$L^1$-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

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

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

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.