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Two-Weighted Inequalities for the Riesz Potential in p-Convex Weighted Modular Banach Function Spaces

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We prove the property of two-weight boundedness for the Riesz potential from one weighted Banach function space to another weighted Banach function space. In particular, we establish the two-weight boundedness for the Riesz potential and determine sufficient conditions on weights for the boundedness of the Riesz potential in weighted Musielak–Orlicz spaces.

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References

  1. 1.

    R. A. Bandaliev, “On an inequality in Lebesgue space with mixed norm and with variable summability exponent,” Math. Notes, 84, No. 3, 303–313 (2008) (correction in Math. Notes, 99, No. 2, 319–320 (2016)).

  2. 2.

    R. A. Bandaliev, “The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces,” Czechoslovak Math. J., 60, No. 2, 327–337 (2010) (corrections in Czechoslovak Math. J., 63, No. 4, 1149–1152 (2013)).

  3. 3.

    R. A. Bandaliev, “The boundedness of multidimensional Hardy operator in the weighted variable Lebesgue spaces,” Lith. Math. J., 50, No. 3, 249–259 (2010).

  4. 4.

    R. A. Bandaliev and K. K. Omarova, “Two-weight norm inequalities for certain singular integrals,” Taiwanese J. Math., 16, No. 2, 713–732 (2012).

  5. 5.

    R. A. Bandaliev, “On Hardy-type inequalities in weighted variable Lebesgue space L p(x), for 0 < p(x) < 1,” Eurasian Math. J., 4, 5–16 (2013).

  6. 6.

    R. A. Bandaliev, “Criteria of two-weighted inequalities for multidimensional Hardy type operator in weighted Musielak–Orlicz spaces and some application,” Math. Stat., 1, No. 3, 144–156 (2013).

  7. 7.

    R. A. Bandaliev, “On the two-weight boundedness of multidimensional Hardy operator in p-convex Banach function spaces and some applications,” Ukr. Math. J., 67, No. 3, 357–371 (2015).

  8. 8.

    C. Bennett and R. Sharpley, “Interpolation of operators,” Pure Appl. Math., 129 (1988).

  9. 9.

    E. I. Berezhnoi, “Sharp estimates of operators on the cones of ideal spaces,” Proc. Steklov Inst. Math., 204, 3–36 (1994).

  10. 10.

    E. I. Berezhnoi, “Two-weighted estimations for the Hardy–Littlewood maximal function in ideal Banach spaces,” Proc. Amer. Math. Soc., 127, 79–87 (1999).

  11. 11.

    S. Bloom and R. Kerman, “Weighted L Φ integral inequalities for operators of Hardy type,” Studia Math., 110, No. 1, 35–52 (1994).

  12. 12.

    J. Bradley, “Hardy inequalities with mixed norms,” Canad. Math. Bull., 21, 405–408 (1978).

  13. 13.

    A. Cianchi, “Hardy inequalities in Orlicz spaces,” Trans. Amer. Math. Soc., 351, 2459–2478 (1999).

  14. 14.

    J. A. Cochran and C. S. Lee, “Inequalities related to Hardy’s and Heinig’s,” Math. Proc. Cambridge Phil. Soc., 96, 1–7 (1984).

  15. 15.

    D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, “The maximal function on variable L p spaces,” Ann. Acad. Sci. Fenn. Math., 28, No. 1, 223–238 (2003) (corrections in Ann. Acad. Sci. Fenn. Math., 29, No. 2, 247–249 (2004)).

  16. 16.

    D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Perez, “The boundedness of classical operators on variable L p spaces,” Ann. Acad. Sci. Fenn. Math., 31, 239–264 (2006).

  17. 17.

    D. Cruz-Uribe and F. I. Mamedov, “On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces,” Rev. Mat. Complut., 25, No. 2, 335–367 (2012).

  18. 18.

    D. Cruz-Uribe and A. Fiorenza, “Variable Lebesgue spaces,” Found. Harmon. Anal., Ser. Appl. Numer. Harmon. Anal. (2013).

  19. 19.

    L. Diening and S. Samko, “Hardy inequality in variable exponent Lebesgue spaces,” Fract. Calc. Appl. Anal., 10, No. 1, 1–18 (2007).

  20. 20.

    L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, “Lebesgue and Sobolev spaces with variable exponents,” Springer Lect. Notes, 2017 (2011).

  21. 21.

    D. E. Edmunds, V. Kokilashvili, and A. Meskhi, “On the boundedness and compactness of weighted Hardy operators in spaces L p(x),” Georgian Math. J., 12, No. 1, 27–44 (2005).

  22. 22.

    V. S. Guliev, “Two-weight inequalities for integral operators in L p -spaces and their applications,” Proc. Steklov Inst. Math., 204, 97–116 (1994).

  23. 23.

    G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge Univ. Press (1988).

  24. 24.

    P. Harjulehto, P. Hasto, and M. Koskenoja, “Hardy’s inequality in a variable exponent Sobolev space,” Georgian Math. J., 12, No. 1, 431–442 (2005).

  25. 25.

    H. P. Heinig, “Some extensions of inequalities,” J. SIAM Math. Anal., 6, 698–713 (1975).

  26. 26.

    P. Jain, L. E. Persson, and A. Wedestig, “From Hardy to Carleman and general mean-type inequalities, in: Function Spaces and Applications, CRC Press, New York, etc. (2000), pp. 117–130.

  27. 27.

    P. Jain, L. E. Persson, and A. Wedestig, “Carleman–Knopp type inequalities via Hardy’s inequality,” Math. Inequal. Appl., 4, No. 3, 343–355 (2001).

  28. 28.

    K. Knopp, “Uber Reihen mit positiven Gliedern,” J. Lond. Math. Soc. (2), 3, 205–211 (1928).

  29. 29.

    T. S. Kopaliani, “On some structural properties of Banach function spaces and boundedness of certain integral operators, Czechoslovak Math. J., 54, No. 129, 791–805 (2004).

  30. 30.

    M. Krbec, B. Opic, L. Pick, and J. Rakosnik, “Some recent results on Hardy type operators in weighted function spaces and related topics, in: Function Spaces, Differential Operators, and Nonlinear Analysis, Teubner, Stuttgart (1993), pp. 158–184.

  31. 31.

    A. Kufner and L. E. Persson, Integral Inequalities with Weights, World Scientific Publ., Singapore (2002).

  32. 32.

    J. Lindenstrauss and L. Tzafriri, “Classical Banach spaces II,” Ergeb. Math. Grenzgeb. (3), 97 (1979).

  33. 33.

    E. Lomakina and V. Stepanov, “On the Hardy-type integral operator in Banach function spaces,” Publ. Mat., 42, 165–194 (1998).

  34. 34.

    E. R. Love, “Inequalities related to those of Hardy and of Cochran and Lee,” Math. Proc. Cambridge Philos. Soc., 99, 395–408 (1986).

  35. 35.

    W. A. J. Luxemburg, Banach Function Spaces: Thesis, Delfi (1955).

  36. 36.

    F. I. Mamedov and A. Harman, “On a weighted inequality of Hardy type in spaces L p(·),” J. Math. Anal. Appl., 353, No. 2, 521–530 (2009).

  37. 37.

    R. A. Mashiyev, B. Cekic, F. I. Mamedov, and S. Ogras, “Hardy’s inequality in power-type weighted L p(·)(0,∞) spaces,” J. Math. Anal. Appl., 334, No. 1, 289–298 (2007).

  38. 38.

    V. G. Maz’ya, Sobolev Spaces, Springer-Verlag, Berlin (1985).

  39. 39.

    B. Muckenhoupt, “Hardy’s inequality with weights,” Studia Math., 44, 31–38 (1972).

  40. 40.

    J. Musielak, “Orlicz spaces and modular spaces,” Lect. Notes Math., 1034 (1983).

  41. 41.

    L. Quinsheng, “Two weight Φ-inequalities for the Hardy operator, Hardy–Littlewood maximal operator, and fractional integrals,” Proc. Amer. Math. Soc., 118, No. 1, 129–142 (1993).

  42. 42.

    A. Schep, “Minkowski’s integral inequality for function norms,” Oper. Theory: Adv. Appl., 75, 299–308 (1995).

  43. 43.

    G. Tomaselli, “A class of inequalities,” Boll. Unione Mat. Ital., 2, 622–631 (1969).

  44. 44.

    A. Wedestig, “Some new Hardy type inequalities and their limiting inequalities,” J. Inequal. Pure Appl. Math., 61, No. 4, 1–33 (2003).

  45. 45.

    Y. Zeren and V. S. Guliyev, “Two-weight norm inequalities for some anisotropic sublinear operators,” Turkish Math. J., 30, 329–355 (2006).

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Correspondence to R. A. Bandaliyev.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 11, pp. 1443–1454, November, 2017.

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Bandaliyev, R.A., Guliyev, V.S. & Hasanov, S.G. Two-Weighted Inequalities for the Riesz Potential in p-Convex Weighted Modular Banach Function Spaces. Ukr Math J 69, 1673–1688 (2018). https://doi.org/10.1007/s11253-018-1463-6

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