A geometric approach to the Littlewood-Paley theory

S. Klainerman; I. Rodnianski

doi:10.1007/s00039-006-0551-1

A geometric approach to the Littlewood-Paley theory

S. Klainerman, I. Rodnianski

Mathematics

Princeton University

Research output: Contribution to journal › Article › peer-review

40 Scopus citations

Abstract

We develop a geometric invariant Littlewood-Paley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited regularity assumptions on the metric. We give invariant descriptions of Sobolev and Besov spaces and prove some sharp product inequalities. This theory has being developed in connection with the work of the authors on the geometry of null hypersurfaces with a finite curvature flux condition, see [KR1,2]. We are confident however that it can be applied, and extended, to many different situations.

Original language	American English
Pages (from-to)	126-163
Number of pages	38
Journal	Geometric and Functional Analysis
Volume	16
Issue number	1
DOIs	https://doi.org/10.1007/s00039-006-0551-1
State	Published - Feb 2006

ASJC Scopus subject areas

Analysis
Geometry and Topology

Keywords

Bochner identity
Heat flow for tensors
Paraproducts

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10.1007/s00039-006-0551-1

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abstract = "We develop a geometric invariant Littlewood-Paley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited regularity assumptions on the metric. We give invariant descriptions of Sobolev and Besov spaces and prove some sharp product inequalities. This theory has being developed in connection with the work of the authors on the geometry of null hypersurfaces with a finite curvature flux condition, see [KR1,2]. We are confident however that it can be applied, and extended, to many different situations.",

keywords = "Bochner identity, Heat flow for tensors, Paraproducts",

author = "S. Klainerman and I. Rodnianski",

note = "Funding Information: Keywords and phrases: Heat flow for tensors, paraproducts, Bochner identity 2000 Mathematics Subject Classification: 35J10 The first author is partially supported by NSF grant DMS-0070696. The second author is a Clay Prize Fellow and is partially supported by NSF grant DMS-01007791.",

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AU - Rodnianski, I.

N1 - Funding Information: Keywords and phrases: Heat flow for tensors, paraproducts, Bochner identity 2000 Mathematics Subject Classification: 35J10 The first author is partially supported by NSF grant DMS-0070696. The second author is a Clay Prize Fellow and is partially supported by NSF grant DMS-01007791.

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N2 - We develop a geometric invariant Littlewood-Paley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited regularity assumptions on the metric. We give invariant descriptions of Sobolev and Besov spaces and prove some sharp product inequalities. This theory has being developed in connection with the work of the authors on the geometry of null hypersurfaces with a finite curvature flux condition, see [KR1,2]. We are confident however that it can be applied, and extended, to many different situations.

AB - We develop a geometric invariant Littlewood-Paley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited regularity assumptions on the metric. We give invariant descriptions of Sobolev and Besov spaces and prove some sharp product inequalities. This theory has being developed in connection with the work of the authors on the geometry of null hypersurfaces with a finite curvature flux condition, see [KR1,2]. We are confident however that it can be applied, and extended, to many different situations.

KW - Bochner identity

KW - Heat flow for tensors

KW - Paraproducts

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JO - Geometric and Functional Analysis

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A geometric approach to the Littlewood-Paley theory

Abstract

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