On p dependent boundedness of singular integral operators

We study the classical Calderón Zygmund singular integral operator with homogeneous kernel. Suppose that Ω is an integrable function with mean value 0 on S ¹. 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| < 1 in S ¹ gives L ^p boundedness of T _Ω in the range ${|1/2-1/p| < \frac \alpha {2(\alpha+1)}}$ . This condition is weaker than the conditions from Grafakos and Stefanov (Indiana Univ Math J 47:455–469, 1998) and Fan et al. (Math Inequal Appl 2:73–81, 1999). We also construct an example of an integrable Ω which satisfies (0.1) such that T _Ω is not L ^p bounded for ${|1/2-1/p| > \frac {3\alpha +1}{6(\alpha +1)}}$ .