On the integrability of a conjugate function in $L^p$ with the polynomial weight

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 $L_\alpha^p$.
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