A maximal function for families of Hilbert transforms along homogeneous curves

Let H ( u ) be the Hilbert transform along the parabola ( t , u t 2 ) where u ∈ R . For a set U of positive numbers consider the maximal function H U f = sup { | H ( u ) f | : u ∈ U } . We obtain an (essentially) optimal result for the L p operator norm of H U when 2 < p < ∞ . The results are...

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Bibliographic Details
Published in:Mathematische annalen 2020-06, Vol.377 (1-2), p.69-114
Main Authors: Guo, Shaoming, Roos, Joris, Seeger, Andreas, Yung, Po-Lam
Format: Article
Language:English
Subjects:
Hilbert transformation
Mathematics
Mathematics and Statistics
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Summary:Let H ( u ) be the Hilbert transform along the parabola ( t , u t 2 ) where u ∈ R . For a set U of positive numbers consider the maximal function H U f = sup { | H ( u ) f | : u ∈ U } . We obtain an (essentially) optimal result for the L p operator norm of H U when 2 < p < ∞ . The results are proved for families of Hilbert transforms along more general nonflat homogeneous curves.
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-019-01915-3