Ricerche di Matematica

, Volume 68, Issue 2, pp 661–677 | Cite as

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

  • Kwok-Pun HoEmail author


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.


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

Mathematics Subject Classification

26D10 26D15 42B35 44A05 46E30 



  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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Busby, R., Smith, H.: Product-convolution operators and mix-norm spaces. Trans. Am. Math. Soc. 263, 309–341 (1981)CrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Carton-Lebrun, C., Heinig, H., Hofmann, H.: Integral operators on weighted amalgams. Stud. Math. 109, 133–157 (1994)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ho, K.-P.: Fourier type transforms on rearrangement-invariant quasi-Banach function spaces. Glasg. Math. J. 61, 231–248 (2019). 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)CrossRefGoogle Scholar
  22. 22.
    Kellogg, C.: An extension of the Hausdorff–Young theorem. Mich. Math. J. 18, 121–127 (1971)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lerner, A., Liflyand, E.: Multidimensional Hausdorff operators on the real Hardy spaces. J. Aust. Math. Soc. 83, 79–86 (2007)MathSciNetCrossRefGoogle 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)CrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Weisz, F.: Local Hardy spaces and summability of Fourier transforms. J. Math. Anal. Appl. 362, 275–285 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wiener, N.: On the representation of functions by trigonometrical integral. Math. Z. 24, 575–616 (1926)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Wiener, N.: Tauberian theorems. Ann. Math. 33, 1–100 (1932)MathSciNetCrossRefGoogle 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

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