Weak-type $(1,1)$ bounds for oscillatory singular integrals with rational phases
Volume 210 / 2012
Studia Mathematica 210 (2012), 57-76
MSC: 42B15.
DOI: 10.4064/sm210-1-4
Abstract
We consider singular integral operators on ${\Bbb R}$ given by convolution with a principal value distribution defined by integrating against oscillating kernels of the form $e^{i R(x)}/x$ 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 $H^1({\Bbb R})$ to $L^1({\Bbb R})$ and we will characterise those rational phases $R(x) = P(x)/Q(x)$ which do map $H^1$ to $L^1$ (and even $H^1$ to $H^1$).
