Grušin operators, Riesz transforms and nilpotent Lie groups

Derek W. Robinson, Adam Sikora*

*Corresponding author for this work

    Research output: Contribution to journalArticle

    Abstract

    We establish that the Riesz transforms of all orders corresponding to the Grušin operator (Formula presented.), and the first-order operators (∇x,xνy) where x∈Rn, y∈Rm, N∈N+, and ν∈{1,…,n}N, are bounded on Lp(Rn+m) for all p∈⟨1,∞⟩ and are also weak-type (1, 1). Moreover, the transforms of order less than or equal to N+1 corresponding toHN and the operators (∇x,|x|Ny) are bounded on Lp(Rn+m) for all p∈⟨1,∞⟩. But if N is odd all transforms of order N+2 are bounded if and only if p∈⟨1,n⟩. The proofs are based on the observation that the (∇x,xνy) generate a finite-dimensional nilpotent Lie algebra, the corresponding connected, simply connected, nilpotent Lie group is isometrically represented on the spaces Lp(Rn+m) and HN is the corresponding sublaplacian.

    Original languageEnglish
    Pages (from-to)461-472
    Number of pages12
    JournalMathematische Zeitschrift
    Volume282
    Issue number1-2
    DOIs
    Publication statusPublished - 1 Feb 2016

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