ON CERTAIN ESTIMATES FOR PARABOLIC MARCINKIEWICZ INTEGRAL AND EXTRAPOLATION

M Ali,E Abo-Shgair

doi:10.12732/ijpam.v96i3.9

Abstract

In this article, we establish Lp boundedness of the parametric Marcinkiewicz integral operators with rough kernels. These estimates and extrapolation arguments improve and extend some known results on parabolic Marcinkiewicz integrals. AMS Subject Classification: 40B15, 40B20, 40B25

Highlights

Throughout this article, let Rn, n ≥ 2, be the n-dimensional Euclidean space, and let Sn−1 be the unit sphere in Rn which is equipped with the normalizedLebesgue surface measure dσ = dσ(·)
The parabolic LittlewoodPaley operator μΩ was introduced by Xue, Ding and Yabuta in [20] in which they proved that μΩ is bounded for p ∈ (1, ∞) provided that Ω ∈ Lq(Sn−1) for q > 1
The authors of [6] established that μΩ is bounded under the condition Ω ∈ L(log L)1/2(Sn−1) for 1 < p < ∞

Summary

Introduction

Throughout this article, let Rn, n ≥ 2, be the n-dimensional Euclidean space, and let Sn−1 be the unit sphere in Rn which is equipped with the normalized. Define the parabolic Marcinkiewicz integral operator MτΩ,h for f ∈ S(Rn) by. The parabolic LittlewoodPaley operator μΩ was introduced by Xue, Ding and Yabuta in [20] in which they proved that μΩ is bounded for p ∈ (1, ∞) provided that Ω ∈ Lq(Sn−1) for q > 1. The authors of [6] established that μΩ is bounded under the condition Ω ∈ L(log L)1/2(Sn−1) for 1 < p < ∞. Our main interest in this paper is to study the Lp boundedness of the parabolic Marcinkiewicz integral under weak conditions on Ω and h, and apply an extrapolation method to establish new improved results. We let ∆γ(R+) ( for γ > 1) denote the collection of all measurable functions h : [0, ∞) → C satisfying h. Throughout this paper, the letter C denotes a bounded positive constant that may vary at each occurrence but independent of the essential variables

Some Lemmas

Proof of Theorem 1