On the solvability of systems of pseudodifferential operators
Nils Dencker
备注：
Changed Definition 2.5 and corrected the proof of Proposition 2.12. Rewrote Section 2, corrected errors and misprints. Corrected some references and the formulation of Theorem 2.7 and Remark 6.1. The paper has 40 pages
The paper studies the solvability for square systems of pseudodifferential operators. We assume that the system is of principal type, i.e., the principal symbol vanishes of first order on the kernel. We shall also assume that the eigenvalues of the principal symbol close to zero have constant multiplicity. We prove that local solvability for the system is equivalent to condition (PSI) on the eigenvalues of the principal symbol. This condition rules out any sign changes from - to + of the imaginary part of the eigenvalue when going in the positive direction on the bicharacteristics of the real part. Thus we need no conditions on the lower order terms. We obtain local solvability by proving a localizable a priori estimate for the adjoint operator with a loss of 3/2 derivatives (compared with the elliptic case).
The paper studies the local solvability and subellipticity for square systems of principal type. These are the systems for which the principal symbol vanishes of first order on its kernel. For systems of principal type having constant characteristics, local solvability is equivalent to condition (PSI) on the eigenvalues, see arXiv:0801.4043. This is a condition on the sign changes of the imaginary part of the eigenvalue along the oriented bicharacteristics of the real part. In the generic case when the principal symbol does not have constant characteristics, condition (PSI) is not sufficient, …
On the solvability of the mixed problem for a fractional-order equation with a time-delay argument and pseudodifferential operators with non-local boundary conditions in Sobolev classes
We provide sufficient conditions of local solvability for partial differential operators with variable Colombeau coefficients. We mainly concentrate on operators which admit a right generalized pseudodifferential parametrix and on operators which are a bounded perturbation of a differential operator with constant Colombeau coefficients. The local solutions are intended in the Colombeau algebra $\G(\Om)$ as well as in the dual $\LL(\Gc(\Om),\wt{\C})$.
For a principal type pseudodifferential operator, we prove that condition (psi) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker's paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from 2 (Dencker's result) to 3/2 (the present paper). It is already known that condition (psi) does not imply local solvability with a loss of 1 derivative, so we have to content ourselves with a loss >1.
Following the work of Duistermaat-Singer \cite{DS} on isomorphisms of algebras of global pseudodifferential operators, we classify isomorphisms of algebras of microlocally defined semiclassical pseudodifferential operators. Specifically, we show that any such isomorphism is given by conjugation by $A = BF$, where $B$ is a microlocally elliptic semiclassical pseudodifferential operator, and $F$ is a microlocal $h$-FIO associated to the graph of a local symplectic transformation.
A new symbol theory for pseudodifferential operators in the complex analytic category is given. This theory provides a cohomological foundation of symbolic calculus.
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.
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.
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 construct parametrices for a class of pseudodifferential operators of infinite order acting on spaces of tempered ultradistributions of Beurling and Roumieu type. As a consequence we obtain a result of hypoellipticity in these spaces.
In this paper we investigate $L^p$ and Sobolev boundedness of a certain class of pseudodifferential operators with non-regular symbols. We employ regularisation methods, namely convolution with a net of mollifiers $(ρ_\eps)_\eps$, and we study the corresponding net of pseudodifferential operators providing $L^p$ and Sobolev estimates which relate the parameter $\eps$ with the non-regularity of the symbol.
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 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.
A generalization of differential operators are pseudodifferential operators which are used for reasoning about partial differential equations with variable coefficients. A lot of useful properties about classical pseudodifferential operators are already discovered by the works of H. Kumano-Go and L. Hörmander. In this work, we like to transfer their ideas on those operators which symbols are merely Hölder-continous. We focus on the composition of operators, symbol smoothing and operators on manifolds.
The aim of these notes is to describe some recent results concerning dispersive estimates for principally normal pseudodifferential operators. The main motivation for this comes from unique continuation problems. Such estimates can be used to prove Carleman inequalities, which in turn yield unique continuation results for various partial differential operators with rough potentials.
We characterize the action of isotropic pseudodifferential operators on functions in terms of their action on Hermite functions. We show that an operator $A : S(\mathbb{R}) \to S(\mathbb{R})$ is an isotropic pseudodifferential operator of order r if and only if its "matrix" $(K(A))_{m,n} := < Aϕ_n,ϕ_m>_{L^2(\mathbb{R})}$ is rapidly decreasing away from the diagonal $\{m = n\}$, order $\frac {r}{2}$ in $m + n$, and where applying the discrete difference operator along the diagonal decreases the order by one. Additionally, we use this result to prove an analogue of Beal's theorem for isotropic p…
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.
We study the boundedness of rough Fourier integral and pseudodifferential operators, defined by general rough Hörmander class amplitudes, on Banach and quasi-Banach $L^p$ spaces. Thereafter we apply the aforementioned boundedness in order to improve on some of the existing boundedness results for Hörmander class bilinear pseudodifferential operators and certain classes of bilinear (as well as multilinear) Fourier integral operators. For these classes of amplitudes, the boundedness of the aforementioned Fourier integral operators turn out to be sharp. Furthermore we also obtain results for roug…
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.
We consider toroidal pseudodifferential operators with operator-valued symbols, their mapping properties and the generation of analytic semigroups on vector-valued Besov and Sobolev spaces. We show that a parabolic toroiodal pseudodifferential operator generates an analytic semigroup both on Banach space-valued Besov spaces and on Banach space-valued Sobolev spaces. Here, no condition on the Banach space is imposed. For the proof of the Sobolev space result, we establish a uniform estimate on the kernel which is given as an infinite parameter-dependent sum. An application to abstract non-auton…
In the first part of the paper the authors study the minimal and maximal extension of a class of weighted pseudodifferential operators in the Fréchet space $L^p_{\rm loc}(Ω)$. 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.
We define classes of pseudodifferential operators on $G$-bundles with compact base and give a generalized $L^2$ Fredholm theory for invariant operators in these classes in terms of von Neumann's $G$-dimension. We combine this formalism with a generalized Paley-Wiener theorem, valid for bundles with unimodular structure groups, to provide solvability criteria for invariant operators. This formalism also gives a basis for a $G$-index for these operators. We also define and describe a transversal dimension and its corresponding Fredholm theory in terms of anisotropic Sobolev estimates, valid also…
In this paper, we consider pseudodifferential operators on the torus with operator-valued symbols and prove continuity properties on vector-valued toroidal Besov spaces, without assumptions on the underlying Banach spaces. The symbols are of limited smoothness with respect to $x$ and satisfy a finite number of estimates on the discrete derivatives. The proof of the main result is based on a description of the operator as a convolution operator with a kernel representation which is related to the dyadic decomposition appearing in the definition of the Besov space.
An approximation Ansatz for the operator solution, $U(z',z)$, of a hyperbolic first-order pseudodifferential equation, $\d_z + a(z,x,D_x)$ with $\Re (a) \geq 0$, is constructed as the composition of global Fourier integral operators with complex phases. An estimate of the operator norm in $L(H^{(s)},H^{(s)})$ of these operators is provided which allows to prove a convergence result for the Ansatz to $U(z',z)$ in some Sobolev space as the number of operators in the composition goes to $\infty$.
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 obtain new semiclassical estimates for pseudodifferential operators with low regular symbols. Such symbols appear naturally in a Cauchy Problem related to recent weak solutions to the unstable Muskat problem constructed via convex integration in [CCF16]. In particular, our new estimates reveal the tight relation between the speed of opening of the mixing zone and the regularity of the interphase.
Motivated by the work of Colin de Verdière and Saint-Raymond on spectral theory 0th order pseudodifferential operators on tori we consider viscosity limits in which 0th order operators $ P $ are replaced by $ P + i νΔ$, $ ν> 0 $. By adapting the Helffer--Sjöstrand theory of scattering resonances we show that in a complex neighbourhood of the continuous spectrum eigenvalues of $ P + i νΔ$ have limits as viscosity $ ν$ goes to 0. In the simplified setting of tori this justifies claims made in the physics literature.
We will present versions of the Rellich-Kondrachov theorem for pseudo-differential operators acting on localizable Hardy spaces. One of the techniques includes boundedness properties for pseudodifferential operators with symbols in the Hörmander class, that might be of interest in his own right, extending results previously obtained by Goldberg, Alvarez and Hounie, Taylor - among others. As application, we obtain compact embedding results for distributions in nonhomogeneous localizable Hardy-Sobolev spaces.
In this paper we establish some endpoint estimates for bilinearpseudodifferential operators with symbol in the class BS^m_{1,1}, involving the space of functions with local bounded mean oscillation bmo. As a consequence we also obtain an endpoint estimate of Kato-Ponce type.
Let $P$ be a classical pseudodifferential operator of complex order $m$ on an $n$-dimensional smooth manifold $Ω_1$. For the truncation $P_Ω$ to a smooth subset $Ω$ there is a well-known theory of boundary value problems when $P_Ω$ has the transmission property (preserves $C^\infty (\barΩ)$) and is of integer order; the calculus of Boutet de Monvel. Many interesting operators, such as for example complex powers of the Laplacian $(-Δ)^μ$ with noninteger mu, are not covered. They have instead the mu-transmission property defined in Hörmander's books, mapping $x_n^μC^\infty (\barΩ)$ into $C^\inft…
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…
In this paper we consider the solvability of pseudodifferential operators in the case when the principal symbol vanishes of order $k \ge 2 $ at a nonradial involutive manifold $Σ_2$. We shall assume that the operator is of subprincipal type, which means that the $ k$:th inhomogeneous blowup at $Σ_2$ of the refined principal symbol is of principal type with Hamilton vector field parallel to the base $Σ_2$, but transversal to the symplectic leaves of $Σ_2$ at the characteristics. When $k = \infty $ this blowup reduces to the subprincipal symbol. We also assume that the blowup is essentially cons…
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…
We shall study the solvability of pseudodifferential operators which are not of principal type. The operator will have real principal symbol and we shall consider the limits of bicharacteristics at the set where the principal symbol vanishes of at least second order. The convergence shall be as smooth curves, then the limit bicharacteristic is a smooth curve. We shall also need uniform bounds on the curvature of the characteristics at the bicharacteristics, but only along the tangents of a given Lagrangean manifold. This gives uniform bounds on the linearization of the normalized Hamilton flow…
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…
We use microlocal radial estimates to prove the full limiting absorption principle for $P$, a self-adjoint 0th order pseudodifferential operator satisfying hyperbolic dynamical assumptions as of Colin de Verdière and Saint-Raymond. We define the scattering matrix for $P-ω$ with generic $ω\in \mathbb R$ and show that the scattering matrix extends to a unitary operator on appropriate $L^2$ spaces. After conjugation with natural reference operators, the scattering matrix becomes a $0$th order Fourier integral operator with a canonical relation associated to the bicharacteristics of $P-ω$. The ope…
For a class of zero order pseudodifferential operators we find the asymptotics of eigenvalues converging to a non-isolated tip of the essential spectrum.
We investigate properties of pseudodifferential operators on $L^2$ space on manifold with ends including asymptotically conical or hyperbolic ends. Our pseudodifferential operators are a generalization of the canonical quantization which naturally appears in the quantum mechanics on curved spaces. We prove a Calderón-Vaillancourt type theorem for our pseudodifferential operators and discuss a construction of parametrix of elliptic differential operators on manifolds with ends.
Motivated by the recent paper of Boggiatto-Garello in J. Pseudo-Differ. Oper. Appl. \textbf{11} (2020), 93-117, where a Gabor operator is regarded as pseudodifferential operator with symbol $p(x,ω)$ periodic on both the variables, we study the continuity and invertibility, on general time frequency invariant spaces, of pseudodifferential operators with completely periodic symbol and general $τ$ quantization.
We investigate global microlocal properties of localization operators and Shubin pseudodifferential operators. The microlocal regularity is measured in terms of a scale of Shubin-type Sobolev spaces. In particular, we prove microlocality and microellipticity of these operators.
We establish the existence of a bounded $H_\infty$-calculus for a large class of hypoelliptic pseudodifferential operators on R^n and closed manifolds.
We construct a parametrix of a resolvent of elliptic differential operators acting on half-densities on manifolds with ends. The construction is carried out by introducing suitable pseudodifferential operators compatible with the end structure. Our class of pseudodifferential operators and symbols is independent of the choice of Riemannian metric on the manifold and applicable to both of asymptotically conical and hyperbolic manifolds. As an application, we prove the essential self-adjointness of elliptic symmetric differential operators on manifolds.
Smooth pseudodifferential operators on $\mathbb{R}^n$ can be characterized by their mapping properties between $L^p-$Sobolev spaces due to Beals and Ueberberg. In applications such a characterization would also be useful in the non-smooth case, for example to show the regularity of solutions of a partial differential equation. Therefore, we will show that every linear operator $P$, which satisfies some specific continuity assumptions, is a non-smooth pseudodifferential operator of the symbol-class $C^τ S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$. The main new difficulties are the limited mapp…
In this paper we discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in Hölder spaces. In analogy to the proof in the smooth case of Beals and Ueberberg, we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results …
We study the dynamics of resonances of analytic perturbations of 0th order pseudodifferential operators $P(s)$. In particular, we prove a Fermi golden rule for resonances of $P(s)$ at embedded eigenvalues of $P=P(0)$. We also study the dynamics of eigenvalues of $P(t)=P+itΔ$ as the eigenvalues converge to simple eigenvalues of $P$. The 0th order pseudodifferential operators we consider satisfy natural dynamical assumptions and are used as microlocal models of internal waves.
The L2-norm closure of the algebra of all zero-order classical (or polyhomogeneous) pseudodifferential operators on a closed manifold $X$ is proven not to contain Hörmander's class $Ψ^0(X)$. A set of generators (for that closed algebra) smaller than all zero-order classical pseudos is also described.
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…
The paper concerns algebras of almost periodic pseudodifferential operators on $\mathbb R^d$ with symbols in Hörmander classes. We study three representations of such algebras, one of which was introduced by Coburn, Moyer and Singer and the other two inspired by results in probability theory by Gladyshev. Two of the representations are shown to be unitarily equivalent for nonpositive order. We apply the results to spectral theory for almost periodic pseudodifferential operators acting on $L^2$ and on the Besicovitch Hilbert space of almost periodic functions.
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.
We give a complete characterization of the continuity of pseudodifferential operators with symbols in modulation spaces $M^{p,q}$, acting on a given Lebesgue space $L^r$. Namely, we find the full range of triples $(p,q,r)$, for which such a boundedness occurs. More generally, we completely characterize the same problem for operators acting on Wiener amalgam space $W(L^r,L^s)$ and even on modulation spaces $M^{r,s}$. Finally the action of pseudodifferential operators with symbols in $W(\Fur L^1,L^\infty)$ is also investigated.
