Percolation of Estimates for ∂¯ by the Method of Alternating Projections

The method of alternating projections is used to examine how regularity of operators associated to the ∂ ¯ -Neumann problem percolates up the ∂ ¯ -complex. The approach revolves around operator identities—rather than estimates—that hold on any Lipschitz domain in C n , not necessarily bounded or pse...

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Bibliographic Details
Published in:The Journal of geometric analysis 2021-07, Vol.31 (7), p.6922-6940
Main Authors: Koenig, Kenneth D., McNeal, Jeffery D.
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Algorithms
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Elias M. Stein: In Memoriam
Estimates
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Mathematics
Mathematics and Statistics
Neumann problem
Operators
Percolation
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Summary:The method of alternating projections is used to examine how regularity of operators associated to the ∂ ¯ -Neumann problem percolates up the ∂ ¯ -complex. The approach revolves around operator identities—rather than estimates—that hold on any Lipschitz domain in 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 ∂ ¯ having closed range. This implies that compactness of the ∂ ¯ -Neumann operator percolates up the ∂ ¯ -complex whenever ∂ ¯ has closed range at the corresponding form levels.
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-020-00532-w