Behavior of the singular integral along the real axis with the density vanishing at infinity near infinity (Q1646300)

Let \[ U(\tau):= \int\limits_{-\infty}^{+\infty}\frac{\varphi(t)\,dt}{t-\tau} \] be a singular integral over the real axis of the complex plane. The author studies its asymptotic at infinity under the assumption that \(\varphi(t)\) decreases near infinity as \((\ln| t|)^{-\beta}\), \(\beta>1\), and certain restrictions on its derivative.