L1L^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 TΩT_\Omega be the singular integral operator with a homogeneous kernel Ω\Omega. In 2006, Janakiraman showed that if Ω\Omega has mean value zero on Sn1\mathbb S^{n-1} and satisfies the condition

()supξ=1§n1Ω(θ)Ω(θ+δξ)dσ(θ)CnδSn1Ω(θ)dσ(θ),(\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<δ<1/n0<\delta<{1}/{n}, then the following limiting behavior:

()limλ0+λm({xRn:TΩf(x)>λ})=1nΩ1f1(\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 fL1(Rn)f\in L^1(\mathbb R^n) and f0f\geq 0.

In the present paper, we prove that if we replace the condition ()(\ast) by a more general condition, the L1L^1-Dini condition, then the limiting behavior ()(\ast\ast) still holds for the singular integral TΩT_\Omega. In particular, we give an example which satisfies the L1L^1-Dini condition, but does not satisfy ()(\ast). Hence, we improve essentially Janakiraman's above result. To prove our conclusion, we show that the L1L^1-Dini conditions defined respectively via rotation and translation in Rn\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 TΩ,αT_{\Omega,\alpha} with a homogeneous kernel is also established in this paper.

Cite this article

Yong Ding, Xudong Lai, L1L^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