# Questions tagged [micro-local-analysis]

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### Absence of embedded eigenvalues when skew-adjoint part is non-trivial but well-signed

Let $g$ be an asymptotically Euclidean Riemannian metric, and consider the operator $$P_\lambda=D_ig^{ij}D_j+\lambda (A_jD_j+D_jA_j+B)+\lambda^2C,$$ where $A,B,C$ are smooth functions (additional ...
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### Local energy estimate in a semiclassical regime

Let us consider $h_n=(2n+1)^{-1/2}\to 0$ as $n\to \infty$ be a small parameter, which we just write as $h$ for convenience, and $u_h : \mathbb{R} \to \mathbb{R}$ be functions satisfying $Pu_h=0$ (I ...
103 views

### Product of Heavisides: calculus vs Fourier transform vs wavefront set

I decided to ask this question here, since I did not get any answer from MSE and perhaps this topic is somewhat far from MSE's topics. I am following the paper here. I am trying to understand how to ...
232 views

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### Algebraic microlocal analysis and nonlinear PDE

Though originating in the study of linear partial differential equations, microlocal analysis has become an invaluable tool in the study of nonlinear pde. Of particular importance has been the ...
303 views

### What's the definition of a microlocal sheaf?

I'm slowly becoming familiar with what microsupport of a sheaf is, but none of the references I've seen give a definition of what a microlocal sheaf should be in general. In this paper of ...
57 views

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 ...
100 views

### If theorem valid for compactly supported distribution then is it also valid for tempered distribution?

I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution. For instance, Theorem: Any $A \in \Psi^{m}$ ...
238 views

### 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 vote
281 views

### 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, ...
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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 ...
859 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 ...
1 vote
240 views

### Microlocal proof of Wigner semicircle theorem?

Something I really enjoy about Tao's writing is that he proves the same theorem over and over. While I complain a bit sometimes about clarity, this is a heuristic that I very much believe in. This ...
1 vote
### 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, ...