Pseudodifferential operator calculus for generalized Q-rank 1 locally symmetric spaces, I
Daniel Grieser, Eugenie Hunsicker
备注：
44 pages, 2 figures -- Some explanations, references added; changed normalization of index sets in full calculus to make it more natural; made full calculus composition result more complete
This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on Q-rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior of a manifold with boundary, where the boundary has the structure of a tower of fibre bundles. The class of operators we consider on such a space includes those arising naturally from metrics which degenerate to various orders at the boundary, in directions given by the tower of fibrations. As well as Q-rank 1 locally symmetric spaces, examples include Ricci-flat metrics on the complement of a divisor in a smooth variety constructed by Tian and Yau. In this first part of the calculus construction, parametrices are found for "fully elliptic differential \bfa-operators", which are uniformly elliptic operators on these manifolds that satisfy an additional invertibility condition at infinity. In the second part we will consider operators that do not satisfy this condition.
We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements related to the magnetic Schroedinger operator in three and higher dimensions.
We establish the existence of a bounded $H_\infty$-calculus for a large class of hypoelliptic pseudodifferential operators on R^n and closed manifolds.
Let $X$ be a compact manifold with boundary. Suppose that the boundary is fibred, $ϕ:\pa X\longrightarrow Y,$ and let $x\in\CI(X)$ be a boundary defining function. This data fixes the space of `fibred cusp' vector fields, consisting of those vector fields $V$ on $X$ satisfying $Vx=O(x^2)$ and which are tangent to the fibres of $ϕ;$ it is a Lie algebra and $\CI(X)$ module. This Lie algebra is quantized to the `small calculus' of pseudodifferential operators $\PsiF*(X).$ Mapping properties including boundedness, regularity, Fredholm condition and symbolic maps are discussed for this calculus. Th…
This paper deals with positivity properties for a pseudodifferential calculus, generalizing Weyl's classical quantization, and set on an infinite dimensional phase space, the Wiener space. In this frame, we show that a positive symbol does not, in general, give a positive operator. In order to measure the nonpositivity, we establish a Gårding's inequality, which holds for the symbol classes at hand. Nevertheless, for symbols with radial aspects, additional assumptions ensure the positivity of the associated operator.
Polyhomogeneous symbols, defined by Kohn-Nirenberg and Hörmander in the 60's, play a central role in the symbolic calculus of most pseudodifferential calculi. We prove a simple characterisation of polyhomogeneous functions which avoids the use of asymptotic expansions. Specifically, if $U$ is open subset of $\mathbb{R}^d$, then a polyhomogeneous symbol on $U \times \mathbb{R}^d$ is precisely the restriction to $t=1$ of a function on $U \times \mathbb{R}^{d+1}$ which is homogeneous for the dilations of $\mathbb{R}^{d+1}$ modulo Schwartz class functions. This result holds for arbitrary graded di…
Parameter-ellipticity with respect to a closed subsector of the complex plane for pseudodifferential Douglis-Nirenberg systems is discussed and shown to imply the existence of a bounded H_\infty-calculus in suitable scales of Sobolev, Besov, and Hoelder spaces. We also admit non pseudodifferential perturbations. Applications concern systems with coefficients of mild Hoelder regularity and the generalized thermoelastic plate equations.
In this paper, we are interested in the construction of a bilinear pseudodifferential calculus. We define some symbolic classes which contains those of Coifman-Meyer. These new classes allow us to consider operators closely related to the bilinear Hilbert transform. We give a description of the action of our bilinear operators on Sobolev spaces. These classes also have a ``nice'' behavior through the transposition and the composition operations that we will present.
In this paper we expand on B.-W. Schulze's abstract edge pseudodifferential calculus and introduce a larger class of operators that is modeled on Hörmander's $\varrho,δ$ calculus, where $0 \leq δ< \varrho \leq 1$. This expansion is motivated by recent work on boundary value problems for elliptic wedge operators with variable indicial roots by G. Mendoza and the author, where operators of type $1,δ$ for $0 < δ< 1$ appear naturally. Some of the results of this paper also represent improvements over the existing literature on the standard abstract edge calculus of operators of type $1,0$, such as…
We characterize the Schwartz kernels of pseudodifferential operators of Shubin type by means of an FBI transform. Based on this we introduce as a generalization a new class of tempered distributions called Shubin conormal distributions. We study their transformation behavior, normal forms and microlocal properties.
R. B. Melrose's b-calculus provides a framework for dealing with problems of partial differential equations that arise in singular or degenerate geometric situations. This article is a somewhat informal short course introducing many of the basic ideas of this world, assuming little more than a basic analysis and manifold background. As examples, classical pseudodifferential operators on manifolds and b-pseudodifferential (also known as totally characteristic) operators on manifolds with boundary are discussed.
A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential operators act continuously on Sobolev spaces and the loss of derivatives may be controled by the order of the operator. Although a large number of works have been devoted in the past to the construction and the study of algebras of variable-coefficient operators, including some very interesting works on the Heisenberg group, our approach is different, and in particular puts into light microloc…
This paper has been withdrawn by the authors. A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential operators act continuously on Sobolev spaces and the loss of derivatives may be controled by the order of the operator. Although a large number of works have been devoted in the past to the construction and the study of algebras of variable-coefficient operators, including some very interesting works on the Heisenberg group, our approach is differe…
One way to geometrically encode the singularities of a stratified pseudomanifold is to endow its interior with an iterated fibred cusp metric. For such a metric, we develop and study a pseudodifferential calculus generalizing the Φ-calculus of Mazzeo and Melrose. Our starting point is the observation, going back to Melrose, that a stratified pseudomanifold can be `resolved' into a manifold with fibred corners. This allows us to define pseudodifferential operators as conormal distributions on a suitably blown-up double space. Various symbol maps are introduced, leading to the notion of full ell…
We develop a singular pseudodifferential calculus. The symbols that we consider do not satisfy the standard decay with respect to the frequency variables. We thus adopt a strategy based on the Calderon-Vaillancourt Theorem. The remainders in the symbolic calculus are bounded operators on $L^2$, whose norm is measured with respect to some small parameter. Our main improvement with respect to an earlier work by Williams consists in showing a regularization effect for the remainders. Due to a nonstandard decay in the frequency variables, the regularization takes place in a scale of anisotropic, a…
A new symbol theory for pseudodifferential operators in the complex analytic category is given. This theory provides a cohomological foundation of symbolic calculus.
We produce, on general homogeneous groups, an analogue of the usual Hörmander pseudodifferential calculus on Euclidean space, at least as far as products and adjoints are concerned. In contrast to earlier works, we do not limit ourselves to analogues of classical symbols, nor to the Heisenberg group. The key technique is to understand ``multipliers'' of any given order j, and the operators of convolution with their inverse Fourier transforms, which we here call convolution operators of order j. (Here a ``multiplier'' is an analogue of a Hörmander-type symbol a(x,ξ), which is independent of x.)…
We construct an abstract pseudodifferential calculus with operator-valued symbol, adapted to the treatment of Coulomb-type interactions, and we apply it to study the quantum evolution of molecules in the Born-Oppenheimer approximation, in the case where the electronic Hamiltonian admits a local gap in its spectrum. In particular, we show that the molecular evolution can be reduced to the one of a system of smooth semiclassical operators, the symbol of which can be computed explicitely. In addition, we study the propagation of certain wave packets up to long time values of Ehrenfest order. (Thi…
In this paper we establish explicit lower bounds for pseudodifferential operators with a radial symbol. The proofs use classical Weyl calculus techniques and some useful, if not celebrated, properties of the Laguerre polynomials.
The Volterra calculus is a simple and powerful pseudodifferential tool for inverting parabolic equations and it has also found many applications in geometric analysis. On the other hand, an important property in the theory of pseudodifferential operators is the asymptotic completeness, which allows us to construct parametrices modulo smoothing operators. In this paper we present new and fairly elementary proofs the asymptotic completeness of the Volterra calculus.
Specific global symbol classes and corresponding pseudodifferential operators of infinite order that act continuously on the space of tempered ultradistributions of Beurling and Roumieu type are constructed. For these classes, symbolic calculus is developed.
We extend the classical regularity theorem of elliptic operators to maximally hypoelliptic differential operators. More precisely, given vector fields $X_1,\ldots,X_m$ on a smooth manifold which satisfy Hörmander's bracket generating condition, we define a principal symbol for \textit{any} linear differential operator. Our symbol takes into account the vector fields $X_i$ and their commutators. We show that for an arbitrary differential operator, its principal symbol is invertible if and only if the operator is maximally hypoelliptic. This answers affirmatively a conjecture due to Helffer and …
We present in this paper the construction of a pseudodifferential calculus on smooth non-compact manifolds associated to a globally defined and coordinate independant complete symbol calculus, that generalizes the standard pseudodifferential calculus on $\R^n$. We consider the case of manifolds $M$ with linearization in the sense of Bokobza-Haggiag, such that the associated (abstract) exponential map provides global diffeomorphisms of $M$ with $\R^n$ at any point. Cartan--Hadamard manifolds are special cases of such manifolds. The abstract exponential map encodes a notion of infinity on the ma…
Starting out from a new description of a class of parameter-dependent pseudodifferential operators with finite regularity number due to G. Grubb, we introduce a calculus of parameter-dependent, poly-homogeneous symbols whose homogeneous components have a particular type of point-singularity in the covariable-parameter space. Such symbols admit intrinsically a second kind of expansion which is closely related to the expansion in the Grubb-Seeley calculus and permits to recover the resolvent-trace expansion for elliptic pseudodifferential oerators originally proved by Grubb-Seeley. Another appli…
We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the Kohn-Nirenberg calculus we introduce a version of Sjoestrand's class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since "hard analysis" techniques are not available on locally compact abelian groups, a new time-frequency approach is used with the emphasis on modulation spaces, Gabor frames, and Banach algebras of matrices. Sjoestrand's original results are thus understood as a phenomenon of abstract harmonic analysis rather …
Mixed-norm Lebesgue spaces found their place in the study of some questions in the theory of partial differential equations, as can be seen from recent interest in the continuity of certain classes of pseudodifferential operators on these spaces. In this paper, we use some recent advances in the pseudodifferential calculus for nonsmooth symbols to prove the boundedness of pseudodifferential operators with such symbols on mixed-norm Lebesgue spaces.
We report some recent results on analytic pseudodifferential operators, also known as Wick operators. An important tool in our study is the Bargmann transform which provides a coupling between the classical (real) and analytic pseudodifferential calculus. Since the Bargmann transform of Hermite functions gives rise to formal power series in the complex domain, the results are formulated in terms of the Bargmann images of Pilipović spaces.
The aim of this article is to prove a Beals type characterization theorem for pseudodifferential operators in Wiener spaces. The definition of pseudodifferential operators in Wiener spaces and a Calderón-Vaillancourt type result appear in [1]. The set of symbols considered here is the one of [1]. The Weyl calculus in infinite dimension considered here emphasizes the role of the Wick bi-symbols.
We study the change of quantization for a class of global pseudodifferential operators of infinite order in the setting of ultradifferentiable functions of Beurling type. The composition of different quantizations as well as the transpose of a quantization are also analysed, with applications to the Weyl calculus. We also compare global $ω$-hypoellipticity and global $ω$-regularity of these classes of pseudodifferential operators.
We develop a theory of pseudodifferential operators of infinite order for the global classes $\mathcal{S}_ω$ of ultradifferentiable functions in the sense of Björck, following the previous ideas given by Prangoski for ultradifferentiable classes in the sense of Komatsu. We study the composition and the transpose of such operators with symbolic calculus and provide several examples.
In this paper a bisingular pseudodifferential calculus, along the lines of the one introduced by L. Rodino in [12], is developed in the global setting of a product of compact Lie groups. The approach follows that introduced by M. Ruzhansky and V. Turunen [13] (see also V. Fischer [5]), in that it exploits the harmonic analysis of the groups involved.
We establish sharp $L^2$-Sobolev estimates for classes of pseudodifferential operators with singular symbols whose non-pseudodifferential (Fourier integral operator) parts exhibit two-sided fold singularities. The operators considered include both singular integral operators along curves in $R^2$ with simple inflection points and normal operators arising in linearized seismic imaging in the presence of fold caustics.
This paper presents the construction of parametrices for the Gauss-Bonnet and Hodge Laplace operators on noncompact manifolds modelled on Q-rank 1 locally symmetric spaces. These operators are, up to a scalar factor, $ϕ$-differential operators, that is, they live in the generalised $ϕ$-calculus studied by the authors in a previous paper, which extends work of Melrose and Mazzeo. However, because they are not totally elliptic elements in this calculus, it is not possible to construct parametrices for these operators within the $ϕ$-calculus. We construct parametrices for them in this paper using…
Pseudodifferential Weyl calculus on vector bundles
Royal Institute of Technology（皇家理工学院）；University of Vienna, Faculty of Physics（维也纳大学物理学院）；Faculty for Mathematics, University of Regensburg（雷根斯堡大学数学系）；Faculty for Mathematics, TU Chemnitz（开姆尼茨工业大学数学系）；Department of Mathematics, University of Genoa（热那亚大学数学系）；INFN-Section Genoa and INdAM-GNFM（意大利国家核物理研究所热那亚分部及印度数学与应用研究网络）
The gauge covariant magnetic Weyl calculus has been introduced and studied in previous works. We prove criteria in terms of commutators for operators to be magnetic pseudo-differential operators of suitable symbol classes. The approach is completely intrinsic; neither the statements nor the proofs depend on a choice of a vector potential. We apply this criteria to inversion problems, functional calculus, affiliation results and to the study of the evolution group generated by a magnetic pseudo-differential operator.
One can argue that on flat space $\mathbb{R}^d$ the Weyl quantization is the most natural choice and that it has the best properties (e.g. symplectic covariance, real symbols correspond to Hermitian operators). On a generic manifold, there is no distinguished quantization, and a quantization is typically defined chart-wise. Here we introduce a quantization that, we believe, has the best properties for studying natural operators on pseudo-Riemannian manifolds. It is a generalization of the Weyl quantization - we call it the balanced geodesic Weyl quantization. Among other things, we prove that …
We study pseudodifferential boundary value problems in the context of the Boutet de Monvel calculus or Green operators, with nonsmooth coefficients on smooth compact manifolds with boundary. In order to have a definition that is independent of the choice of (smooth) coordinates, we prove that nonsmooth Green operators are invariant under smooth coordinate transformations.
Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue…
By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on $\R^n$ as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof is based on an effective use of the Weyl calculus and the Fefferman-Phong inequality.
This is a review of some coordinate-free calculi of pseudodifferential operators developed in the last years. As an application, we use a coordinate-free calculus to obtain new results on the behaviour of the spectral projections of a self-adjoint elliptic second order differential operator under perturbation of coefficients.
Summarizing basic facts from abstract topological modules over Colombeau generalized complex numbers we discuss duality of Colombeau algebras. In particular, we focus on generalized delta functionals and operator kernels as elements of dual spaces. A large class of examples is provided by pseudodifferential operators acting on Colombeau algebras. By a refinement of symbol calculus we review a new characterization of the wave front set for generalized functions with applications to microlocal analysis.
In previous papers, a generalization of the Weyl calculus was introduced in connection with the quantization of a particle moving in $\mathbb R^n$ under the influence of a variable magnetic field $B$. It incorporates phase factors defined by $B$ and reproduces the usual Weyl calculus for B=0. In the present article we develop the classical pseudodifferential theory of this formalism for the standard symbol classes $S^m_{ρ,δ}$. Among others, we obtain properties and asymptotic developments for the magnetic symbol multiplication, existence of parametrices, boundedness and positivity results, pro…
We study the travel time tomography problem for transversely isotropic media using the artificial boundary method used in works in X-ray tomography and boundary rigidity problems. The analysis involved requires a modification of the scattering pseudodifferential calculus in order to provide parametrices for operators which are degenerately elliptic; such a calculus is constructed in this paper and used to solve the tomography problem of interest.
Boundedness properties for pseudodifferential operators with symbols in the bilinear Hörmander classes of sufficiently negative order are proved. The results are obtained in the scale of Lebesgue spaces and, in some cases, end-point estimates involving weak-type spaces and BMO are provided as well. From the Lebesgue space estimates, Sobolev ones are then easily obtained using functional calculus and interpolation. In addition, it is shown that, in contrast with the linear case, operators associated with symbols of order zero may fail to be bounded on products of Lebesgue spaces.
The general theory of boundary value problems for linear elliptic wedge operators (on smooth manifolds with boundary) leads naturally, even in the scalar case, to the need to consider vector bundles over the boundary together with general smooth fiberwise multiplicative group actions. These actions, essentially trivial (and therefore invisible) in the case of regular boundary value problems, are intimately connected with what passes for Poisson and trace operators, and to pseudodifferential boundary conditions in the more general situation. Here the part of the theory pertaining pseudodifferen…
In this first part of the paper, we define a natural dual object for manifolds with corners and show how pseudodifferential calculus on such manifolds can be constructed in terms of the localization principle in C*-algebras. In the second part, these results will be applied to the solution of Gelfand's problem on the homotopy classification of elliptic operators for the case of manifolds with corners.
We study almost periodic pseudodifferential operators acting on almost periodic functions $G_{\rm ap}^s(\rr d)$ of Gevrey regularity index $s \geq 1$. We prove that almost periodic operators with symbols of Hörmander type $S_{ρ,δ}^m$ satisfying an $s$-Gevrey condition are continuous on $G_{\rm ap}^s(\rr d)$ provided $0 < ρ\leq 1$, $δ=0$ and $s ρ\geq 1$. A calculus is developed for symbols and operators using a notion of regularizing operator adapted to almost periodic Gevrey functions and its duality. We apply the results to show a regularity result in this context for a class of hypoelliptic …
Classically, Gohberg-type Lemmas provide lower bounds for the distance of suitable pseudodifferential operators acting in a Hilbert space to the ideal of compact operators, in terms of "the behavior of the symbol at infinity". In this article the pseudodifferential operators are associated to a compact Abelian group $X$ and an important role is played by its Pontryagin dual $\widehat X$. Hörmander-type classes of symbols are not always available; they will be replaced by crossed product $C^*$-algebras involving a vanishing oscillation condition, which anyway is more general even in the particu…
We consider here pseudo-differential operators whose symbol $σ(x,ξ)$ is not infinitely smooth with respect to $x$. Decomposing such symbols into four -sometimes five- components and using tools of paradifferential calculus, we derive sharp estimates on the action of such pseudo-differential operators on Sobolev spaces and give explicit expressions for their operator norm in terms of the symbol $σ(x,ξ)$. We also study commutator estimates involving such operators, and generalize or improve the so-called Kato-Ponce and Calderon-Coifman-Meyer estimates in various ways.
We study the normal operator to the geodesic X-ray transform on functions in the setting of simple asymptotically hyperbolic manifolds. We construct a parametrix for the normal operator in the 0-pseudodifferential calculus and use it show a stability estimate.
We introduce a calculus for parameter-dependent singular Green operators on the half-space $\mathbb{R}^n_+$ that combines both elements of Grubb's calculus for boundary value problems of finite regularity and techniques of Schulze's calculus for pseudodifferential operators on manifolds with edges.
First, we reconsider the magnetic pseudodifferential calculus and show that for a large class of non-decaying symbols, their corresponding magnetic pseudodifferential operators can be represented, up to a global gauge transform, as generalized Hofstadter-like, bounded matrices. As a by-product, we prove a Calderón-Vaillancourt type result. Second, we make use of this matrix representation and prove sharp results on the spectrum location when the magnetic field strength $b$ varies. Namely, when the operators are self-adjoint, we show that their spectrum (as a set) is at least $1/2$-Hölder conti…
