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

Volume 210 / 2012

Magali Folch-Gabayet, James Wright Studia Mathematica 210 (2012), 57-76 MSC: 42B15. DOI: 10.4064/sm210-1-4


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 $H^1$).


  • Magali Folch-GabayetInstituto de Matemáticas
    Universidad Nacional Autónoma de México
    Ciudad Universitaria
    México D.F., 04510, México
  • James WrightMaxwell Institute of Mathematical Sciences
    and the School of Mathematics
    University of Edinburgh
    JCMB, King's Buildings
    Mayfield Road
    Edinburgh EH9 3JZ, Scotland

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