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

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.