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$L^1$-Dini conditions and limiting behavior of weak type estimates for singular integrals

In 2006, Janakiraman [10] showed that if $Ω$ with mean value zero on $S^{n-1}$ satisfies the condition \[ \sup_{|ξ|=1}\int_{S^{n-1}}|Ω(θ)-Ω(θ+δξ)|dσ(θ)\leq Cnδ\int_{S^{n-1}}|Ω(θ)|dσ(θ),\quad 0<δ<\frac{1}{n},\ (\ast) \] then for the singular integral operator $T_Ω$ with homogeneous kernel, the following limiting behavior holds: \[\lim\limits_{λ\rightarrow 0}λm(\{x\in\mathbb{R}^n:|T_Ωf(x)|>λ\})= \frac{1}{n}\|Ω\|_{1}\|f\|_{1},\quad \text{for}\ f\in L^1(\mathbb{R}^n)\ \text{with}\ f\geq 0.\ (\ast\ast)\] In the present paper, we prove that if replacing the condition $(\ast)$ by more general condition, the $L^1$-Dini condition, then the limiting behavior $(\ast\ast)$ still holds for the singular integral $T_Ω$. In particular, we give an example which satisfies the $L^1$-Dini condition, but does not satisfy $(\ast)$. Hence, we improve essentially the above result given in [10]. To prove our conclusion, we show that the $L^1$-Dini conditions defined respectively via the rotation and translation on $\mathbb{R}^n$ are equivalent (see Theorem 2.5 below), which has its own interest in the theory of singular integrals. Moreover, similar limiting behavior for the fractional integral operator $T_{Ω,α}$ with homogeneous kernel is also established in this paper.