Maximal operators and Hilbert transforms along variable non‐flat homogeneous curves. Issue 1 (5th April 2017)

• Record Type: Journal Article
• Authors: Guo, Shaoming; Hickman, Jonathan; Lie, Victor; Roos, Joris
• Is Part Of: Proceedings of the London Mathematical Society. Volume 115:Issue 1(2017)
• Page Start: 177
• Page End: 219
• Publication Date: 2017-04-05
• Subjects: 42B20 -- 42B25 -- 44A12 (primary); Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms.12037
• Languages: English
• ISSNs: 0024-6115

Abstract

Abstract: We prove that the maximal operator associated with variable homogeneous planar curves ( t, u t 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, u t 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, T T ∗ arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.