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Let $$\BA$$ be the ring of adeles of a number field $$k$$ and $$\pi$$ be an irreducible cuspidal automorphic representation of $$\GL_n(\BA)$$. In Jiang and Luo (Pac J Math 318:339–374. https://doi.org/10.2140/pjm.2022.318.339, 2022, Pac J Math 326: 301–372. https://doi.org/10.2140/pjm.2023.326.301, 2023), the authors introduced $$\pi$$-Schwartz space $$\CS_\pi(\BA^\times)$$ and $$\pi$$-Fourier transform $$\CF_{\pi,\psi}$$ with a non-trivial additive character $$\psi$$ of $$k\bs\BA$$, proved the associated Poisson summation formula over $$\BA^\times$$, based on the Godement-Jacquet theory for the standard $$L$$-functions $$L(s,\pi)$$, and provided interesting applications. In this paper, in addition to the further development of the local theory, we found two global applications. First, we find a Poisson summation formula proof of the Voronoi summation formula for $$\GL_n$$ over a number field, which was first proved by A. Ichino and N. Templier (Am J Math 135:65–101. https://doi.org/10.1353/ajm.2013.0005, 2013, Theorem 1). Then we introduce the notion of the Godement-Jacquet kernels $$H_{\pi,s}$$ and their dual kernels $$K_{\pi,s}$$ for any irreducible cuspidal automorphic representation $$\pi$$ of $$\GL_n(\BA)$$ and show in Theorems \ref{thm:H=FK} and \ref{thm:CTh-pi} that $$H_{\pi,s}$$ and $$K_{\pi,1-s}$$ are related by the nonlinear $$\pi_\infty$$-Fourier transform if and only if $$s\in\BC$$ is a zero of $$L_f(s,\pi_f)=0$$, the finite part of the standard automorphic $$L$$-function $$L(s,\pi)$$, which are the $$(\GL_n,\pi)$$-versions of Clozel (J Number Theory 261: 252–298 https://doi.org/10.1016/j.jnt.2024.02.018, 2024, Theorem 1.1), where the Tate kernel with $n=1$ and $$\pi$$ the trivial character are considered. 

We prove the uniqueness of the Ginzburg–Rallis models over p-adic local fields of characteristic zero, which completes the local uniqueness problem for the Ginzburg–Rallis models, starting from the work of Nien (Models of representations of general linear groups over p-adic fields, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)-University of Minnesota, 2006) that proves the non-split case, and the work of Jiang et al. (Trans Am Math Soc 363(5): 2763–2802, 2011) that proves the general case over Archimedean local fields. Our proof extends the strategy of [16] to the p-adic case with the help of the refined structure of the wavefront sets of z-finite distributions as developed by Aizenbud et al. (Adv Math 285:1376–1414,2015).