Behavior of singular integral along the real axis with density vanishing near infinity point

We consider consider the singular integral behavior for the neighborhood of of infinity. The singular integral density satisfies the Hölder condition on the any finite part of the real axis. The singular integral density at the neighborhood of of infinity is continuous infinitely small function with the same order as a degree of logarithm of coordinates modulus for a real axis point under the non-limiting moving from the origin of coordinates less $-1$.