2015/08/30 by Yong Ding, Ding, Yong, Xudong Lai +1
Mathematics · #42B20 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1508.07519
openalex publication_date 2015/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
In 2006, Janakiraman [10] showed that if Ω with mean value zero on Sn-1 satisfies the condition sup|ξ|=1∫Sn-1|Ω(θ)-Ω(θ+δξ)|dσ(θ)≤ Cnδ∫Sn-1|Ω(θ)|dσ(θ), 0λ\)= (1)/(n)‖Ω‖1‖f‖1, for f∈ L1(ℝn) with f≥ 0. (∗∗) In the present paper, we prove that if replacing the condition (∗) by 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 the above result given in [10]. To prove our conclusion, we show that the L1-Dini conditions defined respectively via the rotation and translation on ℝn are equivalent (see Theorem 2.5 below), which has its own interest in the theory of singular integrals. Moreover, similar limiting behavior for the fractional integral operator TΩ,α with homogeneous kernel is also established in this paper.