Analysis on Lie Groups with Polynomial Growth

Lie groups are manifolds symmetric under the group action and the symmetry places uniform constraints on the global properties of the manifold. The simplest constraint resulting from the group action is on the volume growth. There are only two possibilities. In the first case the volume of a ball grows no faster than a power of its radius. Groups with this characteristic are called Lie groups of polynomial growth. Compact Lie groups fall within this class since the volume is uniformly bounded. Nilpotent Lie groups also have polynomial growth, although this is less evident, and the rate of growth is straightforwardly determined by the nilpotent structure if the group is connected and simply connected. Moreover, all Lie groups of polynomial growth are unimodular. In the second case the volume of a ball grows exponentially with its radius. All non-unimodular Lie groups have exponential growth but non-unimodularity is not essential. For example, each non-compact semisimple Lie group is unimodular but has exponential volume growth.

In this chapter we develop the general background information relevant to the subsequent analysis. Part of the material consists of standard results which are summarized for later reference. A second part consists of the basic definitions of subelliptic operators and the related semigroups together with the description of some preliminary results which motivate the later analysis. Thirdly, we introduce several techniques adapted to the Lie group analysis. Since most of the reference material is quite standard it is summarized in formal statements without proof. Further details and specific references to the literature are, however, given in the Notes and Remarks at the end of the chapter.

Structure theory is an essential ingredient in the analysis of subelliptic semigroup kernels on groups of polynomial growth. In Chapter II we summarized most of the relevant standard results but it is also necessary to establish a number of other results adapted to the analysis.

In this chapter we return to the analysis of complex subelliptic operators H, as defined in Section II.2, and the associated semigroup kernels K on groups G of polynomial growth. The eventual aim is to understand the global properties of the kernels and the global geometry of the group. The starting point is the observation that if G is simply connected, then G is the semidirect product M × Q of a compact Levi subgroup M and the group radical Q. Moreover, it follows from the analysis of Section III.7 that G = (M × Q _N , S*) where Q _N is the nilshadow of Q and the group product S* is defined with the homomorphism S given by (III.44). Therefore the theory can be reformulated on the simpler direct product group G _N = M × Q _N , the shadow of G.

In the previous chapter we applied techniques of homogenization theory to derive global Gaussian bounds on the subelliptic semigroup kernel K. Once these bounds are established, one can use quite different techniques based on L₂-estimates to obtain global bounds on the derivatives of K. The nature of these bounds is sensitive to the direction of the derivatives and, in particular, the local and global singularities are usually quite different. The differences reflect the global geometry of the group.

In this chapter we analyze the asymptotic behaviour of the subelliptic semigroup S and its kernel K on the connected Lie group G. The analysis relies heavily on the Lie group formulation of homogenization theory given in Chapter IV and uses the Gaussian bounds of Theorem IV.7.1. The derivation of the latter bounds relied implicitly on homogenization but in the asymptotics the homogenized operator and the corresponding semigroup and kernel play an explicit role. The derivation of the kernel bounds was based on elliptic estimates which were uniform with respect to dilations. Then global Gaussian bounds could be inferred from local Gaussian bounds by scaling. In this manner one could estimate large-time behaviour through control of the dilation structure. By pursuing this line of reasoning one concludes that for simply connected groups S and K are asymptotically approximated by the semigroup P _M ⊗ \(\widehat S\) and its kernel 11 ⊗ \(\widehat K\) on the group G _N , where \(\widehat S\) is the semigroup generated by the homogenized operator \(\widehat H\) on L ₂ (Q _N ) and P _M = ∫ _M dm L _M (т) is the projection onto the constant functions on M. In fact one can identify the first-order corrections in an asymptotic expansion and obtain estimates on the rate of convergence for general groups. The asymptotic control is in part a byproduct of the Gaussian bounds but for detailed estimates it is necessary to develop more fully the homogenization theory. In particular one needs to examine higher-order correctors. Since the asymptotic approximation of K has the form 11 ⊗ \(\widehat K\) the compact Levi subgroup M plays no role in the asymptotics.