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Estimates of Oscillations of the Hardy Transform

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Abstract

For the Hardy transform of a nonincreasing function we obtain a sharp two-sided estimate of the BLO-norm and sharp inequalities between the BMO- and the BLO-norms of a nonincreasing function. A well-known lower bound for the BMO-norm of the Hardy transform is improved on the basis of these inequalities.

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REFERENCES

  1. G. Hardy, D. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1934.

    Google Scholar 

  2. R. Lyons, “A lower bound on the Cesàro operator,” Proc. Amer. Math. Soc., 86 (1982), 694-699.

    Google Scholar 

  3. P. R. Renaud, “A reversed Hardy inequality,” Bull. Austral. Math. Soc., 34 (1986), 225-232.

    Google Scholar 

  4. M. Milman, “A note on reversed Hardy inequalities and Gehring's lemma,” Comm. Pure Appl. Math., 50 (1997), 311-315.

    Google Scholar 

  5. F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Comm. Pure Appl. Math., 14 (1961), 415-426.

    Google Scholar 

  6. R. R. Coifman and R. Rochberg, “Ar characterization of BMO,” Proc. Amer. Math. Soc., 79 (1980), 249-254.

    Google Scholar 

  7. A. G. Siskakis, “Composition semigroups and the Cesàro operator on H p,” J. London Math. Soc., 36 (1987), no. 2, 153-164.

    Google Scholar 

  8. A. G. Siskakis, “The Cesàro operator is bounded on H p,” Proc. Amer. Math. Soc., 110 (1990), 461-462.

    Google Scholar 

  9. K. Stempak, “Cesàro averaging operators,” Proc. Royal Soc. Edinburgh, 124 (1994), 121-126.

    Google Scholar 

  10. Dang Vu Giang and F. Moricz, “The Cesàro operator is bounded on the Hardy space H 1,” Acta Sci. Math. (Szeged), 61 (1995), 535-544.

    Google Scholar 

  11. B. I. Golubov, “Boundedness of the Hardy and Hardy-Littlewood operators on the spaces ReH 1 and BMO,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 188 (1997), 93-106.

    Google Scholar 

  12. Xiao Jie, “A reverse BMO-Hardy inequality,” Real Analysis Exchange, 25 (1999/2000), no. 2, 673-678.

    Google Scholar 

  13. I. Klemes, “A mean oscillation inequality,” Proc. Amer. Math. Soc., 93 (1985), no. 3, 497-500.

    Google Scholar 

  14. C. Bennett and R. Sharpley, Interpolation of Operators, Acad. Press, 1988.

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Authors and Affiliations

  1. I. I. Mechnikov Odessa National University, Ukraine

    A. A. Korenovskii

  2. T. G. Shevchenko Kiev National University, Ukraine

    A. A. Korenovskii

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  1. A. A. Korenovskii
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Korenovskii, A.A. Estimates of Oscillations of the Hardy Transform. Mathematical Notes 72, 350–361 (2002). https://doi.org/10.1023/A:1020599304631

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