Weak-type (1,1) bounds for oscillatory singular integrals with rational phases

@article{MagaliFolch2012,
abstract = {We consider singular integral operators on ℝ given by convolution with a principal value distribution defined by integrating against oscillating kernels of the form $e^\{iR(x)\}/x$ 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).},
author = {Magali Folch-Gabayet, James Wright},
journal = {Studia Mathematica},
keywords = {singular integral; rational phase; weak-type ; Hardy space},
language = {eng},
number = {1},
pages = {57-76},
title = {Weak-type (1,1) bounds for oscillatory singular integrals with rational phases},
url = {http://eudml.org/doc/285592},
volume = {210},
year = {2012},
}

TY - JOUR
AU - Magali Folch-Gabayet
AU - James Wright
TI - Weak-type (1,1) bounds for oscillatory singular integrals with rational phases
JO - Studia Mathematica
PY - 2012
VL - 210
IS - 1
SP - 57
EP - 76
AB - We consider singular integral operators on ℝ given by convolution with a principal value distribution defined by integrating against oscillating kernels of the form $e^{iR(x)}/x$ 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).
LA - eng
KW - singular integral; rational phase; weak-type ; Hardy space
UR - http://eudml.org/doc/285592
ER -