Geodesic
On the moduli of continuity of equimeasurable functions in the classes $\varphi(L)$
Matematičeskie zametki, Tome 17 (1975) no. 2, pp. 231-244.

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In the note we establish a number of relations between the moduli of continuity of the equimeasurable functions $f(x)$ and $f^*(x)$. In particular, for $f(x)\in L_p(0,1)$, $1\le p\infty$, we have proved the inequality $\omega_p(\delta,f)\ge\frac12\omega_p(\delta,f^*),\quad\delta\in\Bigl[0,\frac12\Bigr]$. 
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     author = {P. Oswald},
     title = {On the moduli of continuity of equimeasurable functions in the classes $\varphi(L)$},
     journal = {Matemati\v{c}eskie zametki},
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P. Oswald. On the moduli of continuity of equimeasurable functions in the classes $\varphi(L)$. Matematičeskie zametki, Tome 17 (1975) no. 2, pp. 231-244. http://geodesic.mathdoc.fr/item/MZM_1975_17_2_a5/