Abstract
The existence and boundedness on Sobolev spaces and Hardy-Sobolev spaces for the hypersingular Marcinkiewicz integrals with variable kernels are derived.
Highlights
The function Ω x, z defined on Rn × Rn is said to belong to L∞ Rn × Lq Sn−1, if it satisfies the following two conditions: 1/q < ∞.Let α ≥ 0 and q > max{1, 2 n − 1 / n 2α }, and let Ω ∈ L∞ Rn × Lq Sn−1 satisfy the following cancellation property:Ω x, z Ym z dσ z 0, Sn−1 for all spherical harmonic polynomials Ym z with degree ≤ α and for any x ∈ Rn
The function Ω x, z defined on Rn × Rn is said to belong to L∞ Rn × Lq Sn−1, if it satisfies the following two conditions: 1 Ω x, λz Ω x, z, for any x, z ∈ Rn and any λ > 0; 2 Ω L∞ Rn ×Lq Sn−1 : supr≥0, y∈Rn Sn−1 |Ω rz y, z |qdσ z
As an application of Theorem 1.2, in Section 5 we will get the L2α − L2 boundedness for a class of the Littlewood-Paley type operators μΩ,α,S and μ∗Ω,α,λ with variable kernels and index α ≥ 0, which relate to the Lusin area integral and the Littlewood-Paley gλ∗ function, respectively
Summary
The function Ω x, z defined on Rn × Rn is said to belong to L∞ Rn × Lq Sn−1 , if it satisfies the following two conditions:. 1.8 with the constant C independent of any f ∈ Lpα Rn ∩ S Rn. Further, we can derive the boundedness of μΩ,α on Hαp for some p ≤ n/ n an additional assumption, the L1,β-Dini condition, on Ω as follows:. Let Ω x, z ∈ L∞ Rn ×Lq Sn−1 , q > max{1, 2 n−1 / n 2α }, satisfy cancellation property 1.1 and the L1,β-Dini condition 1.9 with β ≥ 0. As an application of Theorem 1.2, in Section 5 we will get the L2α − L2 boundedness for a class of the Littlewood-Paley type operators μΩ,α,S and μ∗Ω,α,λ with variable kernels and index α ≥ 0, which relate to the Lusin area integral and the Littlewood-Paley gλ∗ function, respectively. Throughout the paper, C always denotes a positive constant not necessarily the same at each occurrence. we use a ∼ b to mean the equivalence of a and b; that is, there exists a positive constant C independent of a, b such that C−1a ≤ b ≤ Ca
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
