# $L_{1}$-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 $T_{Ω}$ be the singular integral operator with a homogeneous kernel $Ω$. In 2006, Janakiraman showed that if $Ω$ has mean value zero on $S_{n−1}$ and satisfies the condition

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

holds for $f∈L_{1}(R_{n})$ and $f≥0$.

In the present paper, we prove that if we replace the condition $(∗)$ by a more general condition, the $L_{1}$-Dini condition, then the limiting behavior $(∗∗)$ still holds for the singular integral $T_{Ω}$. In particular, we give an example which satisfies the $L_{1}$-Dini condition, but does not satisfy $(∗)$. 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 $R_{n}$ 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 $T_{Ω,α}$ with a homogeneous kernel is also established in this paper.

## Cite this article

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

DOI 10.4171/RMI/971