Nahum Zobin College of William and Mary

Smooth functions in bad domains and Fourier analysis in Fock space

Let $\Omega $ be a bounded connected open subset of $R^n$. Consider the space $W^{k+1}_\infty (\Omega)$ of functions with partial derivatives of order $k+1$ bounded in $\Omega.$ Are these functions extendible to the functions of the same class in the whole $R^n$? In 1934 H. Whitney proposed a geometric condition on $\Omega$ (equivalence of the geodesic distance in $\Omega $ to the Euclidean distance) which implies such extendability. A long standing question was if this condition is also necessary. We introduce techniques ( a kind of Fourier analysis in the Fock spaces) which allowed to give a very complete answer to this question.