Hilbert spaces of sections on which the G-action is unitary. This action allows us to define an trace tr G in the algebra of operators commuting with the action of G. Restricting this trace to orthogonal projections P L onto G-invariant subspaces L provides a dimension function dim G given by dim G (L) = tr G (P L ). Generalizing the previous definition, a G-invariant operator A :With this definition, we prove the following:Theorem 1.1. Let M be a complex manifold with boundary which is strongly pseudoconvex. Let G be a unimodular Lie group acting freely by holomorphic transformations on M so that M/G is compact. Then, for q > 0, the Kohn Laplacian in L 2 (M, Λ p,q ) is G-Fredholm.Corollary 1.2. If M is as before and q > 0, then the reduced Dolbeault cohomologies H p,q (M) have finite G-dimension. Corollary 1.3. If M is as before, let L ⊂ (ker ∂) ⊥ be closed and G-invariant. Then ∂| L : L → ∂L is G-Fredholm. Remark 1.4. Examples of manifolds satisfying the hypotheses of the theorem are Grauert tubes of unimodular Lie groups. The unimodularity of G is necessary for the definition of the G-Fredholm property.
Let G be a unimodular Lie group, X a compact manifold with boundary, and M the total space of a principal bundle G → M → X so that M is also a strongly pseudoconvex complex manifold. In this study, we show that if G acts by holomorphic transformations satisfying a local property, then the space of square-integrable holomorphic functions on M is infinite-dimensional.
Keywords∂-Neumann problem · Subelliptic operators · Harmonic analysis
MR Classification Numbers
Let M be a strongly pseudoconvex complex manifold which is also the total space of a principal G-bundle with compact base M/G. Assume also that G acts on M by holomorphic transformations. For such M , we provide a simple condition on forms α sufficient for the regular solvability of u = α and other problems related to the∂-Neumann problem on M. Similar properties are shared by b .
We prove heat kernel estimates for the∂-Neumann Laplacian acting in spaces of differential forms over noncompact manifolds with a Lie group symmetry and compact quotient. We also relate our results to those for an associated Laplace-Beltrami operator on functions.
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