Weighted strongly elliptic operators on Lie groups
Let \(({\mathcal X},G,U)\) be a continuous representation of a Lie group \(G\) by bounded operators \(g \to U(g)\) on the Banach space \(\mathcal X\) and let \(({\mathcal X},{\mathfrak g},dU)\) denote the representation of the Lie algebra \(\mathfrak g\) obtained by differentiation. Let \(a_ 1, \dots, a_{d'}\) be a Lie algebra basis, \(A_ i = dU(a_ i)\), \(A^ \alpha = A_{i_ 1} \cdots A_{i_ k}\) whenever \(\alpha = (i_ 1, \dots, i_ k)\), and \(H = \sum_ \alpha c_ \alpha A^ \alpha\) where the \(c_ \alpha\) are complex coefficients satisfying a weighted strongly elliptic condition in which different directions may have different weights. The authors prove that the closure \(\overline{H}\) generates a holomorphic semigroup \(S\) with holomorphy sector which contains a non-empty subsector determined by the coefficients and independent of the representation. Also elliptic regularity properties for the operators and their powers are established, and as a corollary optimal growth bounds for the eigenfunctions of the anharmonic oscillators \(P^{2m} + Q^{2n}\).
scientific article; zbMATH DE number 704513