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Riesz transforms on solvable extensions of stratified groups

Volume 259 / 2021

Alessio Martini, Maria Vallarino Studia Mathematica 259 (2021), 175-200 MSC: 22E30, 42B20, 42B30. DOI: 10.4064/sm190927-4-1 Published online: 26 April 2021


Let $G = N \rtimes A$, where $N$ is a stratified group and $A = \mathbb {R}$ 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 $\Delta $ on $G$. Here we prove weak type $(1,1)$, $L^p$-boundedness for $p \in (1,2]$ and $H^1 \to L^1$ boundedness of the Riesz transforms $Y \Delta ^{-1/2}$ and $Y \Delta ^{-1} Z$, 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 $\Delta $ is not elliptic.


  • Alessio MartiniSchool of Mathematics
    University of Birmingham
    Birmingham, B15 2TT, United Kingdom
  • Maria VallarinoDipartimento di Scienze Matematiche
    “Giuseppe Luigi Lagrange”
    Dipartimento di Eccellenza 2018–2022
    Politecnico di Torino
    Corso Duca degli Abruzzi 24
    10129 Torino, Italy

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