Solvability of Transversally Elliptic Differential Operators on Nilpotent Lie Groups

1986 ◽  
Vol 108 (3) ◽  
pp. 589 ◽  
Author(s):  
Lawrence Corwin ◽  
Linda Preiss Rothschild
Keyword(s):  
Lie Groups ◽  
Differential Operators ◽  
Nilpotent Lie Groups ◽  
Elliptic Differential Operators

scholarly journals Pseudo-differential operators with nonlinear quantizing functions

2019 ◽  
Vol 150 (1) ◽  
pp. 103-130
Author(s):  
Massimiliano Esposito ◽  
Michael Ruzhansky
Keyword(s):  
Lie Groups ◽  
Differential Operators ◽  
Nilpotent Lie Groups ◽  
General Function ◽  
Polynomial Functions ◽  
Heisenberg Groups ◽  
Model Case ◽  
Pseudo Differential Operators ◽  
Image Position

AbstractIn this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form $$Au(x)=\int_{{\open R}^n}\int_{{\open R}^n}e^{{\rm i}(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)\,{\rm d}y\,{\rm d}\xi,$$ where $\tau :{\open R}^n\to {\open R}^n$ is a general function. In particular, for the linear choices $\tau (x)=0$, $\tau (x)=x$ and $\tau (x)={x}/{2}$ this covers the well-known Kohn–Nirenberg, anti-Kohn–Nirenberg and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions τ and here we investigate the corresponding calculus in the model case of ${\open R}^n$. We also give examples of nonlinear τ appearing on the polarized and non-polarized Heisenberg groups.

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