2007/01/17 by Simon Haller, Haller, Simon
Mathematics · #35A18 #35A27 #35D10 (Secondary) #46F30 (Primary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #History and Theory of Mathematics #Mathematical and Theoretical Analysis #math.AP #math.FA #msc:35A18 #msc:35A27 #msc:35D10 #msc:46F30
paper · pdf · doi:10.48550/arxiv.math/0701485
29 pages, 1 figure
openalex publication_date 2007/01/17 · arxiv created 2007/02/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In distribution theory the pullback of a general distribution by a C∞-function is well-defined whenever the normal bundle of the C∞-function does not intersect the wavefront set of the distribution. However, the Colombeau theory of generalized functions allows for a pullback by an arbitrary c-bounded generalized function. It has been shown in previous work that in the case of multiplication of Colombeau functions (which is a special case of a C∞ pullback), the generalized wave front set of the product satisfies the same inclusion relation as in the distributional case, if the factors have their wavefront sets in favorable position. We prove a microlocal inclusion relation for the generalized pullback (by a c-bounded generalized map) of Colombeau functions. The proof of this result relies on a stationary phase theorem for generalized phase functions, which is given in the Appendix. Furthermore we study an example (due to Hurd and Sattinger), where the pullback function stems from the generalized characteristic flow of a partial differential equation.