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 $$Bigg| int limits _{I} Omega (theta) , dtheta Bigg| leq C |log|I||^{-1-alpha} quadquadquadquad (0.1)$$ 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 ${|1/2-1/p| > frac {3alpha +1}{6(alpha +1)}}$ .