Geodesic
On majorants of $D$-integrable functions
Sbornik. Mathematics, Tome 38 (1981) no. 3, pp. 407-420

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The following majorants are investigated for functions that are integrable in the Denjoy sense: a maximum function in the sense of Hardy and Littlewood; majorants for the conjugate-function operator and for the Hilbert operator. Results of the following kind are obtained: $$ |\{x\in P:M(x)>\lambda\}|\leqslant\frac C\lambda\biggl((L)\int_P|f|\,dt+\sum_i\omega\biggl(\int f;(a_i,b_i)\biggr)\biggr), $$ where $M$ is the majorant of $f$; $P$ is a closed set with complementary intervals $\{(a_i,b_i)\}$; and $\omega\bigl(\int f;(a_i,b_i)\bigr)$ is the oscillation of an indefinite integral of $f$ on $(a_i,b_i)$. Bibhography: 9 titles. 
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     author = {T. P. Lukashenko},
     title = {On majorants of $D$-integrable functions},
     journal = {Sbornik. Mathematics},
     pages = {407--420},
     publisher = {mathdoc},
     volume = {38},
     number = {3},
     year = {1981},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_38_3_a5/}
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T. P. Lukashenko. On majorants of $D$-integrable functions. Sbornik. Mathematics, Tome 38 (1981) no. 3, pp. 407-420. http://geodesic.mathdoc.fr/item/SM_1981_38_3_a5/