Boundary value problems and layer potentials on manifolds with cylindrical ends

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.

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.

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.

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 pseudo-differential Mellin-edge-approach is a tool for studying differential and pseudo-differential operators on manifolds with corners. The Mellin transform, acting on the corner axis ℝ+, is a substitute for the Fourier transform along edge variables in the calculus of wedge pseudo-differential operators. The basic elements of that theory (cf. Schulze [6,8]) are extended to edges like ℝ+ ∋ t with a control of symbols and smoothing operators near the vertext=0. The authors study the weighted Mellin wedge Sobolev spaces, the operator-valued Mellin convention translating Fourier symbols into Mellin ones under preserved smoothness up tot=0, and develop an operator calculus with its characterization on the level of symbols. Throughout the theory, there are involved one-parameter groups of isomorphisms acting on the Banach spaces that are the abstract analogues of the weighted cone Sobolev spaces.

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.

In this paper we consider the semiclassical version of pseudo-differential operators on the lattice space ħZn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar {{\\mathbb {Z}}^{n}}$$\\end{document}. The current work is an extension of the previous work (Botchway et al. in J Funct Anal 278(11):108473, 33, 2020) and agrees with it in the limit of the parameter ħ→1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar \\rightarrow 1$$\\end{document}. The various representations of the operators will be studied as well as the composition, transpose, adjoint and the link between ellipticity and parametrix of operators. We also give the conditions for the ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^p$$\\end{document}, weighted ℓ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^2$$\\end{document} boundedness and ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^p$$\\end{document} compactness of operators. We investigate the relation between the classical and semi-classical quantization in the spirit of Ruzhansky and Turunen (Pseudo-differential operators and symmetries. Pseudo-differential operators, vol 2. Theory and Applications, Birkhäuser, Basel, 2010; J Fourier Anal Appl 16(6):943–982, 2010) RTspsJFAA and employ its applications to Schatten–von Neumann classes on ℓ2(ħZn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^2( \\hbar \\mathbb {Z}^n)$$\\end{document}. We establish Gårding and sharp Gårding inequalities, with an application to the well-posedness of parabolic equations on the lattice ħZn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar \\mathbb {Z}^n$$\\end{document}. Finally we verify that in the limiting case where ħ→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar \\rightarrow 0$$\\end{document} the semi-classical calculus of pseudo-differential operators recovers the classical Euclidean calculus, but with a twist.

Spaces of generalized functions with exponential asymptotic behaviour are considered. Convolutors in these spaces are completely described. It is shown that a convolution equation is uniquely soluble if and only if there exists a fundamental solution that is a convolutor. The explicit description of convolutors renders this condition effective. In particular, Petrovskii's correctness condition is obtained in the case of differential equations. A calculus of pseudodifferential operators with inhomogeneous symbols of constant strength is constructed; the solubility of the Cauchy problem can be proved by means of this calculus for a certain class of differential equations with variable coefficients.

ν-invariants on manifolds with cylindrical end

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.

Relative zeta-determinants of Dirac Laplacians on a half-infinite cylinder with boundary conditions in the smooth, self-adjoint Grassmannian

Spectral properties of zeroth-order pseudodifferential operators

To a domain with conical points Ω, we associate a natural C*-algebra that is motivated by the study of boundary value problems on Ω, especially using the method of layer potentials. In two dimensions, we allow Ω to be a domain with ramified cracks. We construct an explicit groupoid associated to ∂Ω and use the theory of pseudodifferential operators on groupoids and its representations to obtain our layer potentials C*-algebra. We study its structure, compute the associated K-groups, and prove Fredholm conditions for the natural pseudodifferential operators affiliated to this C*-algebra.

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.

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.