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 $1/p = 1/p_1 + \\cdots + 1/p_m$.