Seminar

We will recall classical results in smooth extension theory discovered by Hassler Whitney in the 1930\'s and present the recent construction of a linear extension operator for functions in the Sobolev space $L^{m,p}(R^n)$; that is, a linear operator that takes a real-valued function f defined on a finite subset E of Euclidean space, and produces a function F defined on $R^n$, which matches f on E, and has Sobolev norm minimal to within a factor of $C=C(m,n,p)$. This generalizes work of C.
Fefferman on the extension of $C^m$ functions, and in fact our method gives a new proof of his theorem. Furthermore, a closer analysis of our construction shows that the extension F can be taken to have a simple dependence on f (assisted bounded depth) though, in general, not an even simpler one (bounded depth). This is joint work with C. Fefferman and G.K.
Luli.