The action of √-Δ on weighted Sobolev spaces

The action of root-Delta on distributions is examined within the context of weighted Sobolev spaces. The results obtained are as follows: (1) root-Delta is a continuous map of S(R-n), the space of rapidly decreasing functions, to L-2, s(R-n) for any s < n/2 + 1; (2) if k is an element of R and s > -n/2 - 1, then root-Delta is a continuous map from H-k,H- s(R-n), the weighted Sobolev space, to Hk-1, (t)(R-n) for some t. The results are optimal in a sense.