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### Abstract

We establish that the Riesz transforms of all orders corresponding to the Grušin operator (Formula presented.), and the first-order operators (∇_{x},x^{ν}∇_{y}) where x∈R^{n}, y∈R^{m}, N∈N+, and ν∈{1,…,n}^{N}, are bounded on L_{p}(R^{n+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 toH_{N} and the operators (∇_{x},|x|^{N}∇_{y}) are bounded on L_{p}(R^{n+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},x^{ν}∇_{y}) generate a finite-dimensional nilpotent Lie algebra, the corresponding connected, simply connected, nilpotent Lie group is isometrically represented on the spaces L_{p}(R^{n+m}) and H_{N} is the corresponding sublaplacian.

Original language | English |
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Pages (from-to) | 461-472 |

Number of pages | 12 |

Journal | Mathematische Zeitschrift |

Volume | 282 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 1 Feb 2016 |

## Fingerprint Dive into the research topics of 'Grušin operators, Riesz transforms and nilpotent Lie groups'. Together they form a unique fingerprint.

## Projects

- 1 Active

## Heat kernel and Riesz transform on non-compact metric measure spaces

Sikora, A. & Coulhon, T.

1/02/13 → …

Project: Research

## Cite this

*Mathematische Zeitschrift*,

*282*(1-2), 461-472. https://doi.org/10.1007/s00209-015-1548-y