Journal of the Mathematical Society of Japan

A Hardy inequality and applications to reverse Hölder inequalities for weights on $\mathbb{R}$

Eleftherios N. NIKOLIDAKIS

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We prove a sharp integral inequality valid for non-negative functions defined on $[0,1]$, with given $L^1$ norm. This is in fact a generalization of the well known integral Hardy inequality. We prove it as a consequence of the respective weighted discrete analogue inequality whose proof is presented in this paper. As an application we find the exact best possible range of $p>q$ such that any non-increasing $g$ which satisfies a reverse Hölder inequality with exponent $q$ and constant $c$ upon the subintervals of $(0,1]$, should additionally satisfy a reverse Hölder inequality with exponent $p$ and in general a different constant $c'$. The result has been treated in [1] but here we give an alternative proof based on the above mentioned inequality.

Article information

J. Math. Soc. Japan, Volume 70, Number 1 (2018), 141-152.

First available in Project Euclid: 26 January 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26D15: Inequalities for sums, series and integrals
Secondary: 42B25: Maximal functions, Littlewood-Paley theory

Hardy inequalities reverse Hölder inequalities weights


NIKOLIDAKIS, Eleftherios N. A Hardy inequality and applications to reverse Hölder inequalities for weights on $\mathbb{R}$. J. Math. Soc. Japan 70 (2018), no. 1, 141--152. doi:10.2969/jmsj/07017323.

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  • L. D'Appuzzo and C. Sbordone, Reverse Hölder inequalities, A sharp result, Rend. Mat. (7), 10 (1990), 357–366.
  • B. Bojarski, Remarks on the stability of reverse Hölder inequalities and quasiconformal mappings, \doihref10.5186/aasfm.1985.1000Ann. Acad. Sci. Fenn. Ser. AI Math., \bf10 (1985), 89–94.
  • E. T. Copson, Note on series of positive terms, \doihref10.1112/jlms/s1-3.1.49J. London Math. Soc., 3 (1928), 49–51.
  • E. T. Copson, Some integral inequalities, \doihref10.1017/S0308210500017868Proc. Roy. Soc. Edinburgh Sect. A, 75 (1975/1976), 157–164.
  • E. B. Elliott, A simple exposition of some recently proved facts as to convergency, \doihref10.1112/jlms/s1-1.2.93J. London Math. Soc., 1 (1926), 93–96.
  • F. W. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, \doihref10.1007/BF02392268Acta Math., 130 (1973), 265–277.
  • G. H. Hardy, Note on a theorem of Hilbert, \doihref10.1007/BF01199965Math Z., 6 (1920), 314–317.
  • G. H. Hardy, Notes on some points in the integral calculus (LXIT), An inequality between integrals, Messenger of Math., 57 (1928), 12–16.
  • G. H. Hardy and J. E. Littlewood, Elementary theorems concerning power series with positive coefficients and moment constants of positive functions, \doihref10.1515/crll.1927.157.141J. Reine Angew. Math., 157 (1927), 141–158.
  • G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1934.
  • A. A. Korenovskii, Sharp extension of a reverse Hölder inequality and the Muckenhoupt condition, Math. Notes, 52 (1992), 1192–1201.
  • A. Kufner, L. Maligranda and L. E. Persson, The prehistory of the Hardy inequality, \doihref10.2307/27642033Amer. Math. Monthly, 113 (2006), 715–732.
  • E. Landau, I. Schur and G. H. Hardy, A note on a theorem concerning series of positive terms: Extract from a letter, \doihref10.1112/jlms/s1-1.1.38J. London Math. Soc., 1 (1926), 38–39.
  • L. Leindler, Generalization of inequalities of Hardy and Littlewood, Acta Sci. Math., 31 (1970), 279–285.
  • N. Levinson, Generalizations of an inequality of Hardy, \doihref10.1215/S0012-7094-64-03137-0Duke Math. J., 31 (1964), 389–394.
  • E. R. Love, Generalizations of Hardy's and Copson's inequalities, \doihref10.1112/jlms/s2-30.3.431J. London Math. Soc., \bf30 (1984), 431–440.
  • J. Nemeth, Generalizations of the Hardy–Littlewood inequality, Acta Sci. Math. (Szeged), 32 (1971), 295–299.
  • E. N. Nikolidakis, A sharp integral Hardy type inequality and applications to Muckenhoupt weights on R, \doihref10.5186/aasfm.2014.3947Ann. Acad. Sci. Fenn. Math., 39 (2014), 887–896.
  • B. G. Pachpatte, Mathematical Inequalities, North Holland Mathematical Library, 67 (2005).
  • J. B. G. Pachpatte, On a new class of Hardy type inequalities, \doihref10.1017/S0308210500022095Proc. Roy. Soc. Edinburgh Sect. A, \bf105 (1987), 265–274.