Taiwanese Journal of Mathematics

TWO-WEIGHT NORM INEQUALITIES FOR CERTAIN SINGULAR INTEGRALS

R. A. Bandaliev and K. K. Omarova

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Abstract

In this paper we prove the boundedness of certain convolution operator in a weighted Lebesgue space with kernel satisfying the generalized Hörmander's condition. The sufficient conditions for the pair of weights ensuring the validity of two-weight inequalities of a strong type and of a weak type for singular integral with kernel satisfying the generalized Hörmander's condition are found.

Article information

Source
Taiwanese J. Math., Volume 16, Number 2 (2012), 713-732.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406611

Digital Object Identifier
doi:10.11650/twjm/1500406611

Mathematical Reviews number (MathSciNet)
MR2892908

Zentralblatt MATH identifier
1257.47035

Subjects
Primary: 46B50: Compactness in Banach (or normed) spaces 26D15: Inequalities for sums, series and integrals 47B38: Operators on function spaces (general)

Keywords
weighted Lebesgue space singular integral kernel generalized Hörmander's condition boundedness

Citation

Bandaliev, R. A.; Omarova, K. K. TWO-WEIGHT NORM INEQUALITIES FOR CERTAIN SINGULAR INTEGRALS. Taiwanese J. Math. 16 (2012), no. 2, 713--732. doi:10.11650/twjm/1500406611. https://projecteuclid.org/euclid.twjm/1500406611


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References

  • \item[1.] R. A. Bandaliev, Two-weight inequalities for convolution operators in Lebesgue spaces, Mat. Zametki, 1 (80) (2006), 3-10 (in Russian); English translation: Math. Notes, 80(1) (2006), 3-10.
  • \item[2.] A. P. Calderòn and A. Zygmund, On the existence of certain singular integrals, Acta Math., 88 (1952), 85-139.
  • \item[3.] A. P. Calderòn and A. Zygmund, On singular integrals, Amer. J. Math., 78(2) (1956), 289-309.
  • \item[4.] K. Davis and Y. Chang, Lectures on Bochner-Riesz means, London Math. Soc., Lecture Note Ser. 114, Cambridge Univ. Press, 1987.
  • \item[5.] J. Garsia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. Studies, Amsterdam, 1985, p. 116.
  • \item[6.] D. J. Grubb and C. N. Moore, A variant of Hörmander's condition for singular integrals, Colloq. Math., 73(2) (1997), 165-172.
  • \item[7.] V. S. Guliyev, Two-weight inequalities for singular integrals satisfying a variant of Hörmander condition, Journal of Function Spaces and Appl., 7(1) (2009), 43-54.
  • \item[8.] L. Hörmander, Estimates for translation invariant operators in $L_p$ spaces, Acta Math., 104 (1960), 93-140.
  • \item[9.] V. P. Kabaila, On the embedding $L_p(\mu)$ into $L_q(\nu)$, Litovsky Mat. Sb., 21 (1981), 143-148 (in Russian).
  • \item[10.] V. Kokilashvili and A. Meskhi, Two-weight inequalities for singular integrals defined on homogeneous groups, Proc. A. Razmadze Math. Inst., 112 (1997), 57-90.
  • \item[11.] V. G. Maz'ya, Sobolev Spaces, Springer-Verlag, Berlin, 1985.
  • \item[12.] B. Muckenhoupt, Weighted norm inequalities for Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226.
  • \item[13.] E. M. Stein, Note on singular integral, Proc. Amer. Math. Soc., 8(2) (1957), 250- 254.
  • \item[14.] R. Trujillo-Gonzàles, Weighted norm inequalities for singular integrals operators satisfying a variant of Hörmander condition, Comment. Math. Univ. Carolinae, 44(1) (2003), 137-152.