We consider singular integral operators on ℝ given by convolution with a principal value distribution defined by integrating against oscillating kernels of the form where R(x) = P(x)/Q(x) is a general rational function with real coefficients. We establish weak-type (1,1) bounds for such operators which are uniform in the coefficients, depending only on the degrees of P and Q. It is not always the case that these operators map the Hardy space H¹(ℝ) to L¹(ℝ) and we will characterise those rational phases R(x) = P(x)/Q(x) which do map H¹ to L¹ (and even H¹ to H¹).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm210-1-4,
author = {Magali Folch-Gabayet and James Wright},
title = {Weak-type (1,1) bounds for oscillatory singular integrals with rational phases},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {57-76},
zbl = {1258.42011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm210-1-4}
}
Magali Folch-Gabayet; James Wright. Weak-type (1,1) bounds for oscillatory singular integrals with rational phases. Studia Mathematica, Tome 209 (2012) pp. 57-76. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm210-1-4/