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On the moduli of continuity of equimeasurable functions in the classes ϕ(L)

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Abstract

In the note we establish a number of relations between the moduli of continuity of the equimeasurable functionsf(x) andf *(x). In particular, forf(x)∈ Lp(0, 1), 1 ≤ p < ∞, we have proved the inequality

$$\omega _p (\delta ,f) \geqslant {\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle {2^{\omega _p } }$}}(\delta ,f*),\delta \in [0,{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}].$$

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Literature cited

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Authors and Affiliations

  1. Odessa State University, USSR

    P. Osval'd

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  1. P. Osval'd
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Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 231–244, February, 1975.

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Osval'd, P. On the moduli of continuity of equimeasurable functions in the classes ϕ(L). Mathematical Notes of the Academy of Sciences of the USSR 17, 134–141 (1975). https://doi.org/10.1007/BF01161869

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  • DOI: https://doi.org/10.1007/BF01161869

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