This paper is devoted to show that the limiting weak type behaviors hold for Calderón-Zygmund singular integral operators, Hardy-Littlewood maximal function, Calderón commutators, and Marcinkiewicz integrals with \(L\log L({\mathbb {S}^{n-1}})\) homogeneous kernels. We indeed give a unified method to deal with the limiting weak type behaviors for rough operators. In this process, improved upper bounds and lower bounds for the best constants of the weak type norm inequalities are obtained. In case of Calderón commutators, since they are operators of non-convolution type, things become more subtle, the main difficulty lies in that, after the first decomposition has been made on the kernel, it may destroy the first-order cancellation moment conditions for n variables, we need to reconstruct the kernel so that the cancellation conditions still hold. This is done by the method of induction. Moreover, we give two characterizations on the intrinsic geometry structure of the limiting weak type behaviors for inhomogeneous convolution type singular integral operators by using the method of wavelet analysis, this may help deeply understanding the nature of the limiting weak type behaviors for these operators.
