Compactness of the solution operator to ∂¯ in weighted L2-spaces

Abstract

In this paper we discuss compactness of the canonical solution operator to ∂ on weigthed L 2 spaces on C n . For this purpose we apply ideas which were used for the Witten Laplacian in the real case and various methods of spectral theory of these operators. We also point out connections to the theory of Dirac and Pauli operators.

On some spectral properties of the weighted ∂¯-Neumann operator

“…Decoupled weights are known from [12] to be an obstruction to compactness of the ∂-Neumann operator on (0, 1)-forms, provided at least one ϕ j gives rise to an infinite dimensional weighted Bergman space (of entire functions on C). By using results of [2,17,18], we will characterize compactness of N ϕ 0,q for this class of weights, under the additional assumption that each ϕ j : C → R is a subharmonic function such that ∆ϕ j defines a nontrivial doubling measure.…”

“…(ii) Let n = 1 and suppose that ϕ : C → R is a subharmonic C 2 function such that the Bergman space A 2 (C, e −ϕ ) is infinite dimensional. The complex Laplacian on (0, 1)-forms equals ϕ 0,1 = ∂∂ * ϕ , and the necessary condition is known to be both necessary and sufficient for compactness of the ∂-Neumann operator, see [12] and [18]. where s 1 is the smallest eigenvalue of M ϕ , then A 2 (C n , e −ϕ ) has infinite dimension.…”

“…In the following Theorem 5.1, we will consider the case where all ∆ϕ j define nontrivial doubling measures. It is known from [18] (or [12,Theorem 2.3], with slightly stronger assumptions) that, under these conditions, N ϕ j 0,1 (for the one-variable weights) is compact if and only if lim z→∞ˆB 1 (z) ∆ϕ j dλ = +∞. (5.4) holds.…”

“…Note that the symbol of its formal adjoint with respect to (1.1), which we denote by ∂ t ϕ , depends on the weight ϕ, and we denote the Hilbert space adjoint of ∂ by ∂ * ϕ . For more details on the weighted ∂-problem, see [12,11].…”