Some Examples of C0-Groups
Werner O. Amrein, Anne Boutet de Monvel, Vladimir Georgescu · Birkhäuser Basel eBooks · 1996
In this chapter we present several examples of C 0-Groups which are frequently used. We shall consider Banach spaces F embedded in L *(ℝ n ). We know (see Section 1.2) two n-parameter groups acting in L *, namely {e i(x,P)} and {e i(x,Q)}. If the Banach space F is invariant under one of these groups, one may define the associated Sobolev and Besov scales according to the theory of Chapter 3. This situation is described in Section 4.1. In Section 1.2 we also introduced the dilation group e iDτ , where D = (P · Q + Q · P)/4. It is interesting to consider the more general one-parameter group induced by a generator of the form A = [P · F(Q) + F(Q) · P]/2, where F is a vector field on ℝ n . This is the topic of Section 4.2.