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Title of Journal: Math Z

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Abbravation: Mathematische Zeitschrift

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Springer Berlin Heidelberg

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DOI

10.1002/tox.20273

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1432-1823

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Grušin operators Riesz transforms and nilpotent L

Authors: Derek W Robinson Adam Sikora
Publish Date: 2015/10/27
Volume: 282, Issue: 1-2, Pages: 461-472
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Abstract

We establish that the Riesz transforms of all orders corresponding to the Grušin operator H N=nabla x2x2Nnabla y2 and the firstorder operators nabla xxnu nabla y where xin mathbfRn yin mathbfRm Nin mathbfN + and nu in 1ldots nN are bounded on L pmathbfRn+m for all pin langle 1infty rangle and are also weaktype 1 1 Moreover the transforms of order less than or equal to N+1 corresponding to H N and the operators nabla x xNnabla y are bounded on L pmathbfRn+m for all pin langle 1infty rangle But if N is odd all transforms of order N+2 are bounded if and only if pin langle 1nrangle The proofs are based on the observation that the nabla xxnu nabla y generate a finitedimensional nilpotent Lie algebra the corresponding connected simply connected nilpotent Lie group is isometrically represented on the spaces L pmathbfRn+m and H N is the corresponding sublaplacian

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