CMA Research Report

MRR02-006

High order regularity for subelliptic operators on Lie groups of polynomial growth

Abstract:  Let G be a Lie group of polynomial volume growth, with Lie algebra g. We consider a second-order, right-invariant, subelliptic differential operator H on G, and the associated semigroup S_t = e^-tH. We identify an ideal n' of g such that H satisfies global regularity estimates for spatial derivatives of all orders, provided that the derivatives are taken in the direction of n'. The regularity is expressed as L₂ estimates on derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel. We also obtain the boundedness in L_p, 1 <p<∞, of some associated Riesz transform operators.

Various algebraic characterizations of n' are given. In particular n' always contains the nilradical of g, and if g is solvable then n' equals the nilradical. Finally, we prove that n' is the largest ideal of g for which the regularity results hold.

AMS Classification:  22E30
Date:  17 April 2002

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