Resumen de L1 -Dini conditions and limiting behavior of weak type estimates for singular integrals

Yong Ding, Xudong Lai

• Let TΩ be the singular integral operator with a homogeneous kernel Ω. In 2006, Janakiraman showed that if Ω has mean value zero on Sn−1 and satisfies the condition (∗)sup|ξ|=1∫§n−1|Ω(θ)−Ω(θ+δξ)|dσ(θ)≤Cnδ∫Sn−1|Ω(θ)|dσ(θ), where 0<δ<1/n, then the following limiting behavior:

  (∗∗)limλ→0+λm({x∈Rn:|TΩf(x)|>λ})=1n∥Ω∥1∥f∥1 holds for f∈L1(Rn) and f≥0.

  In the present paper, we prove that if we replace the condition (∗) by a more general condition, the L1-Dini condition, then the limiting behavior (∗∗) still holds for the singular integral TΩ. In particular, we give an example which satisfies the L1-Dini condition, but does not satisfy (∗). Hence, we improve essentially Janakiraman's above result. To prove our conclusion, we show that the L1-Dini conditions defined respectively via rotation and translation in Rn 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 TΩ,α with a homogeneous kernel is also established in this paper.