## 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,

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:

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.

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The authors wish to express their sincere thanks to the referee for his or her valuable remarks and suggestions which made this paper more readable.

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M. Cao and Q. Xue were supported by NSFC (No. 11471041 and 11671039), the Fundamental Research Funds for the Central Universities (No. 2014KJJCA10) and NCET-13-0065. K. Li was supported by the Basque Government through the BERC 2014-2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323.

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Cao, M., Li, K. & Xue, Q. A Characterization of Two-Weight Norm Inequality for Littlewood–Paley \(g_{\lambda }^{*}\)-Function.
*J Geom Anal* **28**, 842–865 (2018). https://doi.org/10.1007/s12220-017-9844-x

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DOI: https://doi.org/10.1007/s12220-017-9844-x

### Keywords

- Two-weight inequality
- Littlewood–Paley \(g_{\lambda }^*\)-function
- Pivotal condition
- Random dyadic grids