Abstract
We study the classical Calderon Zygmund singular integral operator with homogeneous kernel. Suppose that Ω is an integrable function with mean value 0 on S1. We study the singular integral operator $$T_\Omega f= {\rm p.v.} \, f * \frac {\Omega (x/|x|)}{|x|^2}.$$ We show that for α > 0 the condition $$\Bigg| \int \limits _{I} \Omega (\theta) \, d\theta \Bigg| \leq C |\log|I||^{-1-\alpha} \quad\quad\quad\quad (0.1)$$ for all intervals |I| \frac {3\alpha +1}{6(\alpha +1)}}\) .
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