Advertisement

Dilation operators and integral operators on amalgam space \((L_{p},l_{q})\)

  • Kwok-Pun HoEmail author
Article

Abstract

This paper establishes the Hardy–Littlewood–Pólya inequalities, the Hardy inequalities and the Hilbert inequalities on amalgam spaces. Moreover, it also gives the mapping properties of the Mellin convolutions, the Hadamard fractional integrals and the Hausdorff operators on amalgam spaces. We establish these properties by some estimates for the operator norms of the dilation operators on amalgam spaces.

Keywords

Amalgam spaces Integral operator Hardy inequality Hilbert inequality Hadamard fractional integral Mellin convolution Hausdorff operator 

Mathematics Subject Classification

26D10 26D15 42B35 44A05 46E30 

Notes

References

  1. 1.
    Andersen, K.: Boundedness of Hausdorff operators on \(L^{p}({\mathbb{R}}^{n})\), \(H^{1}({\mathbb{R}}^{n})\), and \(BMO({\mathbb{R}}^{n})\). Acta Sci. Math. (Szeged) 69, 409–418 (2003)MathSciNetGoogle Scholar
  2. 2.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Inc., Orlando (1988)zbMATHGoogle Scholar
  3. 3.
    Brown, G., Móricz, F.: The Hausdorff operator and the quasi Hausdorff operator on the space \(L^{p}\), \(1\le p<\infty \). Math. Inequal. Appl. 3, 105–115 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brown, G., Móricz, F.: Multivariate Hausdorff operators on the spaces \(L^{p}({\mathbb{R}}^{n})\). J. Math. Anal. Appl. 271, 443–454 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Busby, R., Smith, H.: Product-convolution operators and mix-norm spaces. Trans. Am. Math. Soc. 263, 309–341 (1981)CrossRefzbMATHGoogle Scholar
  6. 6.
    Butzer, P., Kilbas, A., Trujillo, J.: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, 1–27 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Butzer, P., Kilbas, A., Trujillo, J.: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269, 387–400 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carton-Lebrun, C., Heinig, H., Hofmann, H.: Integral operators on weighted amalgams. Stud. Math. 109, 133–157 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, J.C., Fan, D.S., Li, J.: Hausdorff operators on function spaces. Chin. Ann. Math. Ser. B 33, 537–556 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, J.C., Fan, D.S., Wang, S.L.: Hausdorff operators on Euclidean spaces. Appl. Math. J. Chin. Univ. 28, 548–564 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Edwards, R.E., Hewitt, E., Ritter, G.: Fourier multipliers for certain spaces of functions with compact supports. Invent. Math. 40, 37–57 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fournier, J., Stewart, J.: Amalgams of \(L^{p}\) and \(l^{q}\). Bull. Am. Math. Soc. 13, 1–22 (1985)CrossRefGoogle Scholar
  13. 13.
    Hadamard, J.: Essai sur l’etude des fonctions donnees par leur developpment de taylor. J. Mat. Pure Appl. 8, 101–186 (1892)zbMATHGoogle Scholar
  14. 14.
    Hardy, G., Littlewood, J., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1934)zbMATHGoogle Scholar
  15. 15.
    Ho, K.-P.: Hardy’s inequality and Hausdorff operators on rearrangement-invariant Morrey spaces. Publ. Math. Debr. 88, 201–215 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ho, K.-P.: Hardy–Littlewood–Pólya inequalities and Hausdorff operators on block spaces. Math. Inequal. Appl. 19, 697–707 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ho, K.-P.: Fourier integrals and Sobolev embedding on rearrangement-invariant quasi-Banach function spaces. Ann. Acad. Sci. Fenn. Math. 41, 897–922 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ho, K.-P.: Fourier type transforms on rearrangement-invariant quasi-Banach function spaces. Glasg. Math. J. 61, 231–248 (2019).  https://doi.org/10.1017/S0017089518000186 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ho, K.-P.: Linear operators, Fourier-integral operators and \(k\)-plane transforms on rearrangement-invariant quasi-Banach function spaces (preprint)Google Scholar
  20. 20.
    Ho, K.-P.: Modular Hadamard, Riemann–Liouville and Weyl fractional integrals (preprint)Google Scholar
  21. 21.
    Holland, F.: Harmonic analysis on amalgams of \(L^{p}\) and \(l^{q}\). J. Lond. Math. Soc. 10, 295–305 (1975)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kellogg, C.: An extension of the Hausdorff–Young theorem. Mich. Math. J. 18, 121–127 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lerner, A., Liflyand, E.: Multidimensional Hausdorff operators on the real Hardy spaces. J. Aust. Math. Soc. 83, 79–86 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liflyand, E.: Boundedness of multidimensional Hausdorff operators on \(H^{1}({\mathbb{R}}^{n})\). Acta Sci. Math. (Szeged) 74, 845–851 (2008)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Liflyand, E., Miyachi, A.: Boundedness of the Hausdorff operators in \(H^{p}\) spaces, \(0<p<1\). Stud. Math. 194, 279–292 (2009)CrossRefzbMATHGoogle Scholar
  26. 26.
    Maligranda, L.: Generalized Hardy inequalities in rearrangement invariant spaces. J. Math. Pures Appl. 59, 405–415 (1980)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Opic, B., Kufner, A.: Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series, vol. 219. Longman Scientific and Technical, Harlow (1990)zbMATHGoogle Scholar
  28. 28.
    Szeptycki, P.: Some remarks on the extended domain of Fourier transform. Bull. Am. Math. Soc. 73, 398–402 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Weisz, F.: Local Hardy spaces and summability of Fourier transforms. J. Math. Anal. Appl. 362, 275–285 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wiener, N.: On the representation of functions by trigonometrical integral. Math. Z. 24, 575–616 (1926)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wiener, N.: Tauberian theorems. Ann. Math. 33, 1–100 (1932)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Information TechnologyThe Education University of Hong KongTai PoChina

Personalised recommendations