Weighted Subcoercive Operators On Lie Groups

Mathematics Research Report MRR95-051

Weighted Subcoercive Operators On Lie Groups

A.F.M. ter Elst and Derek W. Robinson

Abstract: Let U be a continuous representation of a Lie group G on a Banach space $\chi$ and a₁,...,a_d' an algebraic basis of the Lie algebra g of G, i.e., the a₁,...,a_d' together with their multi-commutators span g. Let A_i=dU(a_i) denote the infinitesimal generator of the continuous one-parameter group t->U(exp(-ta_i)) and set A^\alpha=A_{i_1}... A_{i_n} where \alpha=(i₁,...,i_n) with i_j \in {1,...,d'}. We analyze properties of m-th order differential operators
dU(C)=\sum_{\alpha; |\alpha|\leq m}c_\alpha A^\alpha
with coefficients c_\alpha\in C. If L denotes the left regular representation of G in L₂(G) then dL(C) satisfies a Gärding inequality on L₂(G) if and only if the closure of each dU(C) generates a holomorphic semigroup S on {x} which is quasi-contractive, i.e., |S_z | \leq e^{\omega |z|}, in an open representation independent subsector of the sector of holomorphy and the action of S_z is determined by a smooth, representation independent, kernel K_z which, together with its derivatives A^\alpha K_z, satisfies m-th order Gaussian bounds. Alternatively, dL(C) satisfies a Gärding inequality on L₂(G) if, and only if, the closure of dL(C) generates a holomorphic, quasi-contractive, semigroup satisfying bounds |A_iS_t|_2->2 \leq c t^-1/m e^{\omega t} for all t\ge 0 and i\in {1,...,d'}. These results extend to operators for which the directions a₁,...,a_d' are given different weights. The unweighted Gärding inequality is a stability condition on the the principal part, i.e., the highest order part, of dU(C) but in the weighted case the condition is on the part of dU(C) with the highest weighted order.

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