A result about the Hilbert transform along curves

Let GG be a connected and simply connected, nilpotent Lie group and let γ:(−1,1)→G\gamma :( - 1,1) \to G be a (connected) analytic curve such that γ(0)=0\gamma (0) = 0. Then the Hilbert transform along γ\gamma, \[ Tf(x)=p.v.∫0>|t|>1f(xγ(t)−1)dt/t,Tf(x) = p.v.\int _{0 > |t| > 1} {f(x\gamm...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1990, Vol.110 (4), p.905-914
Main Author: Saal, Linda V.
Format: Article
Language:English
Subjects:
Analytic functions
Automorphisms
Coordinate systems
Curves
Integers
Lie groups
Mathematical theorems
Research article
Transference
Zero
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Summary:Let GG be a connected and simply connected, nilpotent Lie group and let γ:(−1,1)→G\gamma :( - 1,1) \to G be a (connected) analytic curve such that γ(0)=0\gamma (0) = 0. Then the Hilbert transform along γ\gamma, \[ Tf(x)=p.v.∫0>|t|>1f(xγ(t)−1)dt/t,Tf(x) = p.v.\int _{0 > |t| > 1} {f(x\gamma {{(t)}^{ - 1}})dt/t} , \] is bounded on Lp(G),1>p>∞{L^p}(G),1 > p > \infty.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-1990-1019281-2