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A Generalization Of An Inequality Of Hardy and Littlewood

Published online by Cambridge University Press:  20 November 2018

K. T. Smith
K. T. Smith*
Affiliation:
University of Kansas
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1. Introduction. A well-known inequality of Hardy-Littlewood reads as follows (4): if p > 1 and f > 0, then

,

where is defined as the supremum of the numbers

the constant depends on p only. The statement obtained by putting p = 1 is false; its substitute reads:

the constants depend on p but not on f.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 8 , 1956 , pp. 157 - 170
Copyright
Copyright © Canadian Mathematical Society 1956

References

1. Aronszajn, N. and Smith, K. T., Functional spaces and functional completion. To appear shortly in Ann. Inst. Fourier, Grenoble.Google Scholar
2. Calderón, A. P., A general ergodic theorem. Ann. Math., 8 (1953), 182–191.Google Scholar
3. Calderón, A. P. and Zygmund, A., On the existence of certain singular integrals, Acta Math., 88 (1952), 85–139.Google Scholar
4. Hardy, G. H. and Littlewood, J. E., A maximal theorem with function-theoretic applications, Acta Math., 54 (1930), 81–116.Google Scholar
5. Keldych, M. and Lavrentieff, M., Sur une évaluation pour la fonction de Green, C.R. Ac. Sci. U.S.S.R., 24 (1939), 22–24.Google Scholar
6. Littlewood, J. E., On functions subharmonic in a circle, Lond. Math. Soc, 2 (1927), 192–196.Google Scholar
7. Privaloff, I. I. and Kouznetzoff, P., Sur les problèmes limites et les classes différentes de fonctions harmoniques et subharmoniques définies dans un domaine arbitraire, Rec. Math. Moscou, 6 (1939), 345–376.Google Scholar
8. Rauch, H. E., Généralisation d'une proposition de Hardy et de Littlewood et de théorèmes ergodiques qui s'y rattachent, C.R. Ac. Sci. Paris, 22 (1948), 887–889.Google Scholar
9. Rauch, H. E., Harmonie and analytic functions of several variables and the maximal theorem of Hardy and Littlewood, Can. J. Math., 8 (1956), 171–183.Google Scholar
10. Riesz, F., Sur les fonctions subharmoniques et leur rapport à la théorie du potentiel, Acta Math., 54 (1930), 321–360.Google Scholar
11. Rosenblatt, A., Sur la fonction de Green d'un domaine borné de Vespace à trois dimensions, C.R. Ac. Sci. Paris, 201 (1935), 22–24.Google Scholar
12. Saks, S., Theory of the Integral (New York, 1937).Google Scholar
13. de la Vallée Poussin, C., Propriétés des fonctions harmoniques dans un domaine ouvert limité par des surfaces à courbure bornée, Ann. Scuola Norm. Sup. Pisa, (2) (1933), 167–197.Google Scholar
14. Wiener, N., The ergodic theorem, Duke Math. J., 5 (1939), 1–18.Google Scholar