Let \(T\) be a singular integral operator in \({\mathbb R}^n\) with kernel \(K(x)= {\Omega(x)\over |x|^n}\), where \(\Omega\) is homogeneous of degree \(0\), bounded, and \(\Omega\) satisfies the conditions \[ \int_{S^{n-1}}\Omega(x) \,d\sigma(x)=0 \] and \[ \int_{S^{n-1}}|\Omega(x-\delta\xi)-\Omega(x)| \,d\sigma(x)\leq c n\delta\int_{S^{n-1}}|\Omega(x)| \,d\sigma(x) \] for \(\xi\in S^{n-1}\), \(0<\delta<\frac{1}{n}\), where \(c\) is independent of \(n\).
The author proves that for \(f\in L^1({\mathbb R}^n)\), \(f\geq 0\), \[ \lim_{\lambda\rightarrow 0} \lambda\;m\{x\in{\mathbb R}^n: |Tf(x)|>\lambda\}= {1\over n}\int_{S^{n-1}} |\Omega(x)| \,d\sigma(x) \|f\|_1. \] For the maximal operator \(M\), the corresponding result is \[ \lim_{\lambda\rightarrow 0} \lambda m\{x\in{\mathbb R}^n: |Mf(x)|>\lambda\}= \|f\|_1. \] This paper is a continuation of some earlier work of the author [Indiana Univ. Math. J. 53, No. 2, 533–555 (2004; Zbl 1064.42011)].