October 19 – 23, 2015
Motivated by boundary value problems for partial differential equations, classical trace and extension theorems characterize traces of spaces of generalized smoothness such as Sobolev and Besov to smooth submanifolds of Euclidean space. The subject originated from Hassler Whitney seminal papers of 1934, which deal with the following problem: given a real function on an arbitrary subset of Euclidean space, determine whether it is extendible to a function of a prescribed smoothness on the entire space.
Whitney developed important analytic and geometric techniques that allowed him to solve this problem for functions defined on subsets of the real line to be extended to m-times continuously differentiable functions on the entire real line. He also formulated and solved similar problems related to jets of functions defined on a subset of Euclidean space in any dimension. In the decades since Whitney’s seminal work, fundamental progress was made by Georges Glaeser, Yuri Brudnyi, Pavel Shvartsman, Edward Bierstone, Pierre Milman, Wieslaw Pawlucki, and Charlie Fefferman.
It is natural also to consider similar extension and trace problems for functions in Sobolev spaces. These results are at a much earlier stage, though there has been significant progress of late. Another problem is to find the Lipschitz constant associated to m-jets.
The objective of the program is bringing together an international group of experts in the areas of function theory and functional and geometric analysis to report on and discuss recent progress and open problems in the area of Whitney type problems.
Alex Brudnyi (University of Calgary)
On Bernstein Classes of Well Approximable Maps
The convex paradigm in optimization: dynamical considerations
A transmission problem across a fractal interface
Whitney problems survey
Computing minimal interpolants in C1,1(Rd)
On the (q, p)-Poincaré inequality, when q < p
Measure density and extension of Besov and Triebel– Lizorkin functions
Interpolation of data in Sobolev spaces
Vectorial Calculus of Variations in L infinity and generalised solutions for fully nonlinear PDE systems
Extremal Extension for $m$-jets of one variable with range in a Hilbert space
A decomposition of functions and weighted Korn in- equality on John domains
Interpolation by Nonnegative Functions
Direct proof of termination of the Kohn algorithm in the real-analytic case
Ck -extendability criterion for functions on a closed set defin- able in any polynomially bounded o-minimal structure
A Whitney-type extension theorem for jets generated by Sobolev functions
Two weight norm inequalities for singular and fractional integral operators in Rn
Rational approximation of singular functions
Some duality considerations in Whitney problems