On the theory of multilinear Littlewood–Paley $g$-function

Abstract

Let $m \ge 2$ and define the multilinear Littlewood–Paley $g$-function by

$$g(\vec{f})(x)=\bigg(\int_{0}^{\infty} \bigg| \frac{1}{t^{mn}}\int_{(\mathbb{R}^n)^m} \psi\bigg(\frac{y_1}{t},\dots,\frac{y_m}{t}\bigg) \prod_{j=1}^mf_j(x-y_j)dy_{j}\bigg|^2 \frac{dt}{t} \bigg)^{1/2}.$$

In this paper, we establish the strong $L^{p_1}(w_1)\times \dots \times L^{p_m}(w_m)$ to $L^p(\nu_{\vec{\omega}}$) boundedness and weak type $L^{p_1}(w_1)\times \dots \times L^{p_m}(w_m)$ to $L^{p,\infty}(\nu_{\vec{\omega}}$) estimate for the multilinear $g$-function. The weighted strong and end-point estimates for the iterated commutators of $g$-function are also given. Here $\nu_{\vec{\omega}} = \prod_{i = 1}^m\omega_i^{{p}/{p_i}}$ and each $w_i$ is a nonnegative function on $\mathbb{R}^n$.