DOI QR코드

DOI QR Code

INTEGRAL OPERATORS ON CESÀRO FUNCTION SPACES

  • Ho, Kwok-Pun (Department of Mathematics and Information Technology The Education University of Hong Kong)
  • Received : 2021.07.14
  • Accepted : 2022.05.31
  • Published : 2022.07.31

Abstract

This paper studies the boundedness of integral operators on the Cesàro function spaces. As applications of the main result, we obtain the Hilbert inequalities, the boundedness of the Erdélyi-Kober fractional integrals and the Mellin fractional integrals on the Cesàro function spaces.

Keywords

References

  1. I. Al Alam, L. Gaillard, G. Habib, P. Lefevre, and F. Maalouf, Essential norm of Cesaro operators on Lp and Cesaro spaces, J. Math. Anal. Appl. 467 (2018), no. 2, 1038-1065. https://doi.org/10.1016/j.jmaa.2018.07.038
  2. S. V. Astashkin and L. Maligranda, Cesaro function spaces fail the fixed point property, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4289-4294. https://doi.org/10.1090/S0002-9939-08-09599-3
  3. S. V. Astashkin and L. Maligranda, Structure of Cesaro function spaces, Indag. Math. (N.S.) 20 (2009), no. 3, 329-379. https://doi.org/10.1016/S0019-3577(10)00002-9
  4. S. V. Astashkin and L. Maligranda, Rademacher functions in Cesaro type spaces, Studia Math. 198 (2010), no. 3, 235-247. https://doi.org/10.4064/sm198-3-3
  5. S. V. Astashkin and L. Maligranda, Geometry of Cesaro function spaces, Funktsional. Anal. i Prilozhen. 45 (2011), no. 1, 79-83; English transl.: Funct. Anal. Appl. 45 (2011), 64-68.
  6. S. V. Astashkin and L. Maligranda, Interpolation of Cesaro sequence and function spaces, Studia Math. 215 (2013), no. 1, 39-69. https://doi.org/10.4064/sm215-1-4
  7. S. V. Astashkin and L. Maligranda, A short proof of some recent results related to Cesaro function spaces, Indag. Math. (N.S.) 24 (2013), no. 3, 589-592. https://doi.org/10.1016/j.indag.2013.03.001
  8. S. V. Astashkin and L. Maligranda, Interpolation of Cesaro and Copson spaces, in: Banach and Function Spaces IV (Kitakyushu, 2012), Yokohama Publ., Yokohama 2014, 123-133.
  9. N. L. Braha, Structure of Cesaro second order function spaces, Miskolc Math. Notes 16 (2015), no. 2, 705-711. https://doi.org/10.18514/MMN.2015.1385
  10. P. Butzer and C. Bardaro, Mellin analysis and exponential sampling. Part I: Mellin fractional integrals, Proceedings of the 10th International Conference on Sampling Theory and Applications, 274-276.
  11. P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), no. 1, 1-27. https://doi.org/10.1016/S0022-247X(02)00001-X
  12. P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002), no. 2, 387-400. https://doi.org/10.1016/S0022-247X(02)00049-5
  13. P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl. 270 (2002), no. 1, 1-15. https://doi.org/10.1016/S0022-247X(02)00066-5
  14. G. P. Curbera and W. J. Ricker, Abstract Cesaro spaces: integral representations, J. Math. Anal. Appl. 441 (2016), no. 1, 25-44. https://doi.org/10.1016/j.jmaa.2016.03.074
  15. G. B. Folland, Real Analysis: modern techniques and their applications, John Wiley & Sons, 1984.
  16. A. Gogatishvili, R. Mustafayev, and T. Unver, Embeddings between weighted Copson and Cesaro function spaces, Czechoslovak Math. J. 67(142) (2017), no. 4, 1105-1132. https://doi.org/10.21136/CMJ.2017.0424-16
  17. A. Gogatishvili, R. Mustafayev, and T. Unver, Pointwise multipliers between weighted Copson and Cesaro function spaces, Math. Slovaca 69 (2019), no. 6, 1303-1328. https://doi.org/10.1515/ms-2017-0310
  18. R. Herrmann, Towards a geometric interpretation of generalized fractional integrals-Erdelyi-Kober type integrals on RN , as an example, Fract. Calc. Appl. Anal. 17 (2014), no. 2, 361-370. https://doi.org/10.2478/s13540-014-0174-4
  19. K.-P. Ho, Integral operators on BMO and Campanato spaces, Indag. Math. (N.S.) 30 (2019), no. 6, 1023-1035. https://doi.org/10.1016/j.indag.2019.05.007
  20. K.-P. Ho, Erdelyi-Kober fractional integrals on Hardy space and BMO, Proyecciones 39 (2020), no. 3, 663-677. https://doi.org/10.22199/issn.0717-6279-2020-03-0041
  21. K.-P. Ho, Erdelyi-Kober fractional integral operators on ball Banach function spaces, Rend. Semin. Mat. Univ. Padova 145 (2021), 93-106. https://doi.org/10.4171/rsmup/72
  22. K.-P. Ho, Hardy's inequalities and Erdelyi-Kober fractional integrals on BMO(ρ), Azerb. J. Math. 11 (2021), no. 1, 92-103.
  23. K.-P. Ho, Hardy's inequalities, Hilbert inequalities and fractional integrals on function spaces of q-integral p-variation, Ann. Polon. Math. 126 (2021), no. 3, 251-263. https://doi.org/10.4064/ap200224-11-1
  24. M. Ilkhan, S. Demiriz, and E. E. Kara, Multiplication operators on Cesaro second order function spaces, Positivity 24 (2020), no. 3, 605-614. https://doi.org/10.1007/s11117-019-00700-5
  25. V. Kiryakova, Generalized fractional calculus and applications, Pitman Research Notes in Mathematics Series, 301, Longman Scientific & Technical, Harlow, 1994.
  26. V. Kiryakova, Transmutation method for solving hyper-Bessel differential equations based on the Poisson-Dimovski transformation, Fract. Calc. Appl. Anal. 11 (2008), no. 3, 299-316.
  27. T. Kiwerski and P. Kolwicz, Rotundity and monotonicity properties of selected Cesaro function spaces, Positivity 22 (2018), no. 1, 357-377. https://doi.org/10.1007/s11117-017-0515-8
  28. A. Kufner, L. Maligranda, and L.-E. Persson, The prehistory of the Hardy inequality, Amer. Math. Monthly 113 (2006), no. 8, 715-732. https://doi.org/10.2307/27642033
  29. 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
  30. K. Lesnik and L. Maligranda, Abstract Cesaro spaces. Duality, J. Math. Anal. Appl. 424 (2015), no. 2, 932-951. https://doi.org/10.1016/j.jmaa.2014.11.023
  31. K. Lesnik and L. Maligranda, Abstract Cesaro spaces. Optimal range, Integral Equations Operator Theory 81 (2015), no. 2, 227-235. https://doi.org/10.1007/s00020-014-2203-4
  32. K. Lesnik and L. Maligranda, Interpolation of abstract Cesaro, Copson and Tandori spaces, Indag. Math. (N.S.) 27 (2016), no. 3, 764-785. https://doi.org/10.1016/j.indag.2016.01.009
  33. A. M. Mathai and H. J. Haubold, Erdelyi-Kober fractional integral operators from a statistical perspective (I), Tbilisi Math. J. 10 (2017), no. 1, 145-159. https://doi.org/10.1515/tmj-2017-0009
  34. A. M. Mathai and H. J. Haubold, Erdelyi-Kober Fractional Calculus from a Statistical Perspective, Inspired by Solar Neutrino Physics, Springer Singapore, 2018.
  35. B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, 219, Longman Scientific & Technical, Harlow, 1990.
  36. G. Pagnini, Erdelyi-Kober fractional diffusion, Fract. Calc. Appl. Anal. 15 (2012), no. 1, 117-127. https://doi.org/10.2478/s13540-012-0008-1
  37. L. Plociniczak, Approximation of the Erdelyi-Kober operator with application to the time-fractional porous medium equation, SIAM J. Appl. Math. 74 (2014), no. 4, 1219-1237. https://doi.org/10.1137/130942450
  38. L. Schwartz, Oeuvres de Jean Delsarte, Vols 1,2, Gothier-Villar, Paris, 1971.
  39. J. Shiue, A note on Cesaro function space, Tamkang J. Math. 1 (1970), no. 2, 91-95.
  40. I. N. Sneddon, The use in mathematical physics of Erdelyi-Kober operators and of some of their generalizations, in Fractional calculus and its applications (Proc. Internat. Conf., Univ. New Haven, West Haven, Conn., 1974), 37-79. Lecture Notes in Math., 457, Springer, Berlin, 1975.
  41. P. W. Sy, W. Y. Zhang, and P. Y. Lee, The dual of Ces'aro function spaces, Glas. Mat. Ser. III 22(42) (1987), no. 1, 103-112.
  42. T.-L. Yee and K.-P. Ho, Hardy's inequalities and integral operators on Herz-Morrey spaces, Open Math. 18 (2020), no. 1, 106-121. https://doi.org/10.1515/math-2020-0008