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

• ### Yong Ding

Beijing Normal University, China
• ### Xudong Lai

Harbin Institute of Technology and Beijing Normal University, China ## Abstract

Let $T_\Omega$ be the singular integral operator with a homogeneous kernel $\Omega$. In 2006, Janakiraman showed that if $\Omega$ has mean value zero on $\mathbb S^{n-1}$ and satisfies the condition

$(\ast)\quad \sup_{|\xi|=1}\int_{\S^{n-1}}|\Omega(\theta)-\Omega(\theta+\delta\xi)|\,d\sigma(\theta)\leq Cn\,\delta\int_{\mathbb{S}^{n-1}}|\Omega(\theta)|\,d\sigma(\theta),$

where $0<\delta<{1}/{n}$, then the following limiting behavior:

$(\ast\ast)\quad \lim\limits_{\lambda\to 0_+}\lambda \, m(\{x\in\mathbb R^n:|T_\Omega f(x)|>\lambda\})= \frac{1}{n}\,\|\Omega\|_{1}\|f\|_{1}$

holds for $f\in L^1(\mathbb R^n)$ and $f\geq 0$.

In the present paper, we prove that if we replace the condition $(\ast)$ by a 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 Janakiraman's above result. To prove our conclusion, we show that the $L^1$-Dini conditions defined respectively via rotation and translation in $\mathbb R^n$ are equivalent (see Theorem 2.5 below), which may have its own interest in the theory of the singular integrals. Moreover, similar limiting behavior for the fractional integral operator $T_{\Omega,\alpha}$ with a homogeneous kernel is also established in this paper.

Yong Ding, Xudong Lai, $L^1$-Dini conditions and limiting behavior of weak type estimates for singular integrals. Rev. Mat. Iberoam. 33 (2017), no. 4, pp. 1267–1284