Microlocal Analysis and Spectral Theory

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.

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.

0. Introduction. 1. General Calculus of Pseudodifferential Operators. 2. Model Classes of Degenerate Elliptic Differential Operators. 3. General Classes of Degenerate Elliptic Differential Operators. 4. Degenerate Elliptic Operators in Non--Power--Like Degeneration Case. 5. Lp-Theory for Degenerate Elliptic Operators. 6. Coerciveness of Degenerate Quadratic Forms. 7. Some Classes of Hypoelliptic Pseudodifferential Operators on a Closed Manifold. 8. Algebra of Boundary Value Problems for Class of Pseudodifferential Operators which Change Order on the Boundary. 9. General Schemes of Investigation of Spectral Asymptotics for Degenerate Elliptic Equations. 10. Spectral Asymptotics of Degenerate Elliptic Operators. 11. Spectral Asymptotics of Hypoelliptic Operators with Multiple Characteristics. A Brief Review of the Bibliography. Bibliography. Index of Notations. Index.

The Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable is , where A is a finitely many orders smoothing linear hypoelliptic operator and is the noise magnitude. The covariance operator CU of U is smoothing of order , self-adjoint, injective and elliptic pseudodifferential operator. If was taking values in L2 then in Gaussian case solving the conditional mean (and maximum a posteriori) estimate is linked to solving the minimisation problem However, Gaussian white noise does not take values in L2 but in where is big enough. A modification of the above approach to solve the inverse problem is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of the conditional mean estimate to the correct solution as is proven in appropriate function spaces using microlocal analysis. Also the frequentist posterior contractions rates are studied.

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.

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 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.

In this paper we describe the propagation of C°° and Sobolev singularities for the wave equation on C°° manifolds with corners M equipped with a Riemannian metric g. That is, for X = M x R t , P = D 2 t - Δ M , and u ∈ H 1 loc (X) solving Pu = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WF b (u) is a union of maximally extended generalized broken bicharacteristics. This result is a C°° counterpart of Lebeau's results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [11]. Our methods rely on b-microlocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if M has a smooth boundary (and no corners).

This article is a survey of Louis Nirenberg’s contributions to linear partial differential equations, focusing on his groundbreaking work on pseudo-differential operators and solvability.

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.”

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.

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.

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

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.

In the first part of the paper we study the minimal and maximal extension of a class of weighted pseudodifferential operators in the Fréchet space . In the second one non homogeneous microlocal properties are introduced and propagation of Sobolev singularities for solutions to (pseudo)differential equations is given. For both the arguments actual examples are provided.