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8
votes
1
answer
455
views
On commutator of bounded operators
Let $\mathbb H$ be a Hilbert space and let $\mathcal B(\mathbb H)$ be the bounded operators on
$\mathbb H$. Let $J,K\in \mathcal B(\mathbb H)$ such that
$
J=J^*, K=-K^*.
$
Then the commutator $[J,K]$ …
1
vote
$L^p$ boundedness for pseudo-differential operators
I must apologize for answering my own question since I should have checked the classical results in the literature before asking these two questions. However, let me summarize the situation: we always …
0
votes
1
answer
122
views
$L^p$ boundedness for pseudo-differential operators
Let $\rho, \delta, m$ be real parameters such that $0\le \delta\le \rho\le 1, \delta<1$. The set $S^m_{\rho, \delta}(\mathbb R^{2n})$ is defined as the set of smooth functions $a$ on $\mathbb R^n\time …
4
votes
How to define Laplacian on $L_2$
(1) Let me answer first to the last question: $\Delta \vert x\vert$ is homogeneous of degree $-1$ and radial. On $\mathbb R^d$ ($d\ge 2$)
it is
$$
(\partial_r^2+\frac{d-1}{r}\partial_r)(r)=\frac{d-1}{ …
6
votes
Accepted
Hormander's bracket condition for the adjoint of an operator
The hypoellipticity result is more precise:
you have
$$
Lu \in H^s_{loc}\Longrightarrow u\in H^{s+2-\delta}_{loc}\quad\text{ for some $\delta\in [0,2)$,}
$$
and that $\delta$ is linked to the number o …
0
votes
ordered exponential of unbounded operators
Certainly, one should pay attention to the domain of the operator. However, the following argument should survive a reasonable assumption. We have
$$
\frac{d}{dt}(E^\ast(t)E(t))=-2 E^\ast(t) A(t) E(t …
6
votes
Accepted
When is the adjoint of a hypoelliptic operator also hypoelliptic?
Hormander's operator $L=X_0+\sum_{1\le j\le k} X_j^2$, where the $X_j$ are real smooth vector fields with the Lie algebra of $\{(X_j)\}_{0\le j\le k}$ generating the tangent space is hypoelliptic as …