Research Article
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Year 2021, Volume: 50 Issue: 5, 1334 - 1346, 15.10.2021
https://doi.org/10.15672/hujms.805157

Abstract

Project Number

Grant № EIF-BGM-4-RFTF-1/2017-21/01/1-M-08

References

  • [1] D.R. Aliyeva, On the boundedness of Hardy type integral operators in weighted Lebesgue space, Caspian J. Appl. Math., Ecol. Econ. 6 (1), 35-41, 2018.
  • [2] K.F. Andersen, Boundedness of Hausdorff operators on $L_p\left(\mathbb{R}^n\right),$ $H^1\left(\mathbb{R}^n\right),$ and $BMO\left(\mathbb{R}^n\right)$, Acta Sci. Math. (Szeged) 69 (1-2), 409-418, 2003.
  • [3] R.A. Bandaliev, On one inequalities for convolution type operator, Hacet. J. Math. Stat. 42 (3), 199-210, 2013.
  • [4] R.A. Bandaliyev and P. Górka, Hausdorff operator in Lebesgue spaces, Math. Inequal. Appl. 22 (2), 657-676, 2019.
  • [5] R.A. Bandaliev and K.K. Omarova, Two-weight norm inequalities for certain singular integrals, Taiwanese J. Math. 16 (2), 713-732, 2012.
  • [6] R.A. Bandaliyev and K.H. Safarova, On boundedness of multidimensional Hausdorff operator in weighted Lebesgue spaces, Tbilisi Math. J. 13 (1), 39-45, 2020.
  • [7] G. Brown and F. Mòricz, The Hausdorff operator and the quasi Hausdorff operators on the space $L^p,$ $1< p < \infty$, Math. Inequal. Appl. 3 (1), 105-115, 2000.
  • [8] G. Brown and F. Mòricz, Multivariate Hausdorff operators on the spaces $L^p\left(\mathbb{R}^n\right)$, J. Math. Anal. Appl. 271 (2), 443-454, 2002.
  • [9] J. Chen, D. Fan and J. Li, Hausdorff operators on function spaces, Chin. Ann. Math. Ser. B, 33 (4), 537-556, 2013.
  • [10] J. Chen and X. Wu, Best constant for Hausdorff operators on n-dimensional product spaces, Sci. China Math. 57 (3), 569-578, 2014.
  • [11] D.E. Edmunds, V. Kokilashvili and A. Meskhi, Bounded and Compact Integral Operators, Kluwer Academic Publishers, 2002.
  • [12] G. Gao, X. Wu and W. Guo, Some results for Hausdorff operators, Math. Inequal. Appl. 18 (1), 155-168, 2015.
  • [13] A. Gogatishvili and V.D. Stepanov, Reduction theorems for weighted integral inequalities on the cone of monotone functions, Russian Math. Surveys 68 (4), 597-664, 2013.
  • [14] V.S. Guliyev, Two-weighted inequalities for integral operators in $L_p$-spaces and their applications, Proc. Steklov Math. Inst. 204 (3), 97-116, 1993.
  • [15] A. Hussain and A. Ajaib, Some weighted inequalities for Hausdorff operators and commutators, J. Inequal. Appl. 2018, Article No. 6, 2018.
  • [16] A. Hussain and G. Gao, Multidimensional Hausdorff operators and commutators on Herz-type spaces, J. Inequal. Appl. 2013, Article No. 594, 2013.
  • [17] A. Kufner, L. Maligranda and L.E. Persson, The Hardy inequality: About its history and some related results, Research report, Department of Mathematics, Luleå University of Technology, Sweden, 2005.
  • [18] S.N Lal and S. Ram, On the absolute Hausdorff summability of a Fourier series, Pacific J. Math. 42 (2), 439-451, 1972.
  • [19] E. Liflyand, Hardy type inequalities in the category of Hausdorff operators, Modern Meth. Oper. Theory Harmonic Anal. OTHA 2018. Springer Proc. Math. & Stat., 291, 81-91, 2019.
  • [20] E. Liflyand and A. Miyachi, Boundedness of the Hausdorff operators in $H^p$ spaces, $0 < p < 1,$ Studia Math. 194 (3), 279-292, 2009.
  • [21] E. Liflyand and A. Miyachi, Boundedness of multidimensional Hausdorff operators in $H^p$ spaces, $0 < p < 1,$ Trans. Amer. Math. Soc. 371 (7), 4793-4814, 2019.
  • [22] V.G. Maz’ya, Sobolev Spaces, Springer-Verlag, Berlin, 1985.
  • [23] G. Talenti, Osservazione sopra una classe di disuguaglianze, Rend. Sem. Mat. Fiz. Milano 39 (1), 171-185, 1969.
  • [24] G. Tomaselli, A class of inequalities, Boll. Unione Mat. Ital. 2 (1), 622-631, 1969.

On two-weight inequalities for Hausdorff operators of special kind in Lebesgue spaces

Year 2021, Volume: 50 Issue: 5, 1334 - 1346, 15.10.2021
https://doi.org/10.15672/hujms.805157

Abstract

In this paper, we establish necessary and sufficient conditions on monotone weight functions for the boundedness for Hausdorff operators of special kind in weighted Lebesgue spaces. In particular, we get similar results for important operators of harmonic analysis which are special cases of the Hausdorff operators. The weights are illustrated by examples at the end of the paper.

Supporting Institution

1st Azerbaijan-Russia Joint Grant Competition Grant N EIF-BGM-4-RFTF-1/2017-21/01/1-M08.

Project Number

Grant № EIF-BGM-4-RFTF-1/2017-21/01/1-M-08

References

  • [1] D.R. Aliyeva, On the boundedness of Hardy type integral operators in weighted Lebesgue space, Caspian J. Appl. Math., Ecol. Econ. 6 (1), 35-41, 2018.
  • [2] K.F. Andersen, Boundedness of Hausdorff operators on $L_p\left(\mathbb{R}^n\right),$ $H^1\left(\mathbb{R}^n\right),$ and $BMO\left(\mathbb{R}^n\right)$, Acta Sci. Math. (Szeged) 69 (1-2), 409-418, 2003.
  • [3] R.A. Bandaliev, On one inequalities for convolution type operator, Hacet. J. Math. Stat. 42 (3), 199-210, 2013.
  • [4] R.A. Bandaliyev and P. Górka, Hausdorff operator in Lebesgue spaces, Math. Inequal. Appl. 22 (2), 657-676, 2019.
  • [5] R.A. Bandaliev and K.K. Omarova, Two-weight norm inequalities for certain singular integrals, Taiwanese J. Math. 16 (2), 713-732, 2012.
  • [6] R.A. Bandaliyev and K.H. Safarova, On boundedness of multidimensional Hausdorff operator in weighted Lebesgue spaces, Tbilisi Math. J. 13 (1), 39-45, 2020.
  • [7] G. Brown and F. Mòricz, The Hausdorff operator and the quasi Hausdorff operators on the space $L^p,$ $1< p < \infty$, Math. Inequal. Appl. 3 (1), 105-115, 2000.
  • [8] G. Brown and F. Mòricz, Multivariate Hausdorff operators on the spaces $L^p\left(\mathbb{R}^n\right)$, J. Math. Anal. Appl. 271 (2), 443-454, 2002.
  • [9] J. Chen, D. Fan and J. Li, Hausdorff operators on function spaces, Chin. Ann. Math. Ser. B, 33 (4), 537-556, 2013.
  • [10] J. Chen and X. Wu, Best constant for Hausdorff operators on n-dimensional product spaces, Sci. China Math. 57 (3), 569-578, 2014.
  • [11] D.E. Edmunds, V. Kokilashvili and A. Meskhi, Bounded and Compact Integral Operators, Kluwer Academic Publishers, 2002.
  • [12] G. Gao, X. Wu and W. Guo, Some results for Hausdorff operators, Math. Inequal. Appl. 18 (1), 155-168, 2015.
  • [13] A. Gogatishvili and V.D. Stepanov, Reduction theorems for weighted integral inequalities on the cone of monotone functions, Russian Math. Surveys 68 (4), 597-664, 2013.
  • [14] V.S. Guliyev, Two-weighted inequalities for integral operators in $L_p$-spaces and their applications, Proc. Steklov Math. Inst. 204 (3), 97-116, 1993.
  • [15] A. Hussain and A. Ajaib, Some weighted inequalities for Hausdorff operators and commutators, J. Inequal. Appl. 2018, Article No. 6, 2018.
  • [16] A. Hussain and G. Gao, Multidimensional Hausdorff operators and commutators on Herz-type spaces, J. Inequal. Appl. 2013, Article No. 594, 2013.
  • [17] A. Kufner, L. Maligranda and L.E. Persson, The Hardy inequality: About its history and some related results, Research report, Department of Mathematics, Luleå University of Technology, Sweden, 2005.
  • [18] S.N Lal and S. Ram, On the absolute Hausdorff summability of a Fourier series, Pacific J. Math. 42 (2), 439-451, 1972.
  • [19] E. Liflyand, Hardy type inequalities in the category of Hausdorff operators, Modern Meth. Oper. Theory Harmonic Anal. OTHA 2018. Springer Proc. Math. & Stat., 291, 81-91, 2019.
  • [20] E. Liflyand and A. Miyachi, Boundedness of the Hausdorff operators in $H^p$ spaces, $0 < p < 1,$ Studia Math. 194 (3), 279-292, 2009.
  • [21] E. Liflyand and A. Miyachi, Boundedness of multidimensional Hausdorff operators in $H^p$ spaces, $0 < p < 1,$ Trans. Amer. Math. Soc. 371 (7), 4793-4814, 2019.
  • [22] V.G. Maz’ya, Sobolev Spaces, Springer-Verlag, Berlin, 1985.
  • [23] G. Talenti, Osservazione sopra una classe di disuguaglianze, Rend. Sem. Mat. Fiz. Milano 39 (1), 171-185, 1969.
  • [24] G. Tomaselli, A class of inequalities, Boll. Unione Mat. Ital. 2 (1), 622-631, 1969.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Rovshan Bandaliyev 0000-0002-5038-7632

Kamala Safarova This is me 0000-0002-3428-8693

Project Number Grant № EIF-BGM-4-RFTF-1/2017-21/01/1-M-08
Publication Date October 15, 2021
Published in Issue Year 2021 Volume: 50 Issue: 5

Cite

APA Bandaliyev, R., & Safarova, K. (2021). On two-weight inequalities for Hausdorff operators of special kind in Lebesgue spaces. Hacettepe Journal of Mathematics and Statistics, 50(5), 1334-1346. https://doi.org/10.15672/hujms.805157
AMA Bandaliyev R, Safarova K. On two-weight inequalities for Hausdorff operators of special kind in Lebesgue spaces. Hacettepe Journal of Mathematics and Statistics. October 2021;50(5):1334-1346. doi:10.15672/hujms.805157
Chicago Bandaliyev, Rovshan, and Kamala Safarova. “On Two-Weight Inequalities for Hausdorff Operators of Special Kind in Lebesgue Spaces”. Hacettepe Journal of Mathematics and Statistics 50, no. 5 (October 2021): 1334-46. https://doi.org/10.15672/hujms.805157.
EndNote Bandaliyev R, Safarova K (October 1, 2021) On two-weight inequalities for Hausdorff operators of special kind in Lebesgue spaces. Hacettepe Journal of Mathematics and Statistics 50 5 1334–1346.
IEEE R. Bandaliyev and K. Safarova, “On two-weight inequalities for Hausdorff operators of special kind in Lebesgue spaces”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 5, pp. 1334–1346, 2021, doi: 10.15672/hujms.805157.
ISNAD Bandaliyev, Rovshan - Safarova, Kamala. “On Two-Weight Inequalities for Hausdorff Operators of Special Kind in Lebesgue Spaces”. Hacettepe Journal of Mathematics and Statistics 50/5 (October 2021), 1334-1346. https://doi.org/10.15672/hujms.805157.
JAMA Bandaliyev R, Safarova K. On two-weight inequalities for Hausdorff operators of special kind in Lebesgue spaces. Hacettepe Journal of Mathematics and Statistics. 2021;50:1334–1346.
MLA Bandaliyev, Rovshan and Kamala Safarova. “On Two-Weight Inequalities for Hausdorff Operators of Special Kind in Lebesgue Spaces”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 5, 2021, pp. 1334-46, doi:10.15672/hujms.805157.
Vancouver Bandaliyev R, Safarova K. On two-weight inequalities for Hausdorff operators of special kind in Lebesgue spaces. Hacettepe Journal of Mathematics and Statistics. 2021;50(5):1334-46.