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Title of Journal: Math Ann

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Abbravation: Mathematische Annalen

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Springer Berlin Heidelberg

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DOI

10.1007/bf01836740

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1432-1807

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On CR Paneitz operators and CR pluriharmonic funct

Authors: ChinYu Hsiao
Publish Date: 2014/12/02
Volume: 362, Issue: 3-4, Pages: 903-929
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Abstract

Let XT10X be a compact orientable embeddable three dimensional strongly pseudoconvex CR manifold and let mathrmP be the associated CR Paneitz operator In this paper we show that I mathrmP is selfadjoint and mathrmP has L2 closed range Let N and Pi be the associated partial inverse and the orthogonal projection onto mathrmKermathrmP respectively then N and Pi enjoy some regularity properties II Let hatmathcal P and hatmathcal P 0 be the space of L2 CR pluriharmonic functions and the space of real part of L2 global CR functions respectively Let S be the associated Szegö projection and let tau tau 0 be the orthogonal projections onto hatmathcal P and hatmathcal P 0 respectively Then Pi =S+overlineS+F 0 tau =S+overlineS+F 1 tau 0=S+overlineS+F 2 where F 0 F 1 F 2 are smoothing operators on X In particular Pi tau and tau 0 are Fourier integral operators with complex phases and hatmathcal Pperp bigcap hbox Ker mathrmP hatmathcal P 0perp bigcap hatmathcal P hatmathcal P 0perp bigcap hbox Ker mathrmP are all finite dimensional subspaces of Cinfty X it is wellknown that hatmathcal P 0subset hatmathcal Psubset hbox Ker mathrmP III hbox Spec mathrmP is a discrete subset of mathbb R and for every lambda in hbox Spec mathrmP lambda ne 0 lambda is an eigenvalue of mathrmP and the associated eigenspace H lambda mathrmP is a finite dimensional subspace of Cinfty X

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