corresponding time tN. The symmetry algebra is a semidirect sum of a Virasoro algebra and a Kac–Moody one. The “positive” part of this algebra is embedded into the known W∞ algebra of KP symmetries and into the free fermion algebra ĝl(∞). The corresponding action on the tau-function is presented. The negative part of the point symmetries does not fit into the free fermion algebra, but is embedded into a P∞ algebra, based on the algebra of pseudodifferential operators.

Pseudo differential operators and integrable systems in AXIOM

We discuss a sub-algebra ¸ of Pseudo Differential Operators of which is Fredholm-closed and commutative modulo the ideal of compact operators. The algebra ¸ contains in particular all LP - Fourier-multipliers of Mihlin type. is shown to be an algebra with symbol, and its symbol space is determined. As an application we get simple criteria for an operator in ¸to be Fredholm. Furthermore, we regard algebras ¸ on Sobolev spaces , which are isometrically isomorphic to ¸ and have equivalent properties. Finally we conclude Fredholm criteria for mroe general Differential and Pseudo Differential Operators.

Sufficient conditions for stability on a finite time interval are derived for a class of evolution variational inequalities with the use of Lyapunov functions and frequency-domain conditions. These inequalities are considered for Hilbert spaces and Sobolev spaces of infinite order in the sense of Yu.A. Dubinskii. We show how to use the stability result on a finite interval to characterize bifurcations. An algorithm for finding observation functionals is presented that uses an isomorphism between the algebra of pseudodifferential operators with constant coefficients whose symbols are real-analytic functions in some domain and an algebra of analytic matrix-valued functions.

Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We employ specific smooth symbols of arbitrary real orders and with parameters, and we show that the associated operators induce isomorphisms between Sobolev spaces on a given manifold with boundary. Such operators for integer orders have the transmission property and belong to the calculus of Boutet de Monvel [1], cf. also [9]. In general, they fit to the algebra of boundary value problems without the transmission property in the sense of [17] and [24]. Order reducing elements of the present kind are useful for constructing parametrices of mixed elliptic problems. We show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We then investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. We finally construct order reducing operators on a compact manifold with conical singularities and boundary.

Algebras of pseudodifferential operators, volume 43 : B.A. Plamenevskii, Mathematics and its Applications (Soviet Series). Kluwer, Dordrecht, 1989. 291 pp., Dfl. 210, US $99, £69, ISBN 0-7923-0231-1

The trace class ideal of a $$H^*$$-algebra of pseudo-differential operators on $$(0,\infty )$$

One considers C*-algebras, generated by pseudodifferential operators in a halfspace. The operators act in functional spaces with weighted norms. One describes the set of all classes of equivalent irreducible representations of algebras and reveals the topology on the spectrum. One investigates, in particular, the dependence of the spectrum on the choice of the “weighted” functional space.

We extend the stability and spectral invariance of convolution-dominated matrices to the case of quasi-Banach algebras p < 1 {p<1} . As an application, we construct a spectrally invariant quasi-Banach algebra of pseudodifferential operators with non-smooth symbols that generalize Sjöstrand’s results.

Ellipticity of (pseudo-differential) operators on a manifold with geometric singularities gives rise to a hierarchy of symbols, associated with the system of lower-dimensional strata of the configuration. Classical examples are boundary value problems with interior and boundary symbols (the latter ones describe Shapiro–Lopatinskij ellipticity of boundary conditions), or operators on manifolds with conical singularities with interior and conormal symbols. Ellipticity on a manifold with smooth edges may be investigated by a suitable combination of ideas from boundary value problems and cone calculus. The present article studies another typical case, namely ellipticity on a manifold that has edges with conical singularities. Locally, we may talk about cones, where the base is a manifold with smooth edges. Parametrices and iterated asymptotics of solutions to elliptic equations are determined by a three-component symbolic hierarchy, with interior, edge and conormal symbols. We construct an operator algebra of 2 × 2-block matrices, where the upper left corners contain the interior operators, together with so-called Green and Mellin operators (caused by analogues of Green's function in boundary value problems as well as by asymptotic phenomena), while the other entries contain trace and potential conditions with respect to the edge and pseudo-differential operators on the edge itself that are of Fuchs type with respect to the conical points. The calculus is organized in an iterative way and can be viewed as a starting point for constructing similar operator algebras with asymptotics for higher polyhedral singularities.

Differential operators with degeneracies of various sorts have been studied by a large number of mathematicians, with results ranging from those concerning very special operators to those concerning analytic and algebraic properties of general classes of operators. Much of the analysis has focused on elliptic operators, and the main concern has been to show that many properties usually associated with elliptic operators or elliptic boundary problems have analogues in a more general context. In this note we describe how a certain class of operators with uniformly controlled degeneracies may be analyzed. These are called edge operators because they arise when nondegenerate elliptic operators are written in polar coordinates around an edge of a domain, or indeed around any distinguished submanifold; however as we shall indicate, they also arise in many other geometrically natural situations. Their analysis is undertaken by constructing an algebra of pseudodifferential operators sharing the same type of degeneracy which is large enough to contain parametrices for many of the elliptic elements of the original ring of edge differential operators. These parametrices are constructed explicitly, and this allows us to conclude a great deal of detailed information about solutions of the original equations. Operators in a somewhat restricted subclass of the class of elliptic operators have very good analytic properties on a scale of weighted Sobolev or Holder spaces for all but a discrete set of values of the weight parameter — for many values of the weight parameter the operators are semi-Fredholm, with infinite dimensional kernel or cokernel, although occasionally it will have neither hence be Fredholm. On the other hand for certain values of the weight parameter it may have both, hence the most we can conclude is that the range is closed. Simultaneously we obtain a structure theory for the generalized inverses for these operators as well as the ‘Bergman’ projectors onto the kernel or cokernel.

Based on an analysis of a supersymmetric extension of the algebra of pseudodifferential operators on ℝ1 an infinite hierarchy of supersymmetric Lax-integrable nonlinear dynamical systems is constructed by means of the Yang-Baxter ℛ-equation method. The structure of these systems on reduced invariant submanifolds specified by a natural invariant Lax-type spectral problem is investigated.

An algebra of pseudodifferential operators on a unimodular Lie group is constructed. It is defined by uniform estimates of the local symbols and by conditions on the decrease of the kernel outside the diagonal, formulated in terms of the weight function, having an intermediate growth between the volume function and a standard exponential. The decrease of the Green function is described, complex powers of elliptic operators of the considered classes are constructed, a meromorphic continuation of the kernels of complex powers is described, from where, in particular, the asymptotic behavior of the spectral function of operators of the considered class is obtained.