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In this paper it is shown that for $\Omega\in L\log L(\mathbb{S}^{d-1})$, therough maximal singular integral operator $T_\Omega^*$ is of weak type$L\log\log L(\mathbb{R}^d)$. Furthermore, for $w\in A_1$ and $\Omega\inL^\infty(\mathbb{S}^{d-1})$, it is shown that $T_\Omega^*$ is of weak type$L\log\log L(w)$ with weight dependence$[w]_{A_1}[w]_{A_{\infty}}\log([w]_{A_{\infty}}+1),$ which is same as the bestknown constant for the singular integral $T_\Omega$.

Weak type bounds for rough maximal singular integrals near $L^1$