Maximal operators and Hilbert transforms along variable non-flat homogeneous curves

Shaoming Guo*, Jonathan Hickman, Victor Lie, Joris Roos

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We prove that the maximal operator associated with variable homogeneous planar curves (t,ut α ) t∈ℝ , α≠1 positive, is bounded on L p (ℝ 2 ) for each p>1, under the assumption that u:ℝ 2 → ℝ is a Lipschitz function. Furthermore, we prove that the Hilbert transform associated with (t,ut α ) t∈ℝ , α≠1 positive, is bounded on L p (ℝ 2 ) for each p>1, under the assumption that u:ℝ 2 →ℝ is a measurable function and is constant in the second variable. Our proofs rely on stationary phase methods, TT∗ arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.

Original languageEnglish
Pages (from-to)177-219
Number of pages43
JournalProceedings of the London Mathematical Society
Volume115
Issue number1
Early online date5 Apr 2017
DOIs
Publication statusPublished - 1 Jul 2017

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