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Questions tagged [pseudo-differential-operators]

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Conjugation of pseudodifferential operators

Let $A$ be a pseudodifferential operator of order $m$ with symbol $a(x,\xi)\in S_{1,0}^m$ where $x,\xi \in \mathbb{R}^n$. We define an operation called conjugation of $A$ given by \begin{equation} A_\...
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Dixmier traces, Wodzicki residue and residues of zeta functions

Let $M$ be an $n$ dimensional closed manifold and consider an elliptic, pseudodifferential operator $P$ of order $-n$. Here are some facts which I had learned so far: 1. There exists a density defined ...
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1answer
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Associating a pseudo-differential operator to the symbol in the SG setting

We all know that given a symbol $a(x,\xi) \in S^{\mu,\rho}(\mathbb{R}^n,\mathbb{R}^n)$, a pseudo-differential operator can be defined as \begin{equation} Op(a)u(x)=(2\pi)^{-n}\int \int e^{i(x-x')\cdot ...
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Extension of elliptic complex to an exact sequence

This questions concerns elliptic complexes and is closely related to Green's operator of elliptic differential operator. Let $T_f:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial ...
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Examples of unbounded pseudo-differential operators in $L^{\infty}$

During my graduate studies I've been told that pseudo-differential operators with symbols in $S^0=S^0_{1,0}$ (the simplest class) are bounded $L^2 \to L^2$, and also $L^p \to L^p$ for all $p \in (1, \...
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115 views

Support of a microlocal defect measure

I'm trying to complete the proof of the Theorem 6.1 in the notes https://www.math.u-psud.fr/~nb/articles/coursX.pdf, which ensures, under certain conditions, that the support of the microlocal defect ...
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1answer
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References for the study of parameter dependent symbols $s(t,x,\xi)$ having low regularity in parameter($t$)

I am currently studying parameter dependent symbols, $s(t,x,\xi)$, where $t\in [0,1],x\in \Omega, \xi \in \mathbb{R^n}$. I wanted to know how the low regularity (for example, $s$ is just continuous w....
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Integrability, quantum ergodicity, and observable algebra

Consider (for simplicity and definiteness) the Laplacian on a compact Riemannian manifold $M$. Let $\phi_k$, $E_k$ be its eigenfunctions and eigenvalues in increasing order. Quantum ergodicity is ...
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Parametrix of external product of elliptic operators

Let $S$ and $T$ be Dirac-type operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. ...
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Is this a pseudodifferential operator?

Let $M$ be a non-compact manifold and $D$ a first-order self-adjoint elliptic differential operator on $M$. Then is the bounded operator $$A:=\sqrt{(D^2+1)^{-1}}:L^2(M)\rightarrow H^1(M)$$ a ...
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Inverse of holomorphic elliptic differential operator

Consider the Beltrami-Laplacian $\Delta$ on $\mathbb{S}^n$ with standard metric. One can define a family of operators $A(z):H^1(\mathbb{S}^n)\to H^1(\mathbb{S}^n)$ as the following $$A(z)=\Delta+z$$ ...
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Barry Simon's decay of eigenfunctions for pseudo differential operators

In his celebrated paper on Schrödinger semigroups, Barry Simon proves the following result. Let $V_{-} \in K_\nu$, $V_{+} \in K_\nu^{\mathrm{loc}}$ and suppose that $Hu=Eu$ where $E$ is the ...
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1answer
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Expressing a matrix operator as a sum of an identity and a compact operator

My problem concerns with the unique solvability of a linear system of integral equations. In my problem, as I was able to write the system in matrix form: $$ M \begin{align} \begin{bmatrix} ...
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Exact sequence of Quillen metrics

Let $M$ be a compact Riemann surface equipped with its Arakelov metric. Let $\xi$ be a holomorphic line bundle on $M$ equipped with an admissble metric. For any point $P\in M$ we get an admissible ...
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1answer
64 views

Positive form for a homogeneous elliptic pde

I have a pde of the following form: \begin{align} &P(x,D)u = f \text{ on } \Omega, \\ &P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha}, \end{align} where one can assume that $f$ ...
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Bound of analytic torsion for a line bundle

Let $E\rightarrow X$ be a line bundle over a compact Riemann surface. Let $\Delta_{E}$ be the Dolbeault Laplacian $\Delta_{\overline{\partial}}$ extended to sections of $E$. Let $g_1, g_2$ be two ...
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One-parameter unitary group preserving invariant domain of infinitesimal generator

Let $\mathcal{H}$ be a separable Hilbert space (e.g. $L^{2}(\mathbb{R}^{d}))$, and let $\mathcal{D}_{1}\subset\mathcal{H}$ be a dense subspace (e.g. $\mathcal{S}(\mathbb{R}^{d})$). Suppose that an ...
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1answer
126 views

Accessible reference for (scattering) $\Psi DO$'s on manifolds

I am currently trying to understand Hassell, Tao, and Wunsch's paper on Strichartz estimates on non-trapping asymptotically conic manifolds, however, my understanding of pseudodifferential operators ...
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250 views

Propagation of Singularities

I'm following the book "Elementary Introduction To The Theory Of Pseudodifferential Operators" by X. S. Raymond and the Joshi Lectures Notes - https://arxiv.org/pdf/math/9906155.pdf - to prove the ...
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1answer
188 views

Do Degree Zero Pseudo-Differential Operators on a Manifold Send Smooth Functions to Smooth Functions?

I'm not an analyst, so forgive me if what I'm asking is not suitable for Mathoverflow. For convenience, let $X$ be a compact complex manifold, and $E$ a holomorphic vector bundle on $X$. Let $H$ be ...
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Production of $H^s$ singularities in the strictly hyperbolic Cauchy problem

This question is a spin-off from Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$ as I am trying to find a solution without ...
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1answer
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Definition of Elliptic pseudodifferential operators

A symbol $p \in S^m(\Omega)$ where $\Omega \subset \mathbb{R^n}$, or its corresponding operator $p(x,D) \in \Psi^m(\Omega)$ is said to be elliptic of order m if for every compact $A\subset \Omega$ ...
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Do pseudo-differential operators form a sheaf of algebras?

Let $M$ be a smooth manifold. I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on ...
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Control of the Besov norm with respect to a nonlocal operator

Let's consider a scalar function $u(x)$ on the circle $S^1$. Assume $u\in\dot{B}^{\frac{3}{2}}_{2,1}(S^1)$, which is the homogeneous Besov space. We know now $\partial_x u \in \dot{B}^{\frac{1}{2}}_{2,...
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Is every non-negative test function the limit of a sequence of sums of squares of test functions?

Let $0\leq f\in\mathscr{D}(\mathbb{R}^n)$. As shown e.g. by J.-M. Bony, F. Broglia, F. Colombini and L. Pernazza, Nonnegative functions as squares or sums of squares, J. Funct. Anal. 232 (2006) 137-...
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1answer
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Resolvent of the Laplacian as a pseudodifferential operator and its single layer potential

In M.Taylor's book "Partial differential equations II" it is shown that the fundamental solution $E(x,y)$ of the Laplacian equation gives rise to an elliptic pseudodifferential operator $S$ on the ...
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2answers
211 views

Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold

I'm trying to close in on a definitive answer to my own question BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?, and think I ...
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1answer
278 views

The elliptic regularity theorem for differential operators with variable coefficients

I'm following the book "Introduction to the theory of distributions" by Friedlander and Joshi. There is the following result p. 109 Theorem (8.6.1). Let $X \subset \mathbb{R}^n$ be an open set, and ...
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BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?

I asked the following question on math.SE (https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d) just over two months ago, and it ...
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578 views

Is every continuous microlocal operator a pseudo-differential operator?

Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space. Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal. By being microlocal I mean that the wave ...
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1answer
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Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?

Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the ...
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1answer
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$L^p$-estimates for elliptic pseudodifferential operators

Assume we have an pseudodifferential operator $P:\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{S}(\mathbb{R}^n), Pf(x) = (2\pi)^{-n/2}\int\mathrm{d}\xi\; p(x,\xi)\,\hat{f}(\xi)e^{i\xi x}$ acting on ...
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Pseudodifferential calculus for the Diffeomorphism Invariant Geometry

In the paper "Local Index Formula in Noncommutative Geometry" Connes and Moscovici build the spectral triple $(A,H,D)$ where $A=C^{\infty}_c(P) \rtimes \Gamma$ where $\Gamma$ is an arbitrary subgroup ...
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Spectrum of the hypoelliptic transverse signature operator

Let $D$ be the transverse signature operator constructed by Connes and Moscovici in the paper "Local index formula in Noncommutative Geometry":this is first order hypoelliptic pseudodifferential ...
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Poisson Equation across a Hypersurface [closed]

Let $\mathbb{B}(0,1) \subset \mathbb{R}^3$ denote the unit ball. Let $\Gamma = \{x_3=0\}$. Let us assume $f \in L^2(B)$ .Consider the problem $ \triangle u = f $ in $\mathbb{B}$ in the weak sense such ...
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1answer
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Smoothness of distributions defined by oscillation integrals

In M.A. Shubin's book Pseudodifferential Operators and Spectral Theory, we have the following statement. Let $X\subset\mathbb{R}^n$ be an open set, and fix a symbol $a\in S_{\varrho,\delta}^m(X\...
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K-homology classes of Dirac operators on Hermitian manifolds

Given a compact Hermitian manifold $M$, we have three canonical pseudo-differential operators on the sections of complexified de Rham complex, namely 1) (d + d$^*,\Omega^{*})$ 2) ($\partial$ + $\...
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1answer
141 views

Choice of parametrix on a non-compact manifold

Let $X$ be a non-compact complete Riemannian manifold and $P$ a first-order elliptic pseudodifferential operator on $X$. Let $Q$ be a parametrix for $P$, so that $PQ - 1 = T$ and $QP - 1 = R$ are ...
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Interpretation of Smoothing Operators as $\Psi$DO's

In most of the computations I've seen involving pseudodifferential operators (referred to as $\Psi$DO's from now on) we do not have true equality. Instead we often only have equivalence of operators, ...
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Does adding a compact operator change the symbol of a pseudodifferential operator?

Suppose $X$ is a non-compact manifold. Let $P$ be an order-$0$ pseudodifferential operator on $X$ and $f:L^2(X)\rightarrow L^2(X)$ a compact operator. I'm wondering: 1) Is $P + f$ always a ...
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Characterisation of the wavefront set

I am totally new to microlocal analysis, and have been studying Jared Wunsch's notes. I have been puzzling over the properties of the wavefront set. Let $X$ be a compact Riemannian manifold, and $\...
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Implicit function theorem for operators

Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this ...
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Proof of Taylor's Schwartz Kernel Estimate of Pseudodifferential Operators

I am interested in the proof of the following result which gives an estimate on the Schwartz Kernel of a $\Psi$DO. There is one aspect the proof that is not clear to me which I would like to ask the ...
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Convergence of PDE/PsiDE - expansion of pseudo-differential operators

I have am working with a nonlinear pseudo-differential evolution equation of the form $$u_t + \mathcal{N}(u) + \mathcal{D}_{\epsilon} u = 0$$ where $\mathcal{N}$ is a nonlinear operator and $\mathcal{...
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Show that if $a$ belongs to the symbol class $S^m_{\rho , \delta}$ for $\rho > 1$, then in fact $a \in S^{-\infty}$

I'm trying to learn more tools from microlocal analysis by reading Grigis and Sjostrand's Microlocal analysis for differential operators. In the first chapter, they define the standard $C^\infty$ ...
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Normal form of Principal type $\Psi$DO's

Suppose we have a pseudo differential operator of principal type with a complex symbol and such that the poisson bracket of the real and imaginary parts on the characteristic set is non-negative.I ...
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Inverse of pseudo differential operator

Let $\operatorname{Op}_h(x,D)(a)$ denote the Weyl-quantisation of a symbol $a$. Is there an explicit way to invert this pseudo-differential operator in an asymptotic series? By this I mean, can we ...
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766 views

Fourier transform of $C^\infty_0$, smooth functions vanishing at infinity

Is there a proper description of the space $$\{\hat f\ | \ f\in C^\infty \ s.t. \forall \alpha\in\mathbb{N}^n,\forall \epsilon>0\exists K\subseteq \mathbb{R}^n\ K\ \text{compact};\ \sup_{x\in \...
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$L^\infty$ bounds for pseudo-differential equations of parabolic type

It is well-known that if the solution of $u_t=u_{xx}$, with $t>0$ and $x\in\mathbb R$, is bounded, then $a(t)=\sup_{x\in \mathbb R}u(x,t)$ is non-increasing, while $b(t)=\inf_{x\in \mathbb R}u(x,t)$...
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Domain of operator

Let be $\lambda\in C^{*}$. Consider the following operator: $ T_{\lambda}=-\Delta_{R^{2}}++\frac{\lambda^{2} }{4} (x^{2}+y^{2})+i\lambda N$, where $N=(x \frac{d }{dy} -y \frac{d }{dx})$ , ...