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

Magali Folch-Gabayet¹ ; James Wright²

¹ Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria México D.F., 04510, México
² Maxwell Institute of Mathematical Sciences and the School of Mathematics University of Edinburgh JCMB, King's Buildings Mayfield Road Edinburgh EH9 3JZ, Scotland

Studia Mathematica, Tome 210 (2012) no. 1, pp. 57-76

Résumé

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 $H^1$).

Détail

DOI : 10.4064/sm210-1-4

Keywords: consider singular integral operators bbb given convolution principal value distribution defined integrating against oscillating kernels form where general rational function real coefficients establish weak type bounds operators which uniform coefficients depending only degrees always these operators map hardy space bbb bbb characterise those rational phases which map even