Boundedness of One-Sided Oscillatory Integral Operators on Weighted Lebesgue Spaces

Zunwei Fu,Shanzhen Lu,Shaoguang Shi,Yibiao Pan

doi:10.1155/2014/291397

Abstract

We consider one-sided weight classes of Muckenhoupt type, but larger than the classical Muckenhoupt classes, and study the boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces using interpolation of operators with change of measures.

Highlights

Introduction and Main ResultsOscillatory integrals in one form or another have been an essential part of harmonic analysis from the very beginnings of that subject; three chapters are devoted to them in the celebrated Stein’s book [1]
Where P (x, y) is a real-valued polynomial defined on R × R and the kernel K is a one-sided Calderon-Zygmund kernel with support in R− and R+, respectively
We say a weight w satisfies the one-sided reverse Holder RHr+ condition [18] if there exists C > 0 such that for any a < b and 1 < r < ∞, b w(x)rdx

Summary

Introduction and Main Results

Oscillatory integrals in one form or another have been an essential part of harmonic analysis from the very beginnings of that subject; three chapters are devoted to them in the celebrated Stein’s book [1]. For any real-valued polynomial P (x, y), the oscillatory integral operator T is of type (Lp, Lp) and its norm depends on the total degree of P, but not on the coefficients of P in other respects. The oscillatory singular integral operator T is of type (Lp(w), Lp(w)) with w ∈ Ap. Here its operator norm is bounded by a constant depending on the total degree of P, but not on the coefficients of P in other respects. Theorem 4 is the one-sided version of weighted norm inequality of singular integral due to Coifman and Fefferman [9]. Where P (x, y) is a real-valued polynomial defined on R × R and the kernel K is a one-sided Calderon-Zygmund kernel with support in R− and R+, respectively.

Preliminaries

Proof of Theorem 5