Complex tangential flows and compactness of the ∂-Neumann operator

Abstract

We provide geometric conditions on the set of boundary points of infinite type of a smooth bounded pseudoconvex domain in ‫ރ‬ n implying that the ∂-Neumann operator is compact. These conditions are formulated in terms of certain short time flows in suitable complex tangential directions. It is noteworthy that compactness is not established via the known potential theoretic sufficient conditions. Our results generalize to ‫ރ‬ n the ‫ރ‬ 2 results of the second author.

Geometric Sufficient Conditions for Compactness of the Complex Green Operator

“…We now discuss several classes of examples where Theorem 1 applies. This discussion is analogous to the corresponding discussion in [20].…”

“…A geometrically simple corollary to theorem 1 is obtained using a suitable cone condition on the set K of weakly pseudoconvex points of M (compare [28], Corollary 2, [20], Corollary 1). We say that M \ K satisfies a complex tangential cone condition, if there is a (possibly small) open cone C in R 2n ≈ C n so that the following holds.…”

“…It suffices to establish (5) for u ∈ C ∞ (0,q) (M), as this space is dense in dom(∂ M ) ∩ dom(∂ * M ) in the graph norm u graph = ∂ M u + ∂ * M u + u (this follows from Friedrichs' Lemma, see for example Appendix D in [4]). The implementation of the strategy outlined in section 2 at first proceeds in exactly the same way as the proof of the main theorem in [28] (or Theorem 4.30 in [29]), with the simplification that there is no need to extend the vector fields Z P,j from M. These arguments (pages 705-708 in [28], bottom of page 120 to page 124 in [29]) give the following estimate, which is analogous to (14) in [28] and (2) in [20].…”

“…The exponent 2 − (C 3 − 1)(2m − 1) is strictly positive (because C 3 < 1 + 2/(2m − 1)); it depends on the dimension (2m − 1) of M. This dimension dependence arises from the comparison of volumes argument in [28] (bottom of page 707 to top of page 708, page 123 in [29]). Note that the respective dimensions in [28] (or [29]) and [20] are 4 and 2n, giving exponents of 6−4C 3 and 2n+2−2nC 3 , respectively. A reference for the subelliptic 1/2-estimates for ∂ M near a strictly pseudoconvex point is [14], Theorem 2.5.…”

“…1 When they are orientable, they are boundaries of complex varieties, but in general only in the sense of currents, see [10]. 1 in the ∂-Neumann problem from [28,20]. We refer the reader to the introductions in these two papers for a detailed discussion of the relevance of this condition vis-à-vis the potential theoretic condition property(P ).…”