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On Some Classical Operators of Variable Order in Variable Exponent Spaces

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Analysis, Partial Differential Equations and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 193))

Abstract

We give a survey of a selection of recent results on weighted and non-weighted estimations of classical operators of Harmonic Analysis in variable exponent Lebesgue, Morrey and Hölder spaces, based on the talk presented at International Conference Analysis, PDEs and Applications on the occasion of the 70th birthday of Vladimir Maz’ya, Rome, June 30–July 3, 2008. We touch both the Euclidean case and the general setting within the frameworks of quasimetric measure spaces. Some of the presented results are new.

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References

  1. D.R. Adams. A note on Riesz potentials. Duke Math. J., 42.

    Google Scholar 

  2. A. Almeida. Inversion of the Riesz Potential Operator on Lebesgue Spaces with Variable Exponent. Frac. Calc. Appl. Anal., 6(3):311–327, 2003.

    MATH  MathSciNet  Google Scholar 

  3. A. Almeida, J. Hasanov and S. Samko. Maximal and potential operators in variable exponent Morrey spaces. Georgian Math. J., 15(2):195–208, 2008.

    MATH  MathSciNet  Google Scholar 

  4. A. Almeida and S. Samko. Embeddings of variable Hajlasz-Sobolev spaces into Hölder spaces of variable order. Manuscript.

    Google Scholar 

  5. A. Almeida and S. Samko. Fractional and hypersingular operators in variable exponent spaces on metric measure spaces. Meditter. J. Math., DOI 10.1007/s00009-003-0000.

    Google Scholar 

  6. A. Almeida and S. Samko. Characterization of Riesz and Bessel potentials on variable Lebesgue spaces. J. Function Spaces and Applic., 4(2):113–144, 2006.

    MATH  MathSciNet  Google Scholar 

  7. A. Almeida and S. Samko. Pointwise inequalities in variable Sobolev spaces and applications. Zeit. Anal. und ihre Anwend., 26(2):179–193, 2007.

    MATH  MathSciNet  Google Scholar 

  8. A.-P. Calderón. Inequalities for the maximal function relative to a metric. Studia Math., 57(3):297–306, 1976.

    MATH  MathSciNet  Google Scholar 

  9. Y. Chen, S. Levine, and Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math., 66:1383–1406, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  10. F. Chiarenza and M. Frasca. Morrey spaces and Hardy-Littlewood maximal function. Rend. Math., 7:273–279, 1987.

    MATH  MathSciNet  Google Scholar 

  11. R.R. Coifman and G. Weiss. Analyse harmonique non-commutative sur certaines espaces homogènes, volume 242. Lecture Notes Math., 1971. 160 pages.

    Google Scholar 

  12. 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(1):239–264, 2006.

    MathSciNet  Google Scholar 

  13. D. Cruz-Uribe, A. Fiorenza, and C.J. Neugebauer. The maximal function on variable L p-spaces. Ann. Acad. Scient. Fennicae, Math., 28:223–238, 2003.

    MATH  MathSciNet  Google Scholar 

  14. D. Cruz-Uribe, A. Fiorenza, and C.J. Neugebauer. Corrections to “The maximal function on variable L p-spaces”. Ann. Acad. Scient. Fennicae, Math., 29:247–249, 2004.

    MATH  MathSciNet  Google Scholar 

  15. D. Cruz-Uribe, J.M. Martell, and C. Pérez. Extrapolation from A ∞ weights and applications. J. Funct. Anal., 213(2):412–439, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  16. D. Cruz-Uribe, J.M. Martell, and C. Pérez. Extensions of Rubio de Francia’s extrapolation theorem. Collect. Math., (Vol. Extra):195–231, 2006.

    Google Scholar 

  17. G.P. Curbera, J. García-Cuerva, J.M. Martell, and C. Pérez. Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals. Adv. Math., 203(1):256–318, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  18. L. Diening. Theoretical and numerical results for electrorheological fluids. Ph.D Thesis, University of Freiburg, Germany, 2002.

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  20. L. Diening. Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p (·) and W k,p (·). Mathem. Nachrichten, 268:31–43, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  21. L. Diening. Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math., 129(8):657–700, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  22. L. Diening, P. Hästö, and A. Nekvinda. Open problems in variable exponent Lebesgue and Sobolev spaces. In “Function Spaces, Differential Operators and Nonlinear Analysis”, Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands, May 28–June 2, 2004. Math. Inst. Acad. Sci. Czech Republick, Praha.

    Google Scholar 

  23. L. Diening and S. Samko. Hardy inequality in variable exponent Lebesgue spaces. Albert-Ludwigs-Universität Freiburg, 2006. Preprint Nr. 1/2006-17.03.2006, 15p.

    Google Scholar 

  24. D.E. Edmunds. Sobolev embeddings with variable exponent, II. Math. Nachr., 246/247:53–67, 2002.

    Article  MathSciNet  Google Scholar 

  25. D.E. Edmunds, V. Kokilashvili, and A. Meskhi. Two-weight estimates for singular integrals defined on spaces of homogeneous type. Canad. J. Math., 52(3):468–502, 2000.

    MATH  MathSciNet  Google Scholar 

  26. D.E. Edmunds, J. Lang, and A. Nekvinda. On L p(x) norms. Proc. R. Soc. Lond., A 455:219–225, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  27. D.E. Edmunds and J. Rákosník. Sobolev embeddings with variable exponent. Studia Math., 143(3):267–293, 2000.

    MATH  MathSciNet  Google Scholar 

  28. X. Fan, J. Shen, and D. Zhao. Sobolev embedding theorems for spaces W k,p(x)(Ω). J. Math. Anal. Appl., 262(2):749–760, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  29. X. Fan and D. Zhao. On the spaces L p(x) (Ω) and W m,p(x) (Ω). J. Math. Anal. Appl., 263(2):424–446, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  30. A. Fiorenza. A mean continuity type result for certain Sobolev spaces with variable exponent. Commun. Contemp. Math., 4(3):587–605, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  31. T. Futamura, P. Harjulehto, P. Hästö, Y. Mizuta, and T. Shimomura. Variable exponent spaces on metric measure spaces. “More Progresses in Analysis”, Proceed. of the 5th Congress of ISAAC, Catania, 2005. World scientific, 2009, 107–122.

    Google Scholar 

  32. T. Futamura and Y. Mizuta. Continuity of weakly monotone Sobolev functions of variable exponent. Adv. Math. Sci. Appl., Maruzen Co. Ltd, 15:571–585, 2005.

    MATH  MathSciNet  Google Scholar 

  33. T. Futamura and Y. Mizuta. Continuity properties of Riesz potentials for functions in L p (·) of variable exponent. Math. Inequal. Appl, 8(1):619–631, 2005.

    MATH  MathSciNet  Google Scholar 

  34. T. Futamura, Y. Mizuta, and T. Shimomura. Sobolev embeddings for Riesz potential space of variable exponent. Math. Nachr., 31(279):495–522, 2006.

    MATH  MathSciNet  Google Scholar 

  35. T. Futamura, Y. Mizuta, and T. Shimomura. Sobolev embeddings for variable exponent Riesz potentials on metric spaces. Ann. Acad. Sci. Fenn. Math., 31(2):495–522, 2006.

    MATH  MathSciNet  Google Scholar 

  36. A.E. Gatto. On fractional calculus associated to doubling and non-doubling measures. In J. Marshall (ed.) et al. Ash, editor, Harmonic analysis. Calderon-Zygmund and beyond. A conference in honor of Stephen Vagi’s retirement, Chicago, IL, USA, December 6–8, 2002, volume 411, pages 15–37. Providence, RI: American Mathematical Society, 2006. Contemporary Mathematics.

    Google Scholar 

  37. A.E. Gatto and J. Garcia-Cuerva. Boundedness properties of fractional integral operators associated to non-doubling measures. Studia Math., 162(3):245–261, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  38. A.E. Gatto, C. Segovia, and S. Vagi. On fractional differentiation on spaces of homogeneous type. Revista Mat. Iberoamer, 12(1), 1996. 1–35.

    MathSciNet  Google Scholar 

  39. A.E. Gatto and S. Vagi. Fractional integrals on spaces of homogeneous type. In Cora Sadosky (ed.), editor, Analysis and Partial Differential Equations, volume 122, pages 171–216. Marcel Dekker, New York and Basel, 1990. Lecture Notes in Pure and Appl. Math.

    Google Scholar 

  40. P. Hästö. Local-to-global results in variable exponent spaces. Math. Res. Letters. To appear.

    Google Scholar 

  41. P. Hajłasz and J. Kinnunen. Hölder quasicontinuity of Sobolev functions on metric spaces. Rev. Mat. Iberoamericana, 14(3):601–622, 1998.

    MATH  MathSciNet  Google Scholar 

  42. P. Hajłasz and P. Koskela. Sobolev met Poincaré. Mem. Amer. Math. Soc., 145(688):x+101, 2000.

    Google Scholar 

  43. P. Harjulehto and P. Hästö. A capacity approach to the Poincaré inequality in the Sobolev imbeddings in variable exponent Sobolev spaces. Rev. Mat. Complut., 17(1):129–146, 2004.

    MATH  MathSciNet  Google Scholar 

  44. P. Harjulehto, P. Hästö, and V. Latvala. Sobolev embeddings in metric measure spaces with variable dimension. Math. Z., 254(3):591–609, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  45. P. Harjulehto, P. Hästö, and M. Pere. Variable Exponent Lebesgue Spaces on Metric Spaces: The Hardy-Littlewood Maximal Operator. Real Anal. Exchange, 30(1):87–104, 2004.

    MathSciNet  Google Scholar 

  46. P. Harjulehto, P. Hästö, and M. Pere. Variable exponent Sobolev spaces on metric measure spaces. Funct. Approx. Comment. Math., 36:79–94, 2006.

    MATH  MathSciNet  Google Scholar 

  47. N.K. Karapetyants and A.I. Ginzburg. Fractional integrals and singular integrals in the Hölder classes of variable order. Integral Transform. Spec. Funct., 2(2):91–106, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  48. N.K. Karapetyants and A.I. Ginzburg. Fractional integro-differentiation in Hölder classes of variable order. Dokl. Akad. Nauk, 339(4):439–441, 1994.

    Google Scholar 

  49. M. Khabazi. The maximal operator in spaces of homogenous type. Proc. A. Razmadze Math. Inst., 138:17–25, 2005.

    MATH  MathSciNet  Google Scholar 

  50. Ya.L. Kobelev, L.Ya. Kobelev, and Yu.L. Klimontovich. Statistical physics of dynamic systems with variable memory. Doklady Physics, 48(6):285–289, 2003.

    Article  MathSciNet  Google Scholar 

  51. V. Kokilashvili. On a progress in the theory of integral operators in weighted Banach function spaces. In “Function Spaces, Differential Operators and Nonlinear Analysis”, Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands, May 28–June 2, 2004. Math. Inst. Acad. Sci. Czech Republick, Praha.

    Google Scholar 

  52. V. Kokilashvili and A. Meskhi. On some weighted inequalities for fractional integrals on nonhomogeneous spaces. Zeitschr. Anal. Anwend., 24(4):871–885, 2005.

    MathSciNet  Google Scholar 

  53. V. Kokilashvili and A. Meskhi. Boundedness of maximal and singular operators in Morrey spaces with variable exponent. Govern. College Univ., Lahore, 72:1–11, 2008.

    Google Scholar 

  54. V. Kokilashvili and A. Meskhi. Boundedness of maximal and singular operators in Morrey spaces with variable exponent. Armenian J. Math., 1(1):18–28, 2008.

    MathSciNet  Google Scholar 

  55. V. Kokilashvili, V. Paatashvili, and Samko S. Boundedness in Lebesgue spaces with variable exponent of the Cauchy singular operators on Carleson curves. In Ya. Erusalimsky, I. Gohberg, S. Grudsky, V. Rabinovich, and N. Vasilevski, editors, “Operator Theory: Advances and Applications”, dedicated to 70th birthday of Prof. I.B. Simonenko, volume 170, pages 167–186. Birkhäuser Verlag, Basel, 2006.

    Google Scholar 

  56. V. Kokilashvili, N. Samko, and S. Samko. Singular operators in variable spaces L p (·) (ω, σ) with oscillating weights. Math. Nachrichten, 280(9–10):1145–1156, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  57. V. Kokilashvili and S. Samko. Operators of harmonic analysis in weighted spaces with non-standard growth. J. Math. Anal. Appl. To appear doi:10.1016/j.jmaa.2008.06.056

    Google Scholar 

  58. V. Kokilashvili and S. Samko. Singular operators and Fourier multipliers in weighted Lebesgue spaces with variable exponents. Vestnik Saint-Petersburg University, ser. 1, issue 2, 56–68, Engl. Transl. in “Vestnik St. Petersburg University. Mathematics”, 2008, Vol. 41, No. 2, pp. 134–144

    Google Scholar 

  59. V. Kokilashvili and S. Samko. Boundedness in Lebesgue spaces with variable exponent of the cauchy singular operator. Izvestija VUZov. Severo-Kavkazskii region. Estestvennie nauki, Special issue “Pseudodifferential equations and some problems of mathematical physics”, dedicated to 70th birthday of Prof. I.B. Simonenko, (Special Issue):152–158, 2005.

    Google Scholar 

  60. V. Kokilashvili and S. Samko. The maximal operator in weighted variable spaces on metric measure spaces. Proc. A. Razmadze Math. Inst., 144:137–144, 2007.

    MATH  MathSciNet  Google Scholar 

  61. V. Kokilashvili and S. Samko. Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent. Acta Mathematica Sinica, 24(11):1775–1800, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  62. O. Kovácîk and J. Rákosnîk. On spaces L p(x) and W k,p(x). Czechoslovak Math. J., 41(116):592–618, 1991.

    MathSciNet  Google Scholar 

  63. A. Lerner. Some remarks on the Hardy-Littlewood maximal function on variable L p spaces. Math. Zeit., 251(3):509–521, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  64. R.A. Macías and C. Segovia. Lipshitz functions on spaces of homogeneous type. Adv. Math., 33:257–270, 1979.

    Article  MATH  Google Scholar 

  65. R.A. Macías and C. Segovia. A Well Behaved Quasidistance for Spaces of Homogeneous Type. Trab. Mat. Inst. Argentina Mat., 32:1–18, 1981.

    Google Scholar 

  66. Y. Mizuta and T. Shimomura. Continuity of Sobolev functions of variable exponent on metric spaces. Proc. Japan Acad., 80, Ser. A:96–99, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  67. Y. Mizuta and T. Shimomura. Sobolev’s inequality for Riesz potentials with variable exponent satisfying a log-Hölder condition at infinity. J. Math. Anal. Appl., 311:268–288, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  68. J. Musielak. Orlicz spaces and modular spaces, volume 1034 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1983.

    Google Scholar 

  69. A. Nekvinda. Hardy-Littlewood maximal operator on L p(x) (ℝn). Math. Inequal. and Appl., 7(2):255–265, 2004.

    MATH  MathSciNet  Google Scholar 

  70. A. Nekvinda. A note on maximal operator on ℓpn and L p(x) (ℝn). J. Funct. Spaces and Appl., 5:49–88, 2007.

    MATH  MathSciNet  Google Scholar 

  71. J. Peetre. On the theory of L p, λ spaces. Function. Analysis, 4:71–87, 1969.

    Article  MATH  MathSciNet  Google Scholar 

  72. H. Rafeiro and S. Samko. On multidimensional analogue of Marchaud formula for fractional Riesz-type derivatives in domains in R n. Fract. Calc. and Appl. Anal., 8(4):393–401, 2005.

    MATH  MathSciNet  Google Scholar 

  73. H. Rafeiro and S. Samko. Characterization of the range of one-dimensional fractional integration in the space with variable exponent. In Operator Algebras, Operator Theory and Applications (WOAT Conference, Lisbon, 2006), Operator Theory. Birkhäuser, Basel, 2007 (to appear).

    Google Scholar 

  74. B. Ross and S.G. Samko. Fractional integration operator of variable order in the Hölder spaces H λ(x). Internat. J. Math. Sci., 18(4):777–788, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  75. J.L. Rubio de Francia. Factorization theory and A p weights. Amer. J. Math., 106(3):533–547, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  76. M. Ružčka. Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Lecture Notes in Math., 2000, vol. 1748, 176 pages.

    Google Scholar 

  77. N. Samko, S. Samko, and Vakulov B. Fractional integrals and hypersingular integrals in variable order Hölder spaces on homogeneous spaces (manuscript).

    Google Scholar 

  78. N. Samko, S. Samko, and Vakulov B. Weighted Sobolev theorem in Lebesgue spaces with variable exponent. J. Math. Anal. and Applic., 335:560–583, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  79. N. Samko, and Vakulov B. Spherical fractional and hypersingular integrals in generalized Hölder spaces with variable characteristic Math. Nachrichten, to appear.

    Google Scholar 

  80. S. Samko. Convolution and potential type operators in L p(x). Integr. Transf. and Special Funct., 7(3–4):261–284, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  81. S. Samko. Hypersingular Integrals and their Applications. London-New York: Taylor & Francis, Series “Analytical Methods and Special Functions”, vol. 5, 2002. 358+xvii pages.

    MATH  Google Scholar 

  82. S. Samko. On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integr. Transf. and Spec. Funct, 16(5–6):461–482, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  83. S.G. Samko, E. Shargorodsky, and B. Vakulov. Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II. J. Math, Anal. Appl., 325(1):745–751, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  84. S.G. Samko and B.G. Vakulov. Weighted Sobolev theorem with variable exponent. J. Math, Anal. Appl., 310:229–246, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  85. I.I. Sharapudinov. The topology of the space L p(t) ([0, 1]) (Russian). Mat. Zametki, 26(4):613–632, 1979.

    MathSciNet  Google Scholar 

  86. B.G. Vakulov. Spherical potentials of complex order in the variable order Hölder spaces. Integral Transforms Spec. Funct., 16(5–6):489–497, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  87. B.G. Vakulov. Spherical convolution operators in Hölder spaces of variable order. Matem. Zametki, 80(5):683–695, 2006. Transl. in Math. Notes, 80:5, 2006, 645–657.

    MathSciNet  Google Scholar 

  88. B.G. Vakulov. Spherical potentials of complex order in the spaces with generalized variable Hölder condition (in Russian). Dokl. Akad. Nauk, 407(1):12–15, 2006. Transl. in Doklady Mathematics, 2006, 73(2): 165–168.

    MathSciNet  Google Scholar 

  89. B.G. Vakulov, N.K. Karapetiants, and L.D. Shankishvili. Spherical hypersingular operators of imaginary order and their multipliers. Fract. Calc. Appl. Anal., 4(1):101–112, 2001.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Algarve, Algarve, Portugal

    Stefan Samko

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  1. Stefan Samko
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Editors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università della Basilicata, Viale dell’Ateneo Lucano 10, 85100, Potenza, Italy

    Alberto Cialdea

  2. Dipartimento di Matematico “Guido Castelnuovo”, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185, Rome, Italy

    Paolo Emilio Ricci  & Flavia Lanzara  & 

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To Vladimir Maz’ya on the occasion of his 70th birthday

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Samko, S. (2009). On Some Classical Operators of Variable Order in Variable Exponent Spaces. In: Cialdea, A., Ricci, P.E., Lanzara, F. (eds) Analysis, Partial Differential Equations and Applications. Operator Theory: Advances and Applications, vol 193. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9898-9_20

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