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

Abstract: In 2006, Janakiraman [10] showed that if $\Omega$ with mean value zero on $S^{n-1}$ satisfies the condition [ \sup_{|\xi|=1}\int_{S^{n-1}}|\Omega(\theta)-\Omega(\theta+\delta\xi)|d\sigma(\theta)\leq Cn\delta\int_{S^{n-1}}|\Omega(\theta)|d\sigma(\theta),\quad 0<\delta<\frac{1}{n},\ (\ast) ] then for the singular integral operator $T_\Omega$ with homogeneous kernel, the following limiting behavior holds: [\lim\limits_{\lambda\rightarrow 0}\lambda m({x\in\mathbb{R}^n:|T_\Omega f(x)|>\lambda})= \frac{1}{n}|\Omega|{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 $T_\Omega$. 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 $T_{\Omega,\alpha}$ with homogeneous kernel is also established in this paper.