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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.

MSC:

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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References:

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