Proceedings of the Japan Academy, Series A, Mathematical Sciences

Hardy’s inequality on Hardy spaces

Kwok-Pun Ho

Full-text: Open access


We extend the Hardy inequalities to the classical Hardy spaces and the rearrangement-invariant Hardy spaces.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 10 (2016), 125-130.

First available in Project Euclid: 2 December 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26D10: Inequalities involving derivatives and differential and integral operators 42B30: $H^p$-spaces 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47B38: Operators on function spaces (general)

Hardy’s inequality Hardy space rearrangement-invariant atomic decomposition interpolation


Ho, Kwok-Pun. Hardy’s inequality on Hardy spaces. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 10, 125--130. doi:10.3792/pjaa.92.125.

Export citation


  • W. Abu-Shammala and A. Torchinsky, The Hardy-Lorentz spaces $H^{p,q}(\mathbf{R}^{n})$, Studia Math. 182 (2007), no. 3, 283–294.
  • C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics, 129, Academic Press, Boston, MA, 1988.
  • L. Diening and S. Samko, Hardy inequality in variable exponent Lebesgue spaces, Fract. Calc. Appl. Anal. 10 (2007), no. 1, 1–18.
  • D. E. Edmunds and W. D. Evans, Hardy operators, function spaces and embeddings, Springer Monographs in Mathematics, Springer, Berlin, 2004.
  • M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 34–170.
  • A. Gogatishvili, B. Opic and W. Trebels, Limiting reiteration for real interpolation with slowly varying functions, Math. Nachr. 278 (2005), no. 1–2, 86–107.
  • J. Gustavsson and J. Peetre, Interpolation of Orlicz spaces, Studia Math. 60 (1977), no. 1, 33–59.
  • A. Harman, On necessary condition for the variable exponent Hardy inequality, J. Funct. Spaces Appl. 2012, Art. ID 385925.
  • A. Harman and F. I. Mamedov, On boundedness of weighted Hardy operator in $L^{p(\cdot)}$ and regularity condition, J. Inequal. Appl. 2010, Art. ID 837951.
  • K.-P. Ho, Characterization of BMO in terms of rearrangement-invariant Banach function spaces, Expo. Math. 27 (2009), no. 4, 363–372.
  • K.-P. Ho, Littlewood-Paley spaces, Math. Scand. 108 (2011), no. 1, 77–102.
  • K.-P. Ho, Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces, Anal. Math. 38 (2012), no. 3, 173–185.
  • K.-P. Ho, Hardy's inequality and Hausdorff operators on rearrangement-invariant Morrey spaces, Publ. Math. Debrecen 88 (2016), no. 1–2, 201–215.
  • K.-P. Ho, Hardy-Littlewood-Pólya inequalities and Hausdorff operators on block spaces, Math. Inequal. Appl. 19 (2016), no. 2, 697–707.
  • K.-P. Ho, Fourier integrals and Sobolev embedding on rearrangement invariant quasi-Banach function spaces, Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 2, 897–922.
  • K.-P. Ho, Hardy's inequality on Hardy-Morrey spaces. (to appear in Georgian Math. J.).
  • K.-P. Ho, Hardy's inequality on Hardy-Morrey spaces with variable exponents. (to appear in Mediterr. J. Math.).
  • K.-P. Ho, Atomic decompositions and Hardy's inequality on weak Hardy-Morrey spaces. (to appear in Sci. China Math.).
  • A. Kufner and L.-E. Persson, Weighted inequalities of Hardy type, World Sci. Publishing, River Edge, NJ, 2003.
  • L. Maligranda, Generalized Hardy inequalities in rearrangement invariant spaces, J. Math. Pures Appl. (9) 59 (1980), no. 4, 405–415.
  • R. A. Mashiyev, B. Çekiç, F. I. Mamedov and S. Ogras, Hardy's inequality in power-type weighted $L^{p(\cdot)}(0,\infty)$ spaces, J. Math. Anal. Appl. 334 (2007), no. 1, 289–298.
  • S. J. Montgomery-Smith, The Hardy operator and Boyd indices, in Interaction between functional analysis, harmonic analysis, and probability (Columbia, MO, 1994), 359–364, Lecture Notes in Pure and Appl. Math., 175, Dekker, New York, 1996.
  • E. Nakai and Y. Sawano, Orlicz-Hardy spaces and their duals, Sci. China Math. 57 (2014), no. 5, 903–962.
  • B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, 219, Longman Sci. Tech., Harlow, 1990.
  • H. Rafeiro and S. Samko, Hardy type inequality in variable Lebesgue spaces, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 1, 279–289.
  • S. Samko, Hardy inequality in the generalized Lebesgue spaces, Fract. Calc. Appl. Anal. 6 (2003), no. 4, 355–362.
  • H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, 18, North-Holland, Amsterdam, 1978.
  • B. E. Viviani, An atomic decomposition of the predual of $\mathrm{BMO}(\rho)$, Rev. Mat. Iberoamericana 3 (1987), no. 3–4, 401–425.