Grušin operators, Riesz transforms and nilpotent Lie groups

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• 作者：Derek W. Robinson ; Adam Sikora
• 关键词：35J70 ; 35H20 ; 43A65 ; 22E45
• 刊名：Mathematische Zeitschrift
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：282
• 期：1-2
• 页码：461-472
• 全文大小：431 KB
• 参考文献：1. Almeida, V., Betancor, J.J., Castro, A.J., Sadarangani, K.: Riesz transforms for the Bessel-Grushin operator. arXiv:​1304.​6199 (2013)
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  9. Robinson, D.W., Sikora, A.: Analysis of degenerate elliptic operators of Grušin type. Math. Z. 260, 475-508 (2008)MathSciNet CrossRef
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  11. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)
  12. ter Elst, A.F.M., Robinson, D.W.: Subcoercivity and subelliptic operators on Lie groups I: free nilpotent groups. Potential Anal. 3, 283-337 (1994)MathSciNet CrossRef
  13. ter Elst, A.F.M., Robinson, D.W.: Subcoercivity and subelliptic operators on Lie groups II: the general case. Potential Anal. 4, 205-243 (1995)MathSciNet CrossRef
  14. ter Elst, A.F.M., Robinson, D.W., Sikora, A.: Heat kernels and Riesz transforms on nilpotent Lie groups. Coll. Math. 74, 191-218 (1997)
  
• 作者单位：Derek W. Robinson (1)
  Adam Sikora (2)
  
  1. Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT, 0200, Australia
  2. Department of Mathematics, Macquarie University, Sydney, NSW, 2109, Australia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823

文摘

We establish that the Riesz transforms of all orders corresponding to the Grušin operator \(H_N=-\nabla _{x}^2-|x|^{2N}\,\nabla _{y}^2\), and the first-order operators \((\nabla _{x},x^\nu \,\nabla _{y})\) where \(x\in \mathbf{R}^n\), \(y\in \mathbf{R}^m\), \(N\in \mathbf{N}_+\), and \(\nu \in \{1,\ldots ,n\}^N\), are bounded on \(L_p(\mathbf{R}^{n+m})\) for all \(p\in \langle 1,\infty \rangle \) and are also weak-type (1, 1). Moreover, the transforms of order less than or equal to \(N+1\) corresponding to \(H_N\) and the operators \((\nabla _{x}, |x|^N\nabla _{y})\) are bounded on \(L_p(\mathbf{R}^{n+m})\) for all \(p\in \langle 1,\infty \rangle \). But if N is odd all transforms of order \(N+2\) are bounded if and only if \(p\in \langle 1,n\rangle \). The proofs are based on the observation that the \((\nabla _{x},x^\nu \,\nabla _{y})\) generate a finite-dimensional nilpotent Lie algebra, the corresponding connected, simply connected, nilpotent Lie group is isometrically represented on the spaces \(L_p(\mathbf{R}^{n+m})\) and \(H_N\) is the corresponding sublaplacian. Mathematics Subject Classification 35J70 35H20 43A65 22E45