Operators with Symbol Hierarchies and Iterated Asymptotics

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.

We construct elliptic elements in the algebra of (classical pseudo-differential) operators on a manifold M with conical singularities. The ellipticity of any such operator A refers to a pair of principal symbols (σ 0, σ 1) where σ 0 is the standard (degenerate) homogeneous principal symbol, and σ 1 is the so-called conormal symbol, depending on the complex Mellin covariable z. The conormal symbol, responsible for the conical singularity, is operator-valued and acts in Sobolev spaces on the base X of the cone. The σ 1-ellipticity is a bijectivity condition for all z of real part (n + 1)/2 − γ, n = dim X, for some weight γ. In general, we have to rule out a discrete set of exceptional weights that depends on A. We show that for every operator A which is elliptic with respect to σ 0, and for any real weight γ there is a smoothing Mellin operator F in the cone algebra such that A + F is elliptic including σ 1. Moreover, we apply the results to ellipticity and index of (operator-valued) edge symbols from the calculus on manifolds with edges.

Linear Partial Differential Equations with Multiple Involutive Characteristics O. Liess, L. Rodino. Gevrey and Analytic Hypoellipticity D.S. Tartakoff. Higher Microlocalization and Propagation of Singularities O. Liess. Conormality and Lagrangian Properties in Diffractive Boundary Value Problems P. Laubin. Parametrized Pseudodifferential Operators and Geometric Invariants G. Grubb. Boundary Value Problems and Edge Pseudodifferential Operators B.-W. Schulze. Wodzicki's Noncommutative Residue and Traces for Operator Algebras on Manifolds with Conical Singularities E. Schrohe. Lower Bounds for Pseudodifferential Operators C. Parenti, A. Parmeggiani. Weyl Formula for Globally Hypoelliptic Operator in Rn E. Buzano. Splitting in Large Dimension and Infrared Estimates B. Helffer. Microlocal Exponential Estimates and Applications to Tunneling A. Martinez. A Trace Formula and Review of Some Estimates for Resonances J. Sjostrand.

The analysis of elliptic operators on a manifold W with boundary V and edge Y is developed within a calculus of pseudo-differential boundary value problems on W \ Y with extra trace and potential conditions on the edge and (edge-degenerate) pseudo-differential operators on V \ Y, together with trace and potential conditions and (standard) pseudo-differential operators on Y. The calculus consists of a 3 × 3 block matrix algebra with a three-component symbolic hierarchy, where the ellipticity of boundary and edge data are analogues of the Shapiro-Lopatinskij condition.

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.

Pseudodifferential operator theory was apparently developed no later than half a century ago (Vishik and Eskin, ; Eskin, ; Mikhlin, ; Mikhlin and Prössdorf, ; Malgrange, ; Calderon and Zygmund, ; Hörmander, ; Treves, ; Taylor, ; Rempel and Schulze, ; Sternin, ), but it is not so young because its basic achievements were invented in 1960s or 1970s. The main point of this theory is a symbolic calculus for pseudodifferential operators and boundary value problems for pseudodifferential equations on manifolds with a smooth boundary. This is not so good for manifolds with a non-smooth boundary although there are many studies on this problem (see, for example, the papers of B.-W. Schulze and his colleagues). The first paper in this direction was Kondratiev’s paper of 1967, in which the general boundary value problem for a partial differential operator in a cone was studied. But in this paper, and almost all of he following studies, the conical singularity is geometrically treated as direct product of a manifold with a smooth boundary and a half-line. The author suggests another point of view (for elliptic equations in this time) related to the concept of wave factorization of elliptic symbol (Vasil’ev, ; Vasilyev, ). From this point of view the conical singularity is something indivisible, and it cannot be reduced to other cases which were studied earlier. “Curiously enough,” V.A. Kondratiev said to me (in answer to my question) “there is nothing interesting any more in the theory of boundary value problems in domains with non-smooth boundary” (December 2004). This urged the author to collect his own outlines into one but still unfinished fragment. The author hopes that in the future it will be possible to obtain a “non-smooth” version of Sternin’s results (Sternin, ) by the wave factorization method.

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.

The calculus of classical pseudo-differential operators has been used in a fundamental way in the study of boudary value problems associated to (systems) of elliptic differential and pseudo-differential operators. This calculus was used by Hörmander in [7] to construct parametrices for the elliptic operators, using which the boundary value problem is reduced to a system of pseudo-differential operators on the boundary. In order to treat the boundary value problems for parabolic operators in bounded cylindrical domains Piriou introduced in [15] and [16] a class of operators which called pseudo-differential operators of Volterra type. The basic idea consists in developping a calculus of an appropriate class of anisotropic pseudo-differential operators as in an earlier note of Hunt and Piriou [8].KeywordsVector BundleParabolic SystemPrincipal SymbolCylindrical DomainVolterra TypeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Artificial boundary conditions, which can be identified as Robin boundary conditions positioned at a complex space coordinate, are introduced in order to obtain pertinent approximations for the Green's functions in grounded layered media. These artificial boundary conditions include perfectly matched layers backed by perfectly electric or magnetic conductors. As a first result, we obtain analytical expressions for the differences of Green's functions subject to different boundary conditions. Since weighted sums of Green's functions are again Green's functions, the need arises to solve an optimization problem, in the sense of obtainingthe optimal weig hted mixture of Green's functions, as compared to the exact Green's function. Comprehensive eigenexpansions for the Green's functions are given in the general case, and a few examples illustrate the goodness of fit between the approximate Green's functions and the exact Green's function.

We study the method of layer potentials for manifolds with boundary and cylindrical ends. The fact that the boundary is non-compact prevents us from using the standard characterization of Fredholm or compact pseudo-differential operators between Sobolev spaces, as, for example, in the works of Fabes-Jodeit-Lewis [10] and Kral-Wedland [18]. We first study the layer potentials depending on a parameter on compact manifolds. This then yields the invertibility of the relevant boundary integral operators in the global, non-compact setting. As an application, we prove a well-posedness result for the non-homogeneous Dirichlet problem on manifolds with boundary and cylindrical ends. We also prove the existence of the Dirichlet-to-Neumann map, which we show to be a pseudodifferential operator in the calculus of pseudodifferential operators that are “almost translation invariant at infinity.”

As an integral part of a pseudodifferential calculus for boundary value problems on manifolds with edges we introduce the algebra of Mellin operators. They represent the typical operators near the edge. In fact we show how to associate an operator-valued Mellin symbol to an arbitrary edge-degenerate pseudodifferential boundary value problem, the so-called ‘Mellin quantization’ procedure. Furthermore, we introduce a class of adequate Sobolev spaces based on the Mellin transform on which these operators act continuously.

We present a computer assisted method for generating existence proofs and a posteriori error bounds for solutions to two point boundary value problems (BVPs). All truncation errors are accounted for and, if combined with interval arithmetic to bound the rounding errors, the computer generated results are mathematically rigorous. The method is formulated for $n$-dimensional systems and does not require any special form for the vector field of the differential equation. It utilizes a numerically generated approximation to the BVP fundamental solution and Green's function and thus can be applied to stable BVPs whose initial value problem is unstable. The utility of the method is demonstrated on a pair of singularly perturbed model BVPs and by using it to rigorously show the existence of a periodic orbit in the Lorenz system.

The concept of hypoellipticity for degenerate elliptic boundary value problems is defined, and its relation with the hypoellipticity of certain pseudo-differential operators on the boundary is discussed (for second order equations). A theorem covering smoothness of solutions of boundary value problems such as a ( x ) ∂ u / ∂ n + b ( x ) u = f ( x ) a(x)\partial u/\partial n + b(x)u = f(x) for the Laplace equation is proved. An almost complete characterization of hypoelliptic boundary value problems for elliptic second order equations in two dimensions is given via analysis of hypoelliptic pseudo-differential operators in one variable.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

In this work, we extend Wigner's original framework to analyze linear operators by examining the relationship between their Wigner and Schwartz kernels. Our approach includes the introduction of (quasi-)algebras of Fourier integral operators (FIOs), which encompass FIOs of type I and II. The symbols of these operators belong to (weighted) modulation spaces, particularly in Sjöstrand's class, known for its favorable properties in time-frequency analysis. One of the significant results of our study is demonstrating the inverse-closedness of these symbol classes.Our analysis includes fundamental examples such as pseudodifferential operators and Fourier integral operators related to Schrödinger-type equations. These examples typically feature classical Hamiltonian flows governed by linear symplectic transformations S∈Sp(d,R). The core idea of our approach is to utilize the Wigner kernel to transform a Fourier integral operator T on Rd into a pseudodifferential operator K on R2d. This transformation involves a symbol σ well-localized around the manifold defined by z=Sw.