Generalized parabolic Marcinkiewicz integrals associated with polynomial compound curves with rough kernels

Mohammed Ali,Qutaibeh Katatbeh

doi:10.1515/dema-2020-0004

Mohammed Ali, Qutaibeh Katatbeh

Open Access

https://doi.org/10.1515/dema-2020-0004

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Abstract

AbstractIn this article, we study the generalized parabolic parametric Marcinkiewicz integral operators { {\mathcal M} }_{{\Omega },h,{\Phi },\lambda }^{(r)} related to polynomial compound curves. Under some weak conditions on the kernels, we establish appropriate estimates of these operators. By the virtue of the obtained estimates along with an extrapolation argument, we give the boundedness of the aforementioned operators from Triebel-Lizorkin spaces to Lp spaces under weaker conditions on Ω and h. Our results represent significant improvements and natural extensions of what was known previously.

Highlights

Throughout this article, let Rn (n ≥ 2) be the n-dimensional Euclidean space and Sn−1 be the unit sphere inRn equipped with the normalized Lebesgue surface measure dσ = dσ(·)
Stein [2] and proved the Lp (1 < p ≤ 2) boundedness of Ω,1 provided that Ω ∈ Lipα(Sn−1) with 0 < α ≤ 1
There has been a considerable amount of mathematicians with respect to the study of the boundedness of the generalized parametric Marcinkiewicz integrals (r,c) Ω,h,Φ, λ. This operator was first introduced by Chen et al [17] and showed that whenever Φ(u) = u, h ≡ 1, and Ω ∈ Lq(Sn−1) for some q > 1, a positive constant C exists such that (r,c) Ω,h,Φ, λ f

Summary

Introduction

Throughout this article, let Rn (n ≥ 2) be the n-dimensional Euclidean space and Sn−1 be the unit sphere in. Stein [2] and proved the Lp (1 < p ≤ 2) boundedness of Ω,1 provided that Ω ∈ Lipα(Sn−1) with 0 < α ≤ 1 This result was investigated and improved by many researchers There has been a considerable amount of mathematicians with respect to the study of the boundedness of the generalized parametric Marcinkiewicz integrals (r,c) Ω,h,Φ, λ. This operator was first introduced by Chen et al [17] and showed that whenever Φ(u) = u, h ≡ 1, and Ω ∈ Lq(Sn−1) for some q > 1, a positive constant C exists such that. [24] proved that the parabolic Littlewood-Paley operator μΩ is of type (p,p) for all p ∈ (1,∞) provided that

46 Mohammed Ali and Qutaibeh Katatbeh

Some notations and lemmas

Findings

Proof of the main results