Geodesic
On functions conjugate to Denjoy--Khinchin integrable functions
Izvestiya. Mathematics , Tome 5 (1971) no. 2, pp. 393-420

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In this paper an example is constructed of a function $f(x)$ that is integrable in the sense of Denjoy–Khinchin for which the conjugate function $\bar f(x)$ is integrable in the same sense and has a positive zeroth Fourier coefficient. 
@article{IM2_1971_5_2_a6,
     author = {T. P. Lukashenko},
     title = {On functions conjugate to {Denjoy--Khinchin} integrable functions},
     journal = {Izvestiya. Mathematics },
     pages = {393--420},
     publisher = {mathdoc},
     volume = {5},
     number = {2},
     year = {1971},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1971_5_2_a6/}
}
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T. P. Lukashenko. On functions conjugate to Denjoy--Khinchin integrable functions. Izvestiya. Mathematics , Tome 5 (1971) no. 2, pp. 393-420. http://geodesic.mathdoc.fr/item/IM2_1971_5_2_a6/