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A best constant for Zygmund's conjugate function inequality


Author: Colin Bennett
Journal: Proc. Amer. Math. Soc. 56 (1976), 256-260
MSC: Primary 42A40
DOI: https://doi.org/10.1090/S0002-9939-1976-0402393-4
MathSciNet review: 0402393
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Abstract: When the space $ L{\log ^ + }L$ is given the Hardy-Littlewood norm the best constant in the corresponding version of Zygmund's conjugate function inequality is shown to be

$\displaystyle {\mathbf{K}} = \frac{{{1^{ - 2}} - {3^{ - 2}} + {5^{ - 2}} - {7^{ - 2}} + \cdots }}{{{1^{ - 2}} + {3^{ - 2}} + {5^{ - 2}} + {7^{ - 2}} + \cdots }}.$

This complements the recent result of Burgess Davis that the best constant in Kolmogorov's inequality is $ {{\mathbf{K}}^{ - 1}}$.


References [Enhancements On Off] (What's this?)

  • [1] C. Bennett, Intermediate spaces and the class $ L{\log ^ + }L$, Ark. Mat. 11 (1973), 215-228. MR 0352966 (50:5452)
  • [2] -, Estimates for weak-type operators, Bull. Amer. Math. Soc. 79 (1973), 933-935. MR 47 #9264. MR 0320730 (47:9264)
  • [3] -, Banach function spaces and interpolation methods. III. Hausdorff-Young estimates, J. Approximation Theory 13 (1975), 267-275. MR 0482121 (58:2208)
  • [4] D. Burkholder, Harmonic analysis and probability (preprint). MR 0463788 (57:3728)
  • [5] B. Davis, On the weak-type (1,1) inequality for conjugate functions, Proc. Amer. Math. Soc. 44 (1974), 307-311. MR 0348381 (50:879)
  • [6] G. G. Lorentz, On the theory of spaces $ \Lambda $, Pacific J. Math. 1 (1951), 411-429. MR 13, 470. MR 0044740 (13:470c)
  • [7] R. O'Neil and G. Weiss, The Hilbert transform and rearrangement of functions, Studia Math. 23 (1963), 189-198. MR 28 #3298. MR 0160084 (28:3298)
  • [8] S. K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. 44 (1972), 165-179. MR 47 #702. MR 0312140 (47:702)
  • [9] E. M. Stein and G. Weiss, An extension of the theorem of Marcinkiewicz and some of its applications, J. Math. Mech. 8 (1959), 263-284. MR 21 #5888. MR 0107163 (21:5888)
  • [10] A. Zygmund, Trigonometric series. Vol. I, 2nd rev. ed., Cambridge Univ. Press, New York, 1959. MR 21 #6498. MR 0107776 (21:6498)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0402393-4
Article copyright: © Copyright 1976 American Mathematical Society

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