Pseudo-differential operators with operator-valued symbols in the Mellin-edge-approach

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

The increasing complexity associated with many modern engineering applications, including autonomous robot guidance and navigation, process control in sensor-rich environments, and control of biological systems, has far-reaching implications for control system design. As an example, reactive, embedded software systems, interacting among themselves and remote users over communication networks, introduce a whole new set of system-level challenges, and classic control design objectives such as stability, performance, and robustness are being complemented with a number of new questions. These include the cost of hardware implementation, measured for example not only by computational requirements such as speed and memory, but also by communication requirements such as available communication bandwidth. Moreover, the complexity associated with specifying the control procedures and with verifying the behavior of the closed-loop system increasingly plays a fundamental role, especially in safety-critical control systems arising in energy and transportation networks, and in medical applications.

The paper deals with mixed boundary value problems and transmission problems for elliptic differential and pseudo-differential operators. These problems are considered as elements of an algebra (where Ω is the considered domain, x=∂Ω, Y⊂X a submanifold of codimension 1, where the conditions have a jump, and x∂ is a certain compactification of X/Y), the elements of which are described on a symbolic level. By means of additional conditions (of trace and potential type, with respect to Y) a given mixed problem is extended to an operator , and the Fredholm property of is established under an ellipticity condition, the parametrix is constructed on symbolic level and ind is expressed. In the second part the authors also study classes of special cases, namely differential operators of second and higher order with mixed boundary conditions for dim Ω≧2, where explicit index expressions are derived.

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 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.

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.

Mapping properties of operator-valued pseudo-differential operators

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.

We introduce the calculus of Mellin pseudodifferential operators with parameters based on "twisted" operator-valued Volterra symbols as well as the abstract Mellin calculus with holomorphic symbols. We establish the properties of the symbolic and operational calculi, and we give and make use of explicit oscillatory integral formulas on the symbolic side, e.g., for the Leibniz-product, kernel cut-off, and Mellin quantization. Moreover, we introduce the notion of parabolicity for the calculi of Volterra Mellin operators, and construct Volterra parametrices for parabolic operators within the calculi.

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.

where λ ( ξ) = (|ξ − ( )|2 + 1)1/2. For simplicity we suppose here that the mass of the particle is equal to 1. All the differential and pseudodifferential operators considered in this paper are, possibly unbounded, operators in 2(R ), defined on the Schwartz space S of rapidly decreasing smooth functions on R . In [14] it was proved that if the derivatives of any positive order of ∈ ∞(R ; R ) are bounded, then (λ ) is essentially selfadjoint on S. Let be its unique selfadjoint extension. In [16] the authors proved that if itself is bounded and if all its derivatives converge to zero at infinity, then the essential spectrum of is equal to the essential spectrum of √ + 1, where is the quantum nonrelativistic magnetic Hamiltonian with vector potential , i.e. the selfadjoint operator generated by the differential operator ( − ( ))2. We shall prove in this paper that the essential spectra of and of √ + 1 are still equal if we drop the condition of boundedness of . Thus, vector potentials which behave at infinity as | |1−e, e positive and arbitrary small, are allowed. More precisely, the main result of the paper is the following theorem.

Let E, F, H be Banach spaces, $$A,B\in L(E,F)$$ such that $$A+\lambda B$$ and its adjoint $$A^*+\lambda B^*$$ have a biorthogonal system of eigenvectors and associated vectors $$(y_i)_{i=1}^\infty$$ and $$(v_i)_{i=1}^\infty$$ . It is assumed that a closed finite codimensional subspace $$E_0$$ of E exists such that $$E_0$$ is dense in H and such that the restriction of B to $$E_0$$ has a continuous extension to H. It is shown that a subsystem $$(y_i)_{i=k+1}^\infty$$ is minimal in H if $$(v_i)_{i=1}^k$$ has a certain basis property. This abstract result can be applied to regular ordinary linear differential operators in Sobolev spaces whose boundary conditions may depend linearly on the eigenvalue parameter for proving minimality in $$L_p$$ . Explicit examples for Sturm–Liouville problems with one or both boundary conditions depending on $$\lambda$$ are considered.

We introduce new classes of modulation spaces over phase space. By means of the Kohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operators that bound their operator norms on Lp–spaces, Sobolev spaces, and modulation spaces.

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].

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.