Abstract <p> Let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb Q}_p\)</EquationSource> </InlineEquation> be the locally compact field of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-adic numbers, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">\(|\cdot|_p\)</EquationSource> </InlineEquation> be the <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-adic norm on <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb Q}_p\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">\(x, \lambda\in {\mathbb Q}_p\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">\(f(x), g(x)\)</EquationSource> </InlineEquation> are complex-valued functions, <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(dx\)</EquationSource> </InlineEquation> be the element of Haar measure on <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb Q}_p\)</EquationSource> </InlineEquation>. We study the problems of the limits of integrals <Equation ID="Equi"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_Equi.gif" Format="GIF" Height="55" Rendition="HTML" Resolution="72" Type="Linedraw" Width="125" /> </MediaObject> <EquationSource Format="TEX">\(\int\limits_{{\mathbb Q}_p} f(\lambda x)\, g(x)\, dx,\)</EquationSource> </Equation> when <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq11.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(|\lambda|_p\to\infty\)</EquationSource> </InlineEquation> or <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq12.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">\(|\lambda|_p\to 0\)</EquationSource> </InlineEquation>, for functions <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq13.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq14.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(g\)</EquationSource> </InlineEquation> from some functional classes. These problems are analogs of classical problems on the limits of integrals over <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1126_Article_IEq15.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb R\)</EquationSource> </InlineEquation> in the classical harmonic analysis. </p>

Limits of Some Integrals on \({\mathbb Q}_p\)

期刊论文
发表日期：

2025年8月1日

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Sergey S. Platonov

摘要

Abstract

Let \({\mathbb Q}_p\) be the locally compact field of \(p\) -adic numbers, \(|\cdot|_p\) be the \(p\) -adic norm on \({\mathbb Q}_p\) , \(x, \lambda\in {\mathbb Q}_p\) , \(f(x), g(x)\) are complex-valued functions, \(dx\) be the element of Haar measure on \({\mathbb Q}_p\) . We study the problems of the limits of integrals \(\int\limits_{{\mathbb Q}_p} f(\lambda x)\, g(x)\, dx,\) when \(|\lambda|_p\to\infty\) or \(|\lambda|_p\to 0\) , for functions \(f\) and \(g\) from some functional classes. These problems are analogs of classical problems on the limits of integrals over \(\mathbb R\) in the classical harmonic analysis.