All Questions
Tagged with pseudo-differential-operators micro-local-analysis
30 questions
1
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0
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97
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Oscillatory integrals and regularity
Let $U\subset\mathbb{R}^{d}$ be open and $N\in\mathbb{N}$. Furthermore, let $a\in\mathcal{S}^{m}_{\rho,\sigma}(U\times\mathbb{R}^{N})$ be a symbol and $\Phi\in C^{\infty}(U\times (\mathbb{R}^{N}\...
- 1,171
3
votes
2
answers
352
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Reference for commutator estimate
I'm interested in Sobolev space estimates for commutators involving a pseudodifferential operator and a Fourier multiplier. More specifically, suppose $p = p(x,\xi) \in S_{1,0}^{m_1}$ and let $q = q(\...
- 683
2
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0
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67
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Asking a reference for a fact about nonlocal operators
Let assume that $u$ is smooth enough and $ -\Delta (u \phi) \in L^1(\Omega)$ for any $\phi \in C_c^{\infty}(\Omega)$. Then it easily follows that $ -\Delta u \in L^1_{\mathrm{loc}}(\Omega)$ by ...
- 371
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244
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Holder continuity relative to Rellic-Kondrachov compactness via the nonlinear Aronsson operator
Connected to the question,
Does Morrey's inequality contextually relate to Rellic-Kondrachov compactness?
An analysis of the well-known nonlinear Aronsson operator gives $C^{(1, \frac{1}{3})}$ ...
1
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0
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323
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Does Morrey's inequality contextually relate to Rellic-Kondrachov compactness?
I have been reflecting on this question, and want to share my thinking thus far. I'd be grateful for the community's inputs.
We refer to Morrey's inequality, Theorem 4 on pp 266 of Evan's book on PDE, ...
5
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1
answer
2k
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Does the 'reproducing kernel formula' for a bounded open set $U$ define an equivalent norm on the Sobolev space $H^1_0(U)$
We refer to the 'reproducing convolution formula with a kernel' for an open bounded domain $U$ of $R^n$, $n \geq 2$ discussed in the paper of G. Talenti (Annali de Matematica, Dec 1976) on Best ...
1
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0
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312
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History of microlocal analysis
Was the study of pseudo-differential operators the basis for the birth of microlocal analysis? I'm trying to find out how this branch of analysis was developed...
- 97
1
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1
answer
852
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Kernel of the composition of operators
Let $X \subset \mathbb{R}^{n}$, $Y \subset \mathbb{R}^{m}$, and $Z \subset \mathbb{R}^{p}$ be open subsets, and let $K_P \in C_0^\infty(X \times Y)$ and $K_Q \in C_0^{\infty}(Y \times Z)$. Then, $K_P$ ...
- 509
0
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1
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165
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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 ...
2
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0
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315
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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 ...
- 509
4
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0
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256
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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 ...
- 873
3
votes
0
answers
588
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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 ...
- 509
2
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0
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40
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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 ...
- 503
7
votes
1
answer
861
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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 ...
- 159
14
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4
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1k
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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 ...
- 8,086
1
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0
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260
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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, ...
- 377
3
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1
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606
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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 $\...
- 914
1
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0
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108
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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 $C^\infty$ ...
- 481
1
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0
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70
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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 ...
- 4,135
2
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1
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499
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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 ...
- 61
2
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1
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301
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Show that a very regular kernel $k(x,y)$ has operator $K : \mathcal{E}'(\Omega) \to \mathcal{D}'(\Omega)$ which is pseudo-local
I am reading Francois Treves' Introduction to pseduodifferential and Fourier integral operators, vol. I. I am having trouble understanding the proof of Lemma 2.1, which is stated as follows.
Let $\...
- 481
8
votes
3
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1k
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Characteristic Variety of the Principal Symbol solves PDE system?
In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see for example, http://www.sciencedirect.com/science/...
- 113
1
vote
1
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184
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Ask the validity of Tauberian lemma in Sogge's book
In C.D.Sogge's Fourier Integrals in Classical Analysis pp.128-129, he proved Lemma4.2.3(Tauberian Lemma):
Lemma. Let$g(\lambda)$ be a piece-wise continuous tempered function of $\mathbb{R}$. Assume ...
- 8,071
2
votes
1
answer
415
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Pseudo-differential evolution equation
I'm looking for results (or some ideas) on the following kind of pseudo-differential evolution equation:
$$
\frac{\partial u(t,x)}{\partial t} = \int_{-\infty}^{t} B(t-s,x)\, A(x,D_{x})u(s,x)\,ds \; ;...
- 55
3
votes
1
answer
811
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Application of Egorov's Theorem for Pseudodifferential Operators
Let $\;P_{0} \in OPS^{m}_{1,0}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})\;$, $\;A \in OPS^{1}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})$ and $S(t)$ the solution operator of the scalar hyperbolic ...
- 55
3
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1
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398
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Elliptic Fourier integral operators
I know what it means for a pseudodifferential operator $A\in\Psi(\mathbb{R}^n)$ to be elliptic at a point $(x,\xi)\in T^*\mathbb{R}^n$: the principal symbol of $A$ is non-vanishing at the point.
But ...
- 31
1
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1
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209
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Estimate the analytical wavefront set $WF_A(u)$ given $WF_A(A_K u)$
Let $X$ and $Y$ be two real-analytic manifolds and $Z \subset X \times Y$ be a real-analytic embedded closed submanifold. Suppose now that $K \in \mathscr D'(X \times Y)$ is a distribution conormal to ...
- 1,329
2
votes
1
answer
240
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Support-preserving pseudodifferential operators
Let $A = F^{-1}\sigma F$ be a pseudodifferential operator acting on functions on $\mathbb R^n$, where $F$, $F^{-1}$ are the direct and inverse Fourier transforms respectively and $\sigma$ is the ...
- 1,329
7
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1
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870
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Pseudo-differential operators which are independent of lower order perturbations
In the area of pseudo-differential operators, we know that for elliptic type or real principal type operators reductions are independent of lower order terms. For example, if $P$ is zeroth order real ...
- 2,239
11
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3
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3k
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Characterization of inverse differential operators
If I have a partial differential operator $p(D)$, where $p$ is a polynomial with constant coefficients and $D$ is the derivative in Euclidean space. Its inverse is easily described in Fourier space: $\...
- 21.5k