The Spaces D and D*

Abstract

This chapter presents some of the basic properties of the spaces D and D* which were introduced at the end of Chapter 3. The most important results are the theorem of Meyer stating the algebraic property of D, and the theorem of Watanabe which says that composites of tempered distributions with non-degenerate elements in D belong to D*. Both results are based on the equivalence of the (L^p)-norms of N^1/2 and | ∇· |₂, which has been established in Meyer (1982, 1983) (cf. also Bakry (1985), Feyel and de la Pradelle (1989,1991), Gundy (1989), Krée and Krée (1983), Shigekawa (1990), and literature cited in these articles). The results of Meyer (1982, 1983) have been worked out further in Sugita (1985a) and Watanabe (1983, 1984). The necessity of estimating the (L^p)-norms of | ∇· |₂ by those of the number operator N, when proving that D is an algebra, comes in because of the product rule for $N:N\varphi \psi = \left( {N\varphi } \right) + \varphi N\psi - 2\left( {\nabla \bar \varphi ,\nabla \psi } \right)$, cf. (5.80).