$L^1$-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.