base change for distributions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T02:07:57Z http://mathoverflow.net/feeds/question/99749 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99749/base-change-for-distributions base change for distributions Rami 2012-06-15T22:17:34Z 2012-06-18T17:41:36Z <p>For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see Hormander The analysis of Linear PD operators I, 8.2. </p> <p>I need a reference for the following fact:</p> <p>Let $\phi:X \to Y$ be a proper map of smooth manifolds. Let $Z \subset Y$ be a closed submanifold. Assume that $\phi$ is transversal to $Z$, i.e. for any $x\in X$ s.t. $\phi(x)\in Z$ we have $d_x(T_x(X))+T_{\phi(x)}Z=T_{\phi(x)}Y$. Let $W=\phi^{-1}(Z)$. Note that it is a submanifold. Let $\xi\in C^{-\infty}(X)$ s.t. $WF(\xi) \cap CN_{W}^X \subset X,$ were $WF$ is the wave front set and $CN$ is the co-normal bundle. Then</p> <ol> <li><p>$WF(\phi_*(\xi))\cap CN_{W}^Y \subset Y.$ (This follows easily from -- Hormander The analysis of Linear PD operators I, 8.2)</p></li> <li><p>$\phi_*(\xi)|_Z=(\phi|_W) _*(\xi|_{W})$. The restrictions are defined because of the conditions on the wave front set.</p></li> </ol> <p>Thank you very much</p>