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On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces

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Abstract

We study the Hardy type, two-weight inequality for the multidimensional Hardy operator in the variable exponent Lebesgue space L p(.)(ℝn). We prove equivalent conditions for L p(.)→L q(.) boundness of the Hardy operator in the case of so called “mixed” exponents: q(0)≥p(0), q(∞)<p(∞) or q(0)<p(0), q(∞)≥p(∞). We show that a necessary and sufficient condition for such an inequality to hold coincides with conditions for the validity of two weight Hardy inequalities with constant exponents, provided that the exponents are regular at zero and at infinity.

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References

  1. Acerbi, E., Mingione, G.: Regularity results for a class of functional with nonstandard growth. Arch. Ration. Mech. Anal. 156(2), 121–140 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aguilar Canestro, M.I., Ortega Salvador, P.: Weighted weak type inequalities with variable exponents for Hardy and maximal operators. Proc. Jpn. Acad. Ser. A 82, 126–130 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alkhutov, Y.A.: On the Holder continuity of p(x)-harmonic functions. Sb. Math. 196(2), 147–171 (2005); translations from Mat. Sb. 196(2), 3–28 (2005) (Russian, English)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boza, S., Soria, J.: Weighted Hardy modular inequalities in variable spaces. J. Math. Anal. Appl. 348(1), 383–388 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bradley, J.S.: Hardy inequalities with mixed norms. Can. Math. Bull. 21(4), 405–408 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.J.: The maximal function on variable L p(.) spaces. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 28, 223–238 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Cruz-Uribe, D., Fiorenza, A., Martell, J.M., Pérez, C.: The boundedness of classical operators on variable L p(.) spaces. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 31, 239–264 (2006)

    Google Scholar 

  8. Cruz-Uribe, D.: A new proof of the two weight norm inequality for the one sided fractional maximal operator. Proc. Am. Math. Soc. 125(5), 1419–1424 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cruz-Uribe, D.: New proofs of two-weight norm inequalities for the maximal operator. Georgian Math. J. 7(1), 33–42 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Diening, L., Samko, S.: Hardy inequality in variable exponent Lebesgue spaces. Fract. Calc. Appl. Anal. 10(1), 1–17 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Diening, L.: Maximal function on generalized Lebesgue spaces L p(.). Math. Inequal. Appl. 7(2), 245–253 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Diening, L.: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p(.) and W k,p(.). Math. Nachr. 268, 31–43 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Diening, L., Ruzicka, M.: Calderon-Zigmund operators on generalized Lebesgue spaces Lp(.) and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197–220 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Diening, L., Hasto, P., Nekvinda, A.: Open problems in variable exponent Lebesgue and Sobolev spaces. In: Drabek, P., Rakosnik, J. (eds.) FSDONA 04 Proceedings, Milovy, Czech Republic, pp. 38–58. Academy of Sciences of the Czech Republic, Prague (2005).

    Google Scholar 

  15. Diening, L., Harjulehto, P., Hasto, P., Ruzicka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  16. Edmunds, D.E., Kokilashvili, V., Meskhi, A.: On the boundedness and compactness of the weighted Hardy operators in L p(.) spaces. Georgian Math. J. 12(1), 27–44 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Fan, X.-L., Zhao, D.: The quasi-minimizer of integral functionals with m(x) growth conditions. Nonlinear Anal., Theory Methods Appl. A 39(7), 807–816 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fan, X.-L., Zhao, D.: A class of De Giorgi type and Holder continuity. Nonlinear Anal. Theory Methods Appl. A 36(3), 295–318 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Harjulehto, P., Hästö, P., Koskenoja, M.: Hardy’s inequality in variable exponent Sobolev spaces. Georgian Math. J. 12(3), 431–442 (2005)

    MathSciNet  MATH  Google Scholar 

  20. Kovacik, O., Rakosnik, J.: On spaces L p(.) and W 1,p(.). Czechoslov. Math. J. 41(116), 592–618 (1991)

    MathSciNet  Google Scholar 

  21. Kokilashvili, V., Samko, S.G.: Maximal and fractional operators in weighted L p(.) spaces. Rev. Mat. Iberoam. 20(2), 145–156 (2004)

    MathSciNet  Google Scholar 

  22. Kokilashvili, V., Samko, S.G.: Singular integrals in weighted Lebesgue spaces with variable exponent. Georgian Math. J. 10(1), 145–156 (2003)

    MathSciNet  MATH  Google Scholar 

  23. Kopaliani, T.: Boundedness of Hardy operator in Banach function spaces. Bull. Ga. Acad. Sci. 163(1), 27–28 (2001)

    MathSciNet  MATH  Google Scholar 

  24. Kufner, A., Persson, L.E.: Weighted Inequalities of Hardy Type. World Scientific, Singapore (2003)

    MATH  Google Scholar 

  25. Kufner, A., Maligranda, L., Persson, L.E.: The Hardy Inequality. About Its History and Some Related Results. Vydavatelski Servis Publisher, Pilzen (2007)

    MATH  Google Scholar 

  26. Lerner, A.K.: On modular inequalities in variable L p(.) spaces. Arch. Math. 85, 538–543 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mamedov, F.I., Harman, A.: On weighted inequality of Hardy type in L p(.). J. Math. Anal. Appl. 353(2), 521–530 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mamedov, F.I., Harman, A.: On a Hardy type general weighted inequality in spaces L p(.). Integral Equ. Oper. Theory 66(4), 565–592 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mamedov, F.I., Harman, A.: On boundedness of weighted Hardy operator in L p(.)(ℝn) and regularity condition. J. Inequal. Appl. 1, 837951 (2010), 14 pages

    Google Scholar 

  30. Mashiyev, R., Çekiç, B., Mamedov, F.I., Ogras, S.: Hardy’s inequality in power-type weighted L p(.)(0,∞). J. Math. Anal. Appl. 334(1), 289–298 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mamedov, F.I., Zeren, Y.: On equivalent conditions for the general weighted Hardy type inequality in the space L p(.). Z. Anal. Ihre Anwend. (to appear)

  32. Maz’ya, V.G.: Sobolev Spaces. Springer, Berlin (1985)

    Google Scholar 

  33. Musielak, J.: Orlicz Spaces and Modular Spaces. Springer, Berlin (1983)

    MATH  Google Scholar 

  34. Nekvinda, A.: Hardy-Littlewood maximal operator on L p(.)(ℝn). Math. Inequal. Appl. 7(2), 255–266 (2004)

    MathSciNet  MATH  Google Scholar 

  35. Okpoti, C.A., Persson, L.-E., Sinnamon, G.: Equivalence theorem for some integral conditions with general measures related to Hardy’s inequality I. J. Math. Anal. Appl. 326, 398–413 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Okpoti, C.A., Persson, L.-E., Sinnamon, G.: Equivalence theorem for some integral conditions with general measures related to Hardy’s inequality II. J. Math. Anal. Appl. 337, 219–230 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Opic, B., Kufner, A.: Hardy Type Inequalities. Wiley, New York (1990)

    MATH  Google Scholar 

  38. Pick, L., Ruzicka, M.: An example of a space L p(.)(ℝn) on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math. 19, 369–371 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rafeiro, H., Samko, S.: Hardy type inequality in variable Lebesgue spaces. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 34(1), 279–289 (2009)

    MathSciNet  MATH  Google Scholar 

  40. Ruzicka, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  41. Samko, S.G.: Hardy–Littlewood–Stein–Weiss inequality in the Lebesgue spaces with variable exponent. Fract. Calc. Appl. Anal. 6(4), 421–440 (2003)

    MathSciNet  MATH  Google Scholar 

  42. Samko, S.G.: Hardy inequality in the generalized Lebesgue spaces. Fract. Calc. Appl. Anal. 6(4), 355–362 (2003)

    MathSciNet  MATH  Google Scholar 

  43. Sinnamon, G., Stepanov, V.: The weighted Hardy inequality: new proofs and the case p=1. J. Lond. Math. Soc. 54(2), 89–101 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  44. Stepanov, V.D.: Weighted norm inequalities of Hardy type for a class of integral operators. J. Lond. Math. Soc. 50(2), 105–120 (1994)

    MATH  Google Scholar 

  45. Zhikov, V.V.: On some variational problems. Russ. J. Math. Phys. 5(1), 105–116 (1997)

    MathSciNet  MATH  Google Scholar 

  46. Wedestig, A.: Weighted inequalities of Hardy-type and their limiting inequalities. Doctoral thesis, Dep. Mathem., Lulea University, 2003

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Authors and Affiliations

  1. Department of Mathematics, Trinity College, Hartford, CT, 06106-3100, USA

    David Cruz-Uribe

  2. Institute Mathematics and Mechanics of Nat. Acad. Sci, Baku, Azerbaijan

    Farman I. Mamedov

  3. Department of Mathematics, Education Faculty, Dicle University, 1280, Diyarbakir, Turkey

    Farman I. Mamedov

Authors
  1. David Cruz-Uribe
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  2. Farman I. Mamedov
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Correspondence to David Cruz-Uribe.

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Cruz-Uribe, D., Mamedov, F.I. On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces. Rev Mat Complut 25, 335–367 (2012). https://doi.org/10.1007/s13163-011-0076-5

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  • DOI: https://doi.org/10.1007/s13163-011-0076-5

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