Parametrized Multilinear Littlewood-Paley Operators on Hardy Spaces

Abstract

In this paper, we study the parametrized multilinear Marcinkiewicz integral $\mu^{\rho}$ and the multilinear Littlewood-Paley $g_{\lambda}^{*}$-function. We proved that if the kernel $\Omega$ associated to parametrized multilinear Marcinkiewicz integral $\mu^{\rho}$ is homogeneous of degree zero and satisfies the Lipschitz continuous condition, or the kernel $K$ associated to the multilinear Littlewood-Paley $g_{\lambda}^{*}$-function satisfies the Hörmander condition, then they are bounded from $H^{p_1} \times \cdots \times H^{p_m}$ to $L^p$ with $mn/(mn+\gamma) \lt p_1, \ldots, p_m \leq 1$ and $1/p = 1/p_1 + \cdots + 1/p_m$.