All Questions
9 questions
2
votes
1
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258
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Generalizing a formula with distributions — Distributional Radon transform
I will try to describe the problem, it will necessarily be incomplete, so please if you have questions or remarks to make it more clear do not hesitate to leave them in comments.
The problem
Let $a$ ...
- 21
4
votes
1
answer
225
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Asymptotics of integral representation of distribution
I initially posted this question at MSE (here), but I have gotten no response, so I figured I would ask it to this community.
Background: I am studying the PDE $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
- 850
2
votes
0
answers
96
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Evolution PDE in dual space : Generalization of a result of Gelfand
The following result is proved in "Generalized functions", Volume 3, Chapter 2.2 by Gelfand :
Let $\Phi$ be a Fréchet space with dual space $\Phi'$ endowed with the weak topology. For ...
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11
votes
1
answer
1k
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Research topics in microlocal analysis
Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution ...
- 589
4
votes
2
answers
481
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Hörmander's hypoellipticity theorem for complex coefficients
Hörmander's theorem says that if $L = \sum _{i=1} ^r X_i ^2+ X_0 + f$ on some open subset $U \subseteq \Bbb R$ has the property that the Lie algebra generated by $\{X_0, \dots, X_r\}$ at every point ...
- 5,407
11
votes
2
answers
712
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Poincaré lemma for distributions
Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form
$$
u=\sum_{1\le j\le n} u_j dx_j,\quad ...
- 16.2k
1
vote
0
answers
95
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Construct a PDE solution from a net of approximations
Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$.
Let $...
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0
votes
1
answer
403
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The hypoellipticity of a heat-like operator
I am aware that the heat operator (on a smooth manifold) is hypoelliptic. I am also aware that there are manifolds on which the Schrödinger's operator (with a $\Bbb i = \sqrt {-1}$ multiplying $\frac {...
- 5,407
4
votes
2
answers
368
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Recognizing Schwartz regular distributions
Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)?
To be more detailed: if I want to show that some ...
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