The solution of the Kato square root problem for second order elliptic operators on ℝn

Pascal Auscher; Steve Hofmann; Michael Lacey; Alan McIntosh; Philippe Tchamitchian

doi:10.2307/3597201

The solution of the Kato square root problem for second order elliptic operators on ℝⁿ

Pascal Auscher^*, Steve Hofmann, Michael Lacey, Alan McIntosh, Philippe Tchamitchian

^*Corresponding author for this work

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

239 Citations (Scopus)

Abstract

We prove the Karo conjecture for elliptic operators on ℝⁿ. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L = -div (A∇) with bounded measurable coefficients in ℝⁿ is the Sobolev space H¹(ℝⁿ) in any dimension with the estimate ||√Lf|| ₂ ∼ ||∇f|| ₂.

Original language	English
Pages (from-to)	633-654
Number of pages	22
Journal	Annals of Mathematics
Volume	156
Issue number	2
DOIs	https://doi.org/10.2307/3597201
Publication status	Published - Sept 2002

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title = "The solution of the Kato square root problem for second order elliptic operators on ℝn",

abstract = "We prove the Karo conjecture for elliptic operators on ℝn. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L = -div (A∇) with bounded measurable coefficients in ℝn is the Sobolev space H1(ℝn) in any dimension with the estimate ||√Lf|| 2 ∼ ||∇f|| 2.",

author = "Pascal Auscher and Steve Hofmann and Michael Lacey and Alan McIntosh and Philippe Tchamitchian",

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AU - Hofmann, Steve

AU - Lacey, Michael

AU - McIntosh, Alan

AU - Tchamitchian, Philippe

PY - 2002/9

Y1 - 2002/9

N2 - We prove the Karo conjecture for elliptic operators on ℝn. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L = -div (A∇) with bounded measurable coefficients in ℝn is the Sobolev space H1(ℝn) in any dimension with the estimate ||√Lf|| 2 ∼ ||∇f|| 2.

AB - We prove the Karo conjecture for elliptic operators on ℝn. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L = -div (A∇) with bounded measurable coefficients in ℝn is the Sobolev space H1(ℝn) in any dimension with the estimate ||√Lf|| 2 ∼ ||∇f|| 2.

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DO - 10.2307/3597201

M3 - Article

SN - 0003-486X

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SP - 633

EP - 654

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -

The solution of the Kato square root problem for second order elliptic operators on ℝⁿ

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