Heat kernels, upper bounds and Hardy spaces associated to the generalized Schrödinger operators

Liangchuan Wu, Lixin Yan*

*Corresponding author for this work

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

Let L=-δ+μ be the generalized Schrödinger operator on Rn, n≥3, where μ≢0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. Based on Shen's work for the fundamental solution of L in [23], we establish the following upper bound for semigroup kernels Kt(x,y), associated to e-tL,0≤Kt(x,y)≤Cht(x-y)e-εdμ(x,y,t), where ht(x)=(4πt)-n/2e-|x|2/(4t), and dμ(x, y, t) is some parabolic type distance function associated with μ. As a consequence,0≤Kt(x,y)≤Cht(x-y)exp (-c0(1+m(x,μ)max {|x-y|,√t})1/k0+1),where m(x, μ) is some auxiliary function associated with μ. We then study a Hardy space HL1 by means of a maximal function associated with the heat semigroup e-tL generated by -L to obtain its characterizations via atomic decomposition and Riesz transforms. Also the dual space BMOL of HL1 is studied in this paper.

Original languageEnglish
Pages (from-to)3709-3749
Number of pages41
JournalJournal of Functional Analysis
Volume270
Issue number10
DOIs
Publication statusPublished - 15 May 2016
Externally publishedYes

Keywords

  • hardy space
  • heat kernel
  • scale-invariant kato conditions
  • schrödinger operators

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