Abstract: I’ll discuss the regularity of the trace operator on various smoothness spaces. In short, this is an operator which reduces (roughly) smoothness as measured in L_p by an order of 1/p. Its behavior on Sobolev spaces, especially on the Hilbert spaces W_2^s (i.e., where smoothness is measured in L_2), plays a critical role in approximation theory when boundaries are present, and in the stability, regularity and existence results for weak solutions for PDEs.

Previously I presented a somewhat negative results: that the trace operator from W_2^(1/2) to L_2 is not bounded (although it is bounded from W_2^{s+1/2} to W_2^s when s>0. In this talk, I’ll explain how a minor correction (a so-called “curiosity” according to Hans Triebel) works – namely, by reducing the domain to a certain Besov space and using results from the atomic decomposition of these spaces.