High order regularity for subelliptic operators on Lie groups of polynomial growth. (English) Zbl 1099.22007

Let \(G\) be a Lie group of polynomial volume growth with Lie algebra g. Consider a second-order, right-invariant, subelliptic differential operator \(H\) on \(G,\) and the associated semigroup \(S_t=e^{-tH}.\) The author identifies an ideal n\('\) of g such that \(H\) satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of n\('\). The regularity is expressed as \(L_2\) estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel. The boundedness in \(L_p,\) \(1<p<\infty,\) is obtained of some associated Riesz transform operators. It is shown that n\('\) is the largest ideal of g for which the regularity results hold. Various algebraic characterizations of n\('\) are given. In particular, n\('\)=s \(\oplus\) n where n is the nilradical of g and s is the largest semisimple ideal of g. Additional features of this article include an exposition of the structure theory for \(G\) in Section 2, and a concept of twisted multiplication on Lie groups which includes semidirect products in the Appendix.


22E30 Analysis on real and complex Lie groups
35B65 Smoothness and regularity of solutions to PDEs
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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