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Percolation of Estimates for \({{\bar{\partial }}}\) by the Method of Alternating Projections

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Abstract

The method of alternating projections is used to examine how regularity of operators associated to the \({{\bar{\partial }}}\)-Neumann problem percolates up the \({{\bar{\partial }}}\)-complex. The approach revolves around operator identities—rather than estimates—that hold on any Lipschitz domain in \({{\mathbb {C}}}^n\), not necessarily bounded or pseudoconvex. We show that a geometric rate of convergence in von Neumann’s alternating projection algorithm, applied to two basic projection operators, is equivalent to \({{\bar{\partial }}}\) having closed range. This implies that compactness of the \({{\bar{\partial }}}\)-Neumann operator percolates up the \({{\bar{\partial }}}\)-complex whenever \({{\bar{\partial }}}\) has closed range at the corresponding form levels.

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Notes

  1. For bounded pseudoconvex domains, the \(\Vert u \Vert ^2\) term is superfluous.

  2. This density result is proved in [15, Proposition 2.1.1] under the assumption that \(b\Omega \) is \(C^2\) smooth. Although it is stated in [22, Proposition 2.3]) for (bounded) domains with \(C^1\) boundary, the proof there uses only the strong interior cone condition, which holds since \(b\Omega \) is Lipschitz. The reduction to smooth \(\eta \) in \({\mathscr {D}}({{\bar{\partial }}}_q)\) is the only argument in this paper that requires any boundary regularity of the domain.

  3. Note that if \(u \in {\mathscr {N}}\!(\square _q)\), then \(\Vert {{\bar{\partial }}}_q u \Vert ^2 + \Vert {{\bar{\partial }}} ^{*}_q u \Vert ^2 = ({{\bar{\partial }}}_q u, {{\bar{\partial }}}_q u) + ({{\bar{\partial }}} ^{*}_q u, {{\bar{\partial }}} ^{*}_q u) = (u, \square _q u) =0\).

  4. Actually, it is true that \(c(A,B)=c(A^{\perp }, B^{\perp })\), but this is not needed here.

  5. Or, equivalently, of the canonical solution operator for \({{\bar{\partial }}}\) or for \({{\bar{\partial }}} ^{*}\).

  6. However, in part (c) of Proposition 5.1, the assumption that \({\mathscr {H}}_{0,q+1} = \{ 0 \}\) and the \(L^2\)-boundedness of \(N_{q+1}\) imply that \({\mathscr {R}}({{\bar{\partial }}}_q)\) and \({\mathscr {R}}({{\bar{\partial }}}_{q+1})\) are closed; see for instance the proof of [12, Proposition 3.5].

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  1. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Kenneth D. Koenig & Jeffery D. McNeal

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  1. Kenneth D. Koenig
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Koenig, K.D., McNeal, J.D. Percolation of Estimates for \({{\bar{\partial }}}\) by the Method of Alternating Projections. J Geom Anal 31, 6922–6940 (2021). https://doi.org/10.1007/s12220-020-00532-w

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