Orateur
Description
A classical theorem of Euler is that a function $f$ on $\mathbb{R}^d$ is homogeneous of degree $m$ if and only if $Ef=mf,$ where $E$ is the Euler vector field. If we relax this to $Ef=mf+g,$ where $g$ is Schwartz-class, we say that $f$ is approximately homogeneous. Classical pseudodifferential operators have a symbol function $\sigma(x,\xi)$ which admits an asymptotic expansion into a series $\sum_{n=0}^{\infty} \sigma_{n}(x,\xi),$ where $\sigma_n(x,\xi)$ is homogeneous in its second argument. Van Erp and Yuncken observed that polyhomogeneous functions can be characterised by their having extension to an approximately homogeneous function on a higher-dimensional space. Similarly the Schwartz kernels of classical pseudodifferential operators can be characterised by their having approximately homogeneous extension to the tangent groupoid. If we relax the condition $(E-m)f = 0$ to $(E-m)^k f=0$ for some $k\geq 1,$ we get the class of log-homogeneous functions. There is a corresponding theory of log-polyhomogeneous pseudodifferential operators and via the tangent groupoid we can say some things about their spectral theory.