The Limiting Weak-type Behaviors for Littlewood–Paley Operators

Abstract

This paper is devoted to studying the limiting weak-type behavior for the Littlewood–Paley g_λ,ψ^* function. For every nonnegative function f ∈ L¹(ℝⁿ), we show that $\mathop {\lim }\limits_{\rho \to {0_ + }} \left| {\left\{ {x \in {\mathbb{R}^n}:\left| {g_{\lambda ,\psi }^*({f_\rho })(x)} \right| > 1} \right\}} \right| = \left| {\left\{ {x \in {\mathbb{R}^n}:\tilde G(x) > 1} \right\}} \right|{\left\| f \right\|_{{L^1}({\mathbb{R}^n})}},$ where $f_{\rho}(z)=f({z\over \rho})\rho^{-n}$ and $\tilde{G}(x)=\left(\int\int_{\mathbb{R}_{+}^{n+1}}({t\over t+|x-y})^{\lambda n}|\psi_{t}(y)|^{2}{dydt\over t^{n+1}}\right)^{{1\over 2}}$. The corresponding results for Lusin area integral S_α,ψ with α > 0 are also given.

Keywords

Limiting weak-type behaviors / Littlewood–Paley g_λψ^* function / the ψ-area integral / 42B20 / 42B35

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