대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제59권3호
  • /
  • Pages.757-780
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  • 2022
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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WEIGHTED INTEGRAL INEQUALITIES FOR MODIFIED INTEGRAL HARDY OPERATORS

  • Chutia, Duranta (Department of Mathematical Sciences Tezpur University) ;
  • Haloi, Rajib (Department of Mathematical Sciences Tezpur University)
  • 투고 : 2021.06.20
  • 심사 : 2021.12.06
  • 발행 : 2022.05.31
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초록

In this article, we study the weak and extra-weak type integral inequalities for the modified integral Hardy operators. We provide suitable conditions on the weights ω, ρ, φ and ψ to hold the following weak type modular inequality $${\mathcal{U}}^{-1}\({\int_{{\mid}{\mathcal{I}}f{\mid}>{\gamma}}}\;{\mathcal{U}}({\gamma}{\omega}){\rho}\){\leq}{\mathcal{V}}^{-1}\({\int}_{0}^{\infty}{\mathcal{V}}(C{\mid}f{\mid}{\phi}){\psi}\),$$ where ${\mathcal{I}}$ is the modified integral Hardy operators. We also obtain a necesary and sufficient condition for the following extra-weak type integral inequality $${\omega}\(\{{\left|{\mathcal{I}}f\right|}>{\gamma}\}\){\leq}{\mathcal{U}}{\circ}{\mathcal{V}}^{-1}\({\int}_{0}^{\infty}{\mathcal{V}}\(\frac{C{\mid}f{\mid}{\phi}}{{\gamma}}\){\psi}\).$$ Further, we discuss the above two inequalities for the conjugate of the modified integral Hardy operators. It will extend the existing results for the Hardy operator and its integral version.

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과제정보

Duranta Chutia is thankful to DST INSPIRE, Govt. of India for the financial support DST/INSPIRE Fellowship/2017/IF170509. Rajib Haloi is thankful to Department of Science and Technology, Govt. of India for the financial support DST MATRICS(SERB/F/12082/2018-2019). We would also like to acknowledge the anonymous reviewer for the valuable suggestions that substantially improved the manuscript.

참고문헌

  1. R. J. Bagby, Weak bounds for the maximal function in weighted Orlicz spaces, Studia Math. 95 (1990), no. 3, 195-204. https://doi.org/10.4064/sm-95-3-195-204
  2. R. J. Bagby and J. D. Parsons, Orlicz spaces and rearranged maximal functions, Math. Nachr. 132 (1987), 15-27. https://doi.org/10.1002/mana.19871320103
  3. S. Bloom and R. Kerman, Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator, Studia Math. 110 (1994), no. 2, 149-167. https://doi.org/10.4064/sm-110-2-149-167
  4. A. Gogatishvili and L. Pick, Weak and extra-weak type inequalities for the maximal operator and the Hilbert transform, Czechoslovak Math. J. 43(118) (1993), no. 3, 547-566 https://doi.org/10.21136/cmj.1993.128412
  5. V. Kokilashvili and M. Krbec, Weighted inequalities in Lorentz and Orlicz spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. https://doi.org/10.1142/9789814360302
  6. A. Kufner, L.-E. Persson, and N. Samko, Weighted inequalities of Hardy type, second edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. https://doi.org/10.1142/10052
  7. Q. Lai, Weighted integral inequalities for the Hardy type operator and the fractional maximal operator, J. London Math. Soc. (2) 49 (1994), no. 2, 244-266. https://doi.org/10.1112/jlms/49.2.244
  8. Q. Lai, Two-weight mixed Φ-inequalities for the one-sided maximal function, Studia Math. 115 (1995), no. 1, 1-22. https://doi.org/10.4064/sm-115-1-1-22
  9. Q. Lai, Weighted modular inequalities for Hardy type operators, Proc. London Math. Soc. (3) 79 (1999), no. 3, 649-672. https://doi.org/10.1112/S0024611599012010
  10. E. Lomakina and V. Stepanov, On the Hardy-type integral operators in Banach function spaces, Publ. Mat. 42 (1998), no. 1, 165-194. https://doi.org/10.5565/PUBLMAT_42198_09
  11. F. J. Martin-Reyes, New proofs of weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Proc. Amer. Math. Soc. 117 (1993), no. 3, 691-698. https://doi.org/10.2307/2159130
  12. F. J. Martin-Reyes and P. Ortega, On weighted weak type inequalities for modified Hardy operators, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1739-1746. https://doi.org/10.1090/S0002-9939-98-04247-6
  13. F. J. Martin-Reyes and E. Sawyer, Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater, Proc. Amer. Math. Soc. 106 (1989), no. 3, 727-733. https://doi.org/10.2307/2047428
  14. P. Ortega Salvador, Weighted generalized weak type inequalities for modified Hardy operators, Collect. Math. 51 (2000), no. 2, 149-155.
  15. P. Ortega Salvador and L. Pick, Two-weight weak and extra-weak type inequalities for the one-sided maximal operator, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 6, 1109-1118. https://doi.org/10.1017/S0308210500029760