# Tagged Questions

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### Zigmund-Besov Spaces and Inverse Function Theorem, is the Inverse Zigmund?

Preliminary Definition Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are allowed ...
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### partial differential operators

As we know, there exists a semigroup for partial differential operators $A = \sum_{i,j=1}^N D_i(a_{ij}(\cdot)D_j)$, see (Klaus-Jochen Engel, Rainer Nagel, one-parameter semigroups). My question is ...
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### Change of variables for linear differential operators

I am trying to find an expression for a change of variables (invertible and $C^{\infty}$) of a linear differential operator in $\mathbb R ^d \newcommand{\t}{\tilde} \newcommand{\p}{\partial}$. i) ...
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### Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
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### Composition of Differential Operators

If I have: $A=\partial_x^2+u(x)$ $B=u(x)\partial_x$ How do I compose: $AB$ and $BA$?
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### Poisson's equation under translation and scaling

Let $u\in C^\infty (\Omega,\mathbb R)$ be a solution of $$\begin{array}{cccl} -\Delta u & = & 0 & \mathrm{in}\ \Omega \\ u & = & g & \mathrm{on}\ \partial\Omega\end{array}$$ ...
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### Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a ...
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### Infinite propagation speed for the Schrodinger operator

Question related to: On the propagation of singularities in PDE and Hypoellipticity and singular support. in what sense is to interpret the sentence the schrodinger operator has infinite propagation ...
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### Fundamental solution of the wave operator

What is the explicit formula for a fundamental solution of the wave operator, where the space variable is in $\mathbb{R}^n$ with $n>1$? Thanks. The operator I'm talking about is ...
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### On the propagation of singularities in PDE

This question might be a little generic, but i wanted to get some idea on the concept of propagation of singularities in PDE. Searching the internet i only found very complicated things about the ...
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### Non-hypoelliptic operator

Consider the following operator $$\partial_t^2-\partial_x^2+\lambda(x,t)$$ where $\lambda$ is a $C^{\infty}$ function on $\mathbb{R}^2$ and the operators above are second partial derivatives. How can ...
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### Cayley Transform (PDE)

I really need your help in solving the following problem: Prove that a unitary operator $U$ acting on a Hilbert space $H$ is the Cayley transform of some self-adjoint operator if and only if $1$ is ...
I'm trying to find the error in my logic here. Let's say we are given the Laplace operator in polar coordinates:  \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + ...