<p>In this paper we will focus on understanding the relationship between Sobolev embedding theorems for Hajłasz–Besov spaces defined on a doubling metric measure space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11117_2025_1115_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$(\Omega ,d,\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the non-collapsing condition of the measure, i.e. <Equation ID="Equ52"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11117_2025_1115_Article_Equ52.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \inf _{x\in \Omega }\mu (B(x,1))>0. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munder> <mo movablelimits="true">inf</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> </mrow> </munder> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>></mo> <mn>0</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>We will also obtain embedding results for Hajłasz–Besov spaces whose modulus of smoothness is generated by a rearrangement invariant quasi-norm.</p>

Non-collapsing condition and Sobolev embeddings for Hajłasz–Besov spaces

期刊论文
出版模式：

开放获取
发表日期：

2025年3月12日

查看全文

Joaquim Martín,
Walter A. Ortiz

摘要

In this paper we will focus on understanding the relationship between Sobolev embedding theorems for Hajłasz–Besov spaces defined on a doubling metric measure space $(\Omega ,d,\mu )$ $(Ω, d, μ)$ and the non-collapsing condition of the measure, i.e. $\begin{aligned} \inf _{x\in \Omega }\mu (B(x,1))>0. \end{aligned}$ $\begin{matrix} inf_{x \in Ω} μ (B (x, 1)) > 0 . \end{matrix}$ We will also obtain embedding results for Hajłasz–Besov spaces whose modulus of smoothness is generated by a rearrangement invariant quasi-norm.