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Abstract
We establish that the Riesz transforms of all orders corresponding to the Grušin operator (Formula presented.), and the firstorder 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 weaktype (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 finitedimensional 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 

Pages (fromto)  461472 
Number of pages  12 
Journal  Mathematische Zeitschrift 
Volume  282 
Issue number  12 
DOIs  
Publication status  Published  1 Feb 2016 
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 1 Active

Heat kernel and Riesz transform on noncompact metric measure spaces
Sikora, A. & Coulhon, T.
1/02/13 → …
Project: Research