A Characterization of Two-Weight Norm Inequality for Littlewood–Paley $g_{\lambda }^{*}$-Function

Abstract

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,

$$\begin{aligned} g_{\lambda }^{*}(f)(x) =\bigg (\iint _{\mathbb {R}^{n+1}_{+}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda } |\nabla P_tf(y,t)|^2 \frac{\mathrm{d}y \mathrm{d}t}{t^{n-1}}\bigg )^{1/2}, \ \quad \lambda > 1, \end{aligned}$$

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 $\big \Vert g_{\lambda }^{*}(f \sigma ) \big \Vert _{L^2(w)} \lesssim \big \Vert f \big \Vert _{L^2(\sigma )}$ if and only if the two-weight Muckenhoupt $A_2$ condition holds, and a testing condition holds:

$$\begin{aligned} \sup _{Q : \text {cubes}~\mathrm{in} \ {\mathbb {R}^n}} \frac{1}{\sigma (Q)} \int _{{\mathbb {R}^n}} \iint _{\widehat{Q}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda }|\nabla P_t(\mathbf {1}_Q \sigma )(y,t)|^2 \frac{w \mathrm{d}x \mathrm{d}t}{t^{n-1}} \mathrm{d}y < \infty , \end{aligned}$$

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.