Another look at Sobolev spaces

Let Ω be a domain in Rⁿ with smooth boundary, it is well-known that the Sobolev spaces W ^1,2(Ω) and W ^s,2 (Ω); 0 < s < 1 are function spaces that are associated with the Laplacian Δ and the fractional Laplacian (-Δ)^s respectively. There are extensions of these concepts to the Besov spaces on certain fractal sets K. In this talk, we will discuss some of the recent developments; in particular we will consider a theorem of Bourgain, Brezis and Mironescu on the limit behavior of W ^s,2(Ω), s à 1^– to W ^1,2, and the possible extension of the theorem to Besov spaces on K.