Geodesic
Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity
Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 651-665.

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We construct an example of a function from the class $H_1^{\omega^*}$ , where $\omega^*(t)=\sqrt{\log\log(t^{-1})/\log(t^{-1})}$, $0$, whose trigonometric Fourier series is divergent almost everywhere. We obtain sharp integrability conditions for the majorants of the partial sums of trigonometric Fourier series in terms of whether the functions in question belong to the classes $H_1^\omega$. 
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     author = {N. Yu. Antonov},
     title = {Integrability of the {Majorants} of {Fourier} {Series} and {Divergence} of the {Fourier} {Series} of {Functions} with {Restrictions} on the {Integral} {Modulus} of {Continuity}},
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N. Yu. Antonov. Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity. Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 651-665. http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a1/

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