On Fundamental Solutions of Generalized Schrödinger Operators

Zhongwei Shen

doi:10.1006/jfan.1999.3455

On Fundamental Solutions of Generalized Schrödinger Operators

Zhongwei Shen

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55 Scopus citations

Abstract

We consider the generalized Schrödinger operator -Δ+μ, where μ is a nonnegative Radon measure in Rⁿ, n≥3. Assuming that μ satisfies certain scale-invariant Kato conditions and doubling conditions we establish the following bounds for the fundamental solution of -Δ+μ in Rⁿ,ce^-ε₂^{d(x, y, μ)}x-y^n-2≤Γ_μ(x, y)≤Ce^-ε₁^{d(x, y, μ)}x-y^n-2, where d(x, y, μ) is the distance function for the modified Agmon metric m(x, μ)dx² associated with μ. We also study the boundedness of the corresponding Riesz transforms ∇(-Δ+μ)^-1/2 on L^p(Rⁿ, dx).

Original language	English
Pages (from-to)	521-564
Number of pages	44
Journal	Journal of Functional Analysis
Volume	167
Issue number	2
DOIs	https://doi.org/10.1006/jfan.1999.3455
State	Published - Oct 1 1999

Bibliographical note

Funding Information:
1Supported in part by the AMS Centennial Research Fellowship and the NSF grand DMS-9732894.

Funding

1Supported in part by the AMS Centennial Research Fellowship and the NSF grand DMS-9732894.

Funders	Funder number
National Science Foundation (NSF)	DMS-9732894
Directorate for Mathematical and Physical Sciences	9732894

Keywords

Fundamental solutions
Riesz transforms
Schrödinger operators

ASJC Scopus subject areas

Analysis

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10.1006/jfan.1999.3455

Cite this

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title = "On Fundamental Solutions of Generalized Schr{\"o}dinger Operators",

abstract = "We consider the generalized Schr{\"o}dinger operator -Δ+μ, where μ is a nonnegative Radon measure in Rn, n≥3. Assuming that μ satisfies certain scale-invariant Kato conditions and doubling conditions we establish the following bounds for the fundamental solution of -Δ+μ in Rn,ce-ε2d(x, y, μ)x-yn-2≤Γμ(x, y)≤Ce-ε1d(x, y, μ)x-yn-2, where d(x, y, μ) is the distance function for the modified Agmon metric m(x, μ)dx2 associated with μ. We also study the boundedness of the corresponding Riesz transforms ∇(-Δ+μ)-1/2 on Lp(Rn, dx).",

keywords = "Fundamental solutions, Riesz transforms, Schr{\"o}dinger operators",

author = "Zhongwei Shen",

note = "Funding Information: 1Supported in part by the AMS Centennial Research Fellowship and the NSF grand DMS-9732894.",

year = "1999",

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AU - Shen, Zhongwei

N1 - Funding Information: 1Supported in part by the AMS Centennial Research Fellowship and the NSF grand DMS-9732894.

PY - 1999/10/1

Y1 - 1999/10/1

N2 - We consider the generalized Schrödinger operator -Δ+μ, where μ is a nonnegative Radon measure in Rn, n≥3. Assuming that μ satisfies certain scale-invariant Kato conditions and doubling conditions we establish the following bounds for the fundamental solution of -Δ+μ in Rn,ce-ε2d(x, y, μ)x-yn-2≤Γμ(x, y)≤Ce-ε1d(x, y, μ)x-yn-2, where d(x, y, μ) is the distance function for the modified Agmon metric m(x, μ)dx2 associated with μ. We also study the boundedness of the corresponding Riesz transforms ∇(-Δ+μ)-1/2 on Lp(Rn, dx).

AB - We consider the generalized Schrödinger operator -Δ+μ, where μ is a nonnegative Radon measure in Rn, n≥3. Assuming that μ satisfies certain scale-invariant Kato conditions and doubling conditions we establish the following bounds for the fundamental solution of -Δ+μ in Rn,ce-ε2d(x, y, μ)x-yn-2≤Γμ(x, y)≤Ce-ε1d(x, y, μ)x-yn-2, where d(x, y, μ) is the distance function for the modified Agmon metric m(x, μ)dx2 associated with μ. We also study the boundedness of the corresponding Riesz transforms ∇(-Δ+μ)-1/2 on Lp(Rn, dx).

KW - Fundamental solutions

KW - Riesz transforms

KW - Schrödinger operators

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DO - 10.1006/jfan.1999.3455

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EP - 564

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 2

ER -

On Fundamental Solutions of Generalized Schrödinger Operators

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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