Estimates of some integral operators with bounded variable kernels on the hardy and weak hardy spaces over ₿sup> n

详细信息    查看全文

• 作者：Hua Wang
• 关键词：Singular integral operators ; fractional integrals ; parametric Marcinkiewicz integrals ; variable kernels ; Hardy spaces H p (R n ) ; weak Hardy spaces WH p (R n ) ; Hardy–Lorentz spaces H p ; q (R n )
• 刊名：Acta Mathematica Sinica
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：32
• 期：4
• 页码：411-438
• 全文大小：375 KB
• 参考文献：[1]Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction, Springer–Verlag, Berlin-New York, 1976CrossRef MATH
  [2]Calderón, A. P., Zygmund, A.: On a problem of Mihlin. Trans. Amer. Math. Soc., 78, 209–224 (1955)MathSciNet CrossRef MATH
  [3]Calderón, A. P., Zygmund, A.: On singular integrals. Amer. J. Math., 78, 289–309 (1956)MathSciNet CrossRef MATH
  [4]Calderón, A. P., Zygmund, A.: On singular integrals with variable kernels. Appl. Anal., 7, 221–238 (1978)MathSciNet CrossRef MATH
  [5]Coifman, R. R.: A real variable characterization of Hp. Studia Math., 51, 269–274 (1974)MathSciNet MATH
  [6]Ding, Y., Lin, C. C., Lin, Y. C.: Erratum: “On Marcinkiewicz integral with variable kernels. Indiana Univ. Math. J, 53, 805–821 (2004)”. Indiana Univ. Math. J., 56, 991–994 (2007)MathSciNet CrossRef MATH
  [7]Ding, Y., Lin, C. C., Shao, S. L.: On the Marcinkiewicz integral with variable kernels. Indiana Univ. Math. J., 53, 805–821 (2004)MathSciNet CrossRef MATH
  [8]Fefferman, C., Rivière, N., Sagher, Y.: Interpolation between Hp spaces: The real method. Trans. Amer. Math. Soc., 191, 75–81 (1974)MathSciNet MATH
  [9]Fefferman, R., Soria, F.: The space weak H1. Studia Math., 85, 1–16 (1987)MathSciNet MATH
  [10]Grafakos, L.: Classical and Modern Fourier Analysis, Pearson Education, Inc., New Jersey, 2004MATH
  [11]Hörmander, L.: Estimates for translation invariant operators in Lp spaces. Acta Math., 104, 93–140 (1960)MathSciNet CrossRef MATH
  [12]Latter, R. H.: A characterization of Hp(Rn) in terms of atoms. Studia Math., 62, 93–101 (1978)MathSciNet MATH
  [13]Lee, J., Rim, K. S.: Estimates of Marcinkiewicz integrals with bounded homogeneous kernels of degree zero. Integral Equations Operator Theory, 48, 213–223 (2004)MathSciNet CrossRef MATH
  [14]Liu, H. P.: The weak Hp spaces on homogeneous groups, Lecture Notes in Math., Vol. 1494, Springer-Verlag, Berlin-Heidelberg, 1991, 113–118
  [15]Lu, S. Z.: Four Lectures on Real Hp Spaces, World Scientific Publishing, River Edge, N.J., 1995CrossRef
  [16]Muckenhoupt, B., Wheeden, R. L.: Weighted norm inequalities for singular and fractional integrals. Trans. Amer. Math. Soc., 161, 249–258 (1971)MathSciNet CrossRef MATH
  [17]Abu-Shammala, W., Torchinsky, A.: The Hardy–Lorentz spaces H p,q(Rn). Studia Math., 182, 283–294 (2007)MathSciNet CrossRef MATH
  [18]Stein, E. M.: On the functions of Littlewood–Paley, Lusin and Marcinkiewicz. Trans. Amer. Math. Soc., 88, 430–466 (1958)MathSciNet CrossRef MATH
  [19]Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, New Jersey, 1993MATH
  [20]Wang, H.: Boundedness of several integral operators with bounded variable kernels on Hardy and weak Hardy spaces. Internat. J. Math., 24, 1–22 (2013)MathSciNet CrossRef
  
• 作者单位：Hua Wang (1)
  
  1. College of Mathematics and Econometrics, Hu’nan University, Changsha, 410082, P. R. China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Chinese Library of Science
  
• 出版者：Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, co-published
• ISSN：1439-7617

文摘

In this paper, we first introduce \({L^{{\sigma _1}}}{\left( {\log L} \right)^{{\sigma _2}}}\) conditions satisfied by the variable kernels Ω(x, z) for 0 ≤ σ ₁ ≤ 1 and σ ₂ ≥ 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators T _Ω, fractional integrals T _Ω,α and parametric Marcinkiewicz integrals μ _Ω ρ with variable kernels on the Hardy spaces H p (R n ) and weak Hardy spaces WH p (R n ). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy–Lorentz spaces H p,q(R n ) for all p < q < ∞.