Weighted Subcoercive Operators on Lie Groups

LetUbe a continuous representation of a Lie groupGon a Banach space X anda1, …, ad′an algebraic basis of the Lie algebra g ofG, i.e., thea1, …, ad′together with their multi-commutators span g. LetAi=dU(ai) denote the infinitesimal generator of the continuous one-parameter groupt↦U(exp(−tai)) and setAα=Ai1 …Ainwhereα=(i1, …, in) withij∈{1, …, d′}. We analyze properties ofmth order differential operatorsdU(C)=∑α; |α|⩽m cαAαwith coefficientscα∈C. IfLdenotes the left regular representation ofGinL2(G) thendL(C) satisfies a Gårding inequality onL2(G) if, and only if, the closure of eachdU(C) generates a holomorphic semigroupSon X, the action ofSzis determined by a smooth, representation independent, kernelKzwhich, together with its derivativesAαKz, satisfiesmth order Gaussian bounds and, ifUis unitary,Sis quasi-contractive in an open representation independent subsector of the sector of holomorphy. Alternatively,dL(C) satisfies a Gårding inequality onL2(G) if, and only if, the closure ofdL(C) generates a holomorphic, quasi-contractive, semigroup satisfying bounds ‖AiSt‖2→2⩽ct−1/meωtfor allt>0 andi∈{1, …, d′}. These results extend to operators for which the directionsa1, …, ad′are given different weights. The unweighted Gårding inequality is a stability condition on the principal part, i.e., the highest-order part, ofdL(C) but in the weighted case the condition is on the part ofdL(C) with the highest weighted order.