On Fundamental Solutions of Generalized Schrödinger Operators

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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,ce2d(x, y, μ)x-yn-2≤Γμ(x, y)≤Ce1d(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).

Original languageEnglish
Pages (from-to)521-564
Number of pages44
JournalJournal of Functional Analysis
Volume167
Issue number2
DOIs
StatePublished - Oct 1 1999

Bibliographical note

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

Keywords

  • Fundamental solutions
  • Riesz transforms
  • Schrödinger operators

ASJC Scopus subject areas

  • Analysis

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