Limiting Weak-Type Behaviors for Certain Littlewood-Paley Functions

Abstract

In this paper, we establish the following limiting weak-type behaviors of Littlewood-Paley g-function g': for nonnegative function f ∈ L¹(Rⁿ), $\mathop {\lim }\limits_{\lambda \to {0_ + }} \lambda m\left( {\left\{ {x \in {\mathbb{R}^n}:|g\varphi f\left( x \right)| > \lambda } \right\}} \right) = m\left( {\left\{ {x \in {\mathbb{R}^n}:{{\left( {\int_0^\infty {{{\left| {\varphi r\left( x \right)} \right|}^2}\frac{{dr}}{r}} } \right)}^{1/2}} > 1} \right\}} \right){\left\| f \right\|_1}$ and $\mathop {\lim }\limits_{t \to {0_ + }} m\left( {\left\{ {x \in {\mathbb{R}^n}:|g\varphi {f_t}\left( x \right) - {{\left( {\int_0^\infty {{{\left| {\varphi r\left( x \right)} \right|}^2}\frac{{dr}}{r}} } \right)}^{1/2}}| > 1} \right\}} \right) = 0$, where f_t(x) = t⁻ⁿf(t⁻¹x) for t > 0. Meanwhile, the corresponding results for Marcinkiewicz integral and its fractional version with kernels satisfying L_α^q -Dini condition are also given.