Joint functional calculi and a sharp multiplier theorem for the Kohn Laplacian on spheres

Alessio Martini

doi:10.1007/s00209-016-1813-8

Joint functional calculi and a sharp multiplier theorem for the Kohn Laplacian on spheres

Alessio Martini

Mathematics

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7 Citations (Scopus)

159 Downloads (Pure)

Abstract

Let b be the Kohn Laplacian acting on (0; j)-forms on the unit sphere in Cn. In a recent paper of Casarino, Cowling, Sikora and the author, a
spectral multiplier theorem of Mihlin{Hormander type for b is proved in the case 0 < j < n 􀀀 1. Here we prove an analogous theorem in the exceptional
cases j = 0 and j = n 􀀀 1, including a weak type (1; 1) endpoint estimate. We also show that both theorems are sharp. The proof hinges on an abstract
multivariate multiplier theorem for systems of commuting operators.

Original language	English
Pages (from-to)	1539-1574
Journal	Mathematische Zeitschrift
Volume	286
Issue number	3-4
Early online date	10 Nov 2016
DOIs	https://doi.org/10.1007/s00209-016-1813-8
Publication status	Published - Aug 2017

Keywords

tangential Cauchy-Riemann complex
multivariable multiplier theorem
Kohn Laplacian
spectral multiplier

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abstract = "Let b be the Kohn Laplacian acting on (0; j)-forms on the unit sphere in Cn. In a recent paper of Casarino, Cowling, Sikora and the author, aspectral multiplier theorem of Mihlin{Hormander type for b is proved in the case 0 < j < n 􀀀 1. Here we prove an analogous theorem in the exceptionalcases j = 0 and j = n 􀀀 1, including a weak type (1; 1) endpoint estimate. We also show that both theorems are sharp. The proof hinges on an abstractmultivariate multiplier theorem for systems of commuting operators.",

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T1 - Joint functional calculi and a sharp multiplier theorem for the Kohn Laplacian on spheres

AU - Martini, Alessio

PY - 2017/8

Y1 - 2017/8

N2 - Let b be the Kohn Laplacian acting on (0; j)-forms on the unit sphere in Cn. In a recent paper of Casarino, Cowling, Sikora and the author, aspectral multiplier theorem of Mihlin{Hormander type for b is proved in the case 0 < j < n 􀀀 1. Here we prove an analogous theorem in the exceptionalcases j = 0 and j = n 􀀀 1, including a weak type (1; 1) endpoint estimate. We also show that both theorems are sharp. The proof hinges on an abstractmultivariate multiplier theorem for systems of commuting operators.

AB - Let b be the Kohn Laplacian acting on (0; j)-forms on the unit sphere in Cn. In a recent paper of Casarino, Cowling, Sikora and the author, aspectral multiplier theorem of Mihlin{Hormander type for b is proved in the case 0 < j < n 􀀀 1. Here we prove an analogous theorem in the exceptionalcases j = 0 and j = n 􀀀 1, including a weak type (1; 1) endpoint estimate. We also show that both theorems are sharp. The proof hinges on an abstractmultivariate multiplier theorem for systems of commuting operators.

KW - tangential Cauchy-Riemann complex

KW - multivariable multiplier theorem

KW - Kohn Laplacian

KW - spectral multiplier

U2 - 10.1007/s00209-016-1813-8

DO - 10.1007/s00209-016-1813-8

M3 - Article

SN - 0025-5874

VL - 286

SP - 1539

EP - 1574

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3-4

ER -

Joint functional calculi and a sharp multiplier theorem for the Kohn Laplacian on spheres

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Keywords

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