Asymptotical behavior of one class of $p$-adic singular Fourier integrals

TL;DR

This paper investigates the asymptotic behavior of a class of $p$-adic singular Fourier integrals involving quasi associated homogeneous distributions, extending Erdélyi's lemma to the $p$-adic setting and revealing a stabilization property.

Contribution

It introduces a $p$-adic analogue of Erdélyi's lemma for singular Fourier integrals and establishes asymptotic expansions with a stabilization property, unlike the real case.

Findings

Derived $p$-adic asymptotics for singular Fourier integrals.

Extended Erdélyi's lemma to $p$-adic distributions.

Discovered stabilization property of asymptotics in $p$-adic context.

Abstract

We study the asymptotical behavior of the $p$-adic singular Fourier integrals $$ J_{\pi_{\alpha},m;\phi}(t) =\bigl< f_{\pi_{\alpha};m}(x)\chi_p(xt), \phi(x)\bigr> =F\big[f_{\pi_{\alpha};m}\phi\big](t), \quad |t|_p \to \infty, \quad t\in \bQ_p, $$ where $f_{\pi_{\alpha};m}\in {\cD}'(\bQ_p)$ is a {\em quasi associated homogeneous} distribution (generalized function) of degree $\pi_{\alpha}(x)=|x|_p^{\alpha-1}\pi_1(x)$ and order $m$, $\pi_{\alpha}(x)$, $\pi_1(x)$, and $\chi_p(x)$ are a multiplicative, a normed multiplicative, and an additive characters of the field $\bQ_p$ of $p$-adic numbers, respectively, $\phi \in {\cD}(\bQ_p)$ is a test function, $m=0,1,2...$, $\alpha\in \bC$. If $Re\alpha>0$ the constructed asymptotics constitute a $p$-adic version of the well known Erd\'elyi lemma. Theorems which give asymptotic expansions of singular Fourier integrals are the Abelian type theorems. In contrast to the real case, all constructed asymptotics have the {\it stabilization} property.