BMOL(H{double-struck}n) spaces and Carleson measures for Schrödinger operators

Chin Cheng Lin, Heping Liu

Research output: Contribution to journalArticlepeer-review

32 Scopus citations

Abstract

Let L=-ΔH{double-struck}n+V be a Schrödinger operator on the Heisenberg group H{double-struck}n, where ΔH{double-struck}n is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class BQ2. Here Q is the homogeneous dimension of H{double-struck}n. In this article we investigate the dual space of the Hardy-type space HL1(H{double-struck}n) associated with the Schrödinger operator L, which is a kind of BMO-type space BMOL(H{double-struck}n) defined by means of a revised sharp function related to the potential V. We give the Fefferman-Stein type decomposition of BMOL-functions with respect to the (adjoint) Riesz transforms R̃jL for L, and characterize BMOL(H{double-struck}n) in terms of the Carleson measure. We also establish the BMOL-boundedness of some operators, such as the (adjoint) Riesz transforms R̃jL, the Littlewood-Paley function sQL, the Lusin area integral SQL, the Hardy-Littlewood maximal function, and the semigroup maximal function. All results hold for stratified groups as well.

Original languageEnglish
Pages (from-to)1631-1688
Number of pages58
JournalAdvances in Mathematics
Volume228
Issue number3
DOIs
StatePublished - 20 Oct 2011

Keywords

  • BMO space
  • Carleson measure
  • Hardy space
  • Heisenberg group
  • Reverse Hölder class
  • Schrödinger operator
  • Stratified group

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