Journal of Inequalities and Applications
Volume 2011 (2011), Article ID 479576, 14 pages
doi:10.1155/2011/479576
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

Received 30 June 2010; Revised 5 December 2010; Accepted 20 January 2011

Academic Editor: Andrei Volodin

Copyright © 2011 Wei Ruiying and Li 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.

Abstract

Let 𝛼 ≥ 0 , the authors introduce in this paper a class of the hypersingular Marcinkiewicz integrals along surface with variable kernels defined by 𝜇 Φ Ω , 𝛼 ( 𝑓 ) ( 𝑥 ) = ( ∫ ∞ 0 | ∫ | 𝑦 | ≤ 𝑡 ( Ω ( 𝑥 , 𝑦 ) / | 𝑦 | 𝑛 − 1 ) 𝑓 ( 𝑥 − Φ ( | 𝑦 | ) 𝑦 ′ ) 𝑑 𝑦 | 2 ( 𝑑 𝑡 / 𝑡 3 + 2 𝛼 ) ) 1 / 2 , where Ω ( 𝑥 , 𝑧 ) ∈ 𝐿 ∞ ( ℝ 𝑛 ) × 𝐿 𝑞 ( 𝕊 𝑛 − 1 ) with 𝑞 > m a x { 1 , 2 ( 𝑛 − 1 ) / ( 𝑛 + 2 𝛼 ) } . The authors prove that the operator 𝜇 Φ Ω , 𝛼 is bounded from Sobolev space 𝐿 𝑝 𝛼 ( ℝ 𝑛 ) to 𝐿 𝑝 ( ℝ 𝑛 ) space for 1 < 𝑝 ≤ 2 , and from Hardy-Sobolev space 𝐻 𝑝 𝛼 ( ℝ 𝑛 ) to 𝐿 𝑝 ( ℝ 𝑛 ) space for 𝑛 / ( 𝑛 + 𝛼 ) < 𝑝 ≤ 1 . As corollaries of the result, they also prove the ̇ 𝐿 2 𝛼 ( 𝑅 𝑛 ) − 𝐿 2 ( 𝑅 𝑛 ) boundedness of the Littlewood-Paley type operators 𝜇 Φ Ω , 𝛼 , 𝑆 and 𝜇 ∗ , Φ Ω , 𝛼 , 𝜆 which relate to the Lusin area integral and the Littlewood-Paley 𝑔 ∗ 𝜆 function.