the relation between a continuous family of distributions and a distribution of 2 variables - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:28:27Z http://mathoverflow.net/feeds/question/99925 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99925/the-relation-between-a-continuous-family-of-distributions-and-a-distribution-of-2 the relation between a continuous family of distributions and a distribution of 2 variables Rami 2012-06-18T17:32:16Z 2012-06-22T15:37:41Z <p>Let X,Y be smooth manifolds and let $f:X \to C^{-\infty}(Y)$ be a continuous map, where $C^{-\infty}(Y)$ is the space of generalized functions on $Y$ equipped with the weak topology. By Schwartz kernel theorem, this gives us a generalized function $\xi_f$ on $X \times Y$. Do you know any reference for this construction and its properties? </p> <p>Specifically, what is the relationship between this construction and the wave front set? For example, if we know the wave front set of $\xi_f$, it seems that we can bound the wave front set of $f(x)$. Is it written anywhere?</p> <p>In general, it is reasonable to say that the restriction of $\xi_f$ to ${x}\times Y$ is $f(y)$. On the other hand there another is a notion of restriction (or more generally pull back) of a distribution (see Hormander The analysis of Linear PD operators I, Theorem 8.2.4 ). Do you know any reference for the fact that those to notions coincide whenever both are defined?</p>