Abstract
Let , the authors introduce in this paper a class of the hypersingular Marcinkiewicz integrals along surface with variable kernels defined by , where with . The authors prove that the operator is bounded from Sobolev space to space for , and from Hardy-Sobolev space to space for . As corollaries of the result, they also prove the boundedness of the Littlewood-Paley type operators and which relate to the Lusin area integral and the Littlewood-Paley function.
Highlights
Let Ên n ≥ 2 be the n-dimensional Euclidean space and Ën−1 be the unit sphere in Ên equipped with the normalized Lebesgue measure dσ dσ ·
We first introduce some definitions about the variable kernel Ω x, z
A function Ω x, z defined on Ên × Ên is said to be in L∞ Ên × Lq Ën−1, q ≥ 1, if Ω x, z satisfies the following two conditions: 1 Ω x, λz Ω x, z, for any x, z ∈ Ên and any λ > 0; Ê Ë 2 Ω L∞ n ×Lq n−1 supr≥0, y∈Ên Ën−1 |Ω rz y, z |qdσ z 1/q < ∞
Summary
Let Ên n ≥ 2 be the n-dimensional Euclidean space and Ën−1 be the unit sphere in Ên equipped with the normalized Lebesgue measure dσ dσ ·. A function Ω x, z defined on Ên × Ên is said to be in L∞ Ên × Lq Ën−1 , q ≥ 1, if Ω x, z satisfies the following two conditions:. Our aim of this paper is to study the hypersingular Marcinkiewicz integral μΦΩ,α along surfaces with variable kernel Ω, and with index α ≥ 0, on the homogeneous Sobolev space. Our result can be extended to the Littlewood-Paley type operators μΦΩ,α,S and μ∗Ω,Φ,α,λ with variable kernels and index α ≥ 0, which relate to the Lusin area integral and the Littlewood-Paley gλ∗ function, respectively. The authors in 9 proved the boundedness of hypersingular Marcinkiewicz integral with variable kernels on homogeneous Sobolev space Lpα Rn for 1 < p ≤ 2 and 0 < α < 1 without any smoothness on Ω. Throughout this paper, the letter C always remains to denote a positive constant not necessarily the same at each occurrence
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