On weighted weak type norm inequalities for one-sided oscillatory singular integrals
We consider one sided weight classes of Muckenhoupt type and study the weighted weak type norm inequalities of a class of one sided oscillatory singular integrals with smooth kernel Introduction Oscillatory integrals have been an essential part of harmonic analysis three chapters are devoted to them in the celebrated Stein s book ST Many important operators in harmonic analysis are some versions of oscil latory integrals such as the Fourier transform the Bochner Riesz means the Radon transform in CT technology and so on For a more complete ac count on oscillatory integrals in classical harmonic analysis we would like to refer the interested reader to G L L LDY LZ PS and refer ences therein Another early impetus for the study of oscillatory integrals came with their application to number theory B In more recent times the operators fashioned from oscillatory integrals such as pseudo di erential operator in PDE become another motivation to study them Based on the estimates of some kinds of oscillatory integrals one can establish the well posedness theory of a class of dispersive equations for some of this works we refer to CM KPV KPV This paper is focused on a class of oscillatory singular integrals related to the one de ned by Ricci and Stein RS Tf x p v Z R e x y K x y f y dy where P x y is a real valued polynomial de ned on R R and K C R n f g is a Calder on Zygmund kernel which satis es jK x j C jxj jrK x j C jxj Mathematics Subject Classi cation Primary B Secondary B