Weighted Estimates for Oscillatory Singular Integrals

H Al-Qassem,L Cheng,Y Pan,A Fukui

doi:10.1155/2013/543748

Abstract

We establish uniform bounds for oscillatory singular integrals as well as oscillatory singular integral operators. We allow the singular kernel to be given by a function in the Hardy space <svg style="vertical-align:-2.3205pt;width:61.875px;" id="M1" height="18.725" version="1.1" viewBox="0 0 61.875 18.725" width="61.875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28
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Highlights

In this paper we establish uniform bounds for oscillatory singular integrals
Let n ≥ 2 and Sn−1 denote the unit sphere in Rn equipped with the induced Lebesgue measure σ
= {P : Rm 󳨀→ R : P be a polynomial with deg (P) ≤ d}. (3)

Summary

Research Article Weighted Estimates for Oscillatory Singular Integrals

We establish uniform bounds for oscillatory singular integrals as well as oscillatory singular integral operators. We allow the singular kernel to be given by a function in the Hardy space H1(Sn−1), while such results were known previously only for kernels in L log L(Sn−1), a proper subspace of H1(Sn−1). One of our results established a Lp(w) → Lp(w) bound for certain weights. It provides a solution to an open problem in Lu (2005)

Introduction

Journal of Function Spaces and Applications

Therefore we have