We establish sharp bounds for oscillatory singular integrals with an arbitrary real polynomial phase P. The kernels are allowed to be rough both on the unit sphere and in the radial direction. We show that the bounds grow no faster than log deg(P), which is optimal and was first obtained by Papadimitrakis and Parissis (2010) for kernels without any radial roughness. Among key ingredients of our methods are an L¹ → L² estimate and extrapolation.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm221-3-4,
author = {Hussain Mohammad Al-Qassem and Leslie Cheng and Yibiao Pan},
title = {Rough oscillatory singular integrals on Rn},
journal = {Studia Mathematica},
volume = {223},
year = {2014},
pages = {249-267},
zbl = {1298.42016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm221-3-4}
}
Hussain Mohammad Al-Qassem; Leslie Cheng; Yibiao Pan. Rough oscillatory singular integrals on ℝⁿ. Studia Mathematica, Tome 223 (2014) pp. 249-267. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm221-3-4/