Hypersingular Marcinkiewicz Integrals along Surface with Variable Kernels on Sobolev Space and Hardy-Sobolev Space
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Research Article Hypersingular Marcinkiewicz Integrals along Surface with Variable Kernels on Sobolev Space and Hardy-Sobolev Space Wei Ruiying and Li Yin School of Mathematics and Information Science, Shaoguan University, Shaoguan 512005, China Correspondence should be addressed to Wei Ruiying, [email protected] Received 30 June 2010; Revised 5 December 2010; Accepted 20 January 2011 Academic Editor: Andrei Volodin Copyright q 2011 W. Ruiying and L. Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let α ≥ 0, the authors introduce in this paper a class of the hypersingular Marcinkiewicz integrals along surface with variable kernels defined by μΦ Ω,α f x ∞ 1/2 2 n−1 32α ∞ n q n−1 0 | |y|≤t Ωx, y/|y| f x − Φ|y|y dy| dt/t , where Ωx, z ∈ L Ê × L Ë
with q > max{1, 2n − 1/n 2α}. The authors prove that the operator μΦ Ω,α is bounded from p p n p n Sobolev space Lα Ê to L Ê space for 1 < p ≤ 2, and from Hardy-Sobolev space Hα Ên to p n 2 n L Ê space for n/n α < p ≤ 1. As corollaries of the result, they also prove the L˙ α R − L2 Rn boundedness of the Littlewood-Paley type operators μΦ and μ∗,Φ which relate to the Lusin Ω,α,λ Ω,α,S ∗ area integral and the Littlewood-Paley gλ function.
1. Introduction 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·. For x ∈ Ên \ {0}, let x x/|x|. Before stating our theorems, we first introduce some definitions about the variable kernel Ωx, z. A function Ωx, z defined on Ên × Ên is said to be in L∞ Ên × Lq Ën−1, q ≥ 1, if Ωx, z satisfies the following two conditions: 1 Ωx, λz Ωx, z, for any x, z ∈ Ên and any λ > 0; 1/q 2 ΩL∞ Ên×Lq Ën−1 supr≥0, y∈Ên Ën−1 |Ωrz y, z |q dσz < ∞. In 1955, Calderon ´ and Zygmund 1 investigated the Lp boundedness of the singular integrals TΩ with variable kernel. They found that these operators connect closely with the
2
Journal of Inequalities and Applications
problem about the second-order linear elliptic equations with variable coefficients. In 2002, Tang and Yang 2 gave Lp boundedness of the singular integrals with variable kernels associated to surfaces of the form {x Φ|y|y }, where y y/|y| for any y ∈ Ên \ {0} n ≥ 2. That is, they considered the variable Calderon-Zygmund singular integral operator TΩΦ ´ defined by TΩΦ f x
Ω x, y p·v· n f x − Φ y y dy. Ên y
1.1
On the other hand, as a related vector-valued singular integral with variable kernel, the Marcinkiewicz singular with rough variable kernel associated with surfaces of the form {x Φ|y|y } is considered. It is defined by μΦ Ω f x
∞ 2 dt 1/2 Φ , FΩ,t x 3 t 0
1.2
where Φ FΩ,t x
|y|≤t
Ë
n−1
Ω x, y n−1 f x − Φ y y dy
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