A characterization of two weight norm inequality for Littlewood-Paley $g_{\lambda}^{*}$-function

Let $n\ge 2$ and $g_{\lambda}^{*}$ be the well-known high dimensional Littlewood-Paley function which was defined and studied by E. M. Stein, \begin{align*} g_{\lambda}^{*}(f)(x) =\bigg(\iint_{\mathbb R^{n+1}_{+}} \Big(\frac{t}{t+|x-y|}\Big)^{n\lambda} |\nabla P_tf(y,t)|^2 \frac{dy dt}{t^{n-1}}\bigg)^{1/2}, \ \quad \lambda > 1, \end{align*} where $P_tf(y,t)=p_t*f(y)$, $p_t(y)=t^{-n}p(y/t)$ and $p(x) = (1+|x|^2)^{-(n+1)/2}$, $\nabla =(\frac{\partial}{\partial y_1},\ldots,\frac{\partial}{\partial y_n},\frac{\partial}{\partial t})$. In this paper, we give a characterization of two-weight norm inequality for $g_{\lambda}^{*}$-function. We show that, $\big\| g_{\lambda}^{*}(f \sigma) \big\|_{L^2(w)} \lesssim \big\| f \big\|_{L^2(\sigma)}$ if and only if the two-weight Muchenhoupt $A_2$ condition holds, and a testing condition holds : \begin{align*} \sup_{Q : cubes \ in \mathbb R^n} \frac{1}{\sigma(Q)} \int_{\mathbb R^n} \iint_{\widehat{Q}} \Big(\frac{t}{t+|x-y|}\Big)^{n\lambda}|\nabla P_t(\mathbf{1}_Q \sigma)(y,t)|^2 \frac{w dx dt}{t^{n-1}} dy < \infty, \end{align*} where $\widehat{Q}$ is the Carleson box over $Q$ and $(w, \sigma)$ is a pair of weights. We actually prove this characterization for $g_{\lambda}^{*}$-function associated with more general fractional Poisson kernel $p^\alpha(x) = (1+|x|^2)^{-{(n+\alpha)}/{2}}$. Moreover, the corresponding results for intrinsic $g_{\lambda}^*$-function are also presented.