Geodesic
On the integrability of a~conjugate function in $L^p$ with the polynomial weight
Sbornik. Mathematics, Tome 64 (1989) no. 2, pp. 339-358

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For any $p>1$ and any real $\alpha$, $-\infty\alpha\infty$, conditions on a function $f\in L_\alpha^p$ ($L_\alpha^p$ is the set of $2\pi$-periodic measurable functions $f$ such that $|f(x)|^p|x|^\alpha$ is integrable on $(-\pi,\pi]$) are found that are necessary and sufficient for its conjugate function $\widetilde f$ to be in $L_\alpha^p$. Bibliography: 16 titles. 
@article{SM_1989_64_2_a3,
     author = {R. I. Gurielashvili},
     title = {On the integrability of a~conjugate function in $L^p$ with the polynomial weight},
     journal = {Sbornik. Mathematics},
     pages = {339--358},
     publisher = {mathdoc},
     volume = {64},
     number = {2},
     year = {1989},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1989_64_2_a3/}
}
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R. I. Gurielashvili. On the integrability of a~conjugate function in $L^p$ with the polynomial weight. Sbornik. Mathematics, Tome 64 (1989) no. 2, pp. 339-358. http://geodesic.mathdoc.fr/item/SM_1989_64_2_a3/