A note on certain integrals along polynomial compound curves

Feng Liu,Pu Zhang

doi:10.1186/s13660-019-2023-9

Abstract

In this paper we consider a singular integral operator and a parametric Marcinkiewicz integral operator with rough kernel. These operators have singularity along sets of the form curves {x=P(varphi (|y|))y'}, where P is a real polynomial satisfying P(0)=0 and φ satisfies certain smooth conditions. Under the conditions that varOmega in H^{1} (mathbf{S}^{n-1}) and hin Delta _{gamma }(mathbf{R}_{+}) for some gamma >1, we prove that the above operators are bounded on the Lebesgue space L^{2}( mathbf{R}^{n}). Moreover, the L^{2}-bounds of the maximal functions related to the above integrals are also established. Particularly, the bounds are independent of the coefficients of the polynomial P. In addition, we also present certain Hardy type inequalities related to these operators.

Highlights

Let Rn (n ≥ 2) be the n-dimensional Euclidean space and Sn–1 denote the unit sphere in Rn equipped with the induced Lebesgue measure dσ
Ω (y)h(|y|) K (y) = |y|n, where h is a suitable function defined on R+ := (0, ∞) and Ω is homogeneous of degree zero, with Ω ∈ L1(Sn–1) and
For a suitable function φ defined on R+, we consider that the singular integral operator Th,Ω,P,φ along the “polynomial compound curve” P(φ(|y|))y on Rn is defined by

Summary

Introduction

Let Rn (n ≥ 2) be the n-dimensional Euclidean space and Sn–1 denote the unit sphere in Rn equipped with the induced Lebesgue measure dσ. During the last several years the Lp mapping properties for singular integral operators with singularity along various sets and with rough kernel in H1(Sn–1) have been actively studied by many authors. In this paper we focus on the Lp-boundedness of the singular integral operator along the polynomial compound curves with rough kernels Ω ∈ H1(Sn–1) and h ∈ γ (R+) for some γ > 1.

Results

Conclusion