Scholarlink

Abstract <p> We consider complete seminormed spaces of functions of one real variable whose seminorm has a finite-dimensional kernel. If the seminorm is invariant under affine changes of the argument, we call such a space interesting. We prove that the maximal interesting space embedded in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_{1,\mathrm{loc}}(\mathbb{R}^n)\)</EquationSource> </InlineEquation> is equivalent to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathrm{BMO}\)</EquationSource> </InlineEquation>, and the maximal interesting space embedded in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal{D}'(\mathbb{R})\)</EquationSource> </InlineEquation> is equivalent to the homogeneous Besov space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\dot{B}^0_{\infty,\infty}\)</EquationSource> </InlineEquation>. We also construct a minimal interesting space that contains the space of smooth functions with compact support. </p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On Translation and Dilation Invariant Seminorms

  • 期刊论文

  • 发表日期：

    2026年3月31日

查看全文
  • Maxim Romanov

摘要

Abstract

We consider complete seminormed spaces of functions of one real variable whose seminorm has a finite-dimensional kernel. If the seminorm is invariant under affine changes of the argument, we call such a space interesting. We prove that the maximal interesting space embedded in \(L_{1,\mathrm{loc}}(\mathbb{R}^n)\) is equivalent to \(\mathrm{BMO}\) , and the maximal interesting space embedded in \(\mathcal{D}'(\mathbb{R})\) is equivalent to the homogeneous Besov space \(\dot{B}^0_{\infty,\infty}\) . We also construct a minimal interesting space that contains the space of smooth functions with compact support.