Mathematical Notes

, Volume 80, Issue 1–2, pp 3–10 | Cite as

Two-weight inequalities for convolution operators in Lebesgue space

  • R. A. Bandaliev


In this paper, we prove a theorem on the boundedness of a convolution operator in a weighted Lebesgue space with kernel satisfying a certain version of Hörmander’s condition.

Key words

convolution operator Lebesgue space measurable function two-weight inequality singular integral Hölder’s inequality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. P. Calderón and A. Zygmund, “On the existence of certain singular integrals,” Acta. Math., 88 (1952), 85–139.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    L. Hörmander, “Estimates for translation-invariant operators in L p spaces,” Acta. Math., 104 (1960), 93–140.MATHMathSciNetGoogle Scholar
  3. 3.
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970; Russian transl.: Moscow, Mir, 1973.MATHGoogle Scholar
  4. 4.
    R. Trujillo-Gonzalez, “Weighted norm inequalities for singular integrals operators satisfying a variant of Hormander condition,” Comment. Math. Univ. Carolinae 44, 1 (2003), 137–152.MathSciNetGoogle Scholar
  5. 5.
    D. J. Grubb and C. N. Moore, “A variant of Hörmander’s condition for singular integrals, ” Colloq. Math., 73 (1997), no. 2, 165–172.MATHMathSciNetGoogle Scholar
  6. 6.
    J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Studies, vol. 116, Amsterdam, 1985.Google Scholar
  7. 7.
    D. K. Watson, “Weighted estimates for singular integrals via Fourier transform estimates,” Duke Math. J., 60 (1990), no. 2, 389–399.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    B. Muckenhoupt, “Weighted norm inequalities for Hardy maximal functions,” Trans. Amer. Math. Soc., 165 (1972), 207–226.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    B. Muckenhoupt, “Hardy’s inequality with weighteds,” Studia Math., 44 (1972), no. 1, 31–38.MATHMathSciNetGoogle Scholar
  10. 10.
    G. Talenti, “Osservazioni sopra una classe di disuguaglianze,” Rend. Sem. Mat. Fis. Milano, 39 (1969), 171–185.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    G. Tomaselli, “A class of inequalities,” Boll.-Un.-Mat.-Ital., 2 (1969), no. 2, 622–631.MATHMathSciNetGoogle Scholar
  12. 12.
    I. A. Aliev and A. D. Gadzhiev, “Weighted estimates for multidimensional singular integrals generated by a generalized shift operator,” Mat. Sb. [Russian Acad. Sci. Sb. Math.],, 183 (1992), no. 9, 45–66.MATHGoogle Scholar
  13. 13.
    V. S. Guliev and R. A. Bandaliev, “Two-weight inequalities for integral operators in L p-spaces of Banach-valued functions and their applications,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],, 243 (2003), 1–19.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • R. A. Bandaliev
    • 1
  1. 1.Institute of Mathematics and Mechanics, National Academy of Sciences of AzerbaijanAzerbaijan

Personalised recommendations