Riesz transforms on solvable extensions of stratified groups

Alessio Martini, Maria Vallarino

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Let G=N⋊A, where N is a stratified group and A=R acts on N via automorphic dilations. Homogeneous sub-Laplacians on N and A can be lifted to left-invariant operators on G and their sum is a sub-Laplacian Δ on G. Here we prove weak type (1,1), Lp-boundedness for p∈(1,2] and H1→L1 boundedness of the Riesz transforms YΔ−1/2 and YΔ−1Z, where Y and Z are any horizontal left-invariant vector fields on G, as well as the corresponding dual boundedness results. At the crux of the argument are large-time bounds for spatial derivatives of the heat kernel, which are new when Δ is not elliptic.
Original languageEnglish
Pages (from-to)175-200
JournalStudia Mathematica
Issue number2
Publication statusPublished - 26 Apr 2021


  • Hardy space
  • Heat kernel
  • Riesz transform
  • Singular integral operator
  • Solvable group
  • Sub-Laplacian

ASJC Scopus subject areas

  • Mathematics(all)


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