<p>For the Reshetnyak-class homeomorphisms <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varphi :\Omega \rightarrow Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a domain in some Carnot group and <i>Y</i> is a metric space, we obtain an equivalent description as the homeomorphisms which induce the bounded composition operator <Equation ID="Equ16"> <EquationSource Format="TEX">$ \varphi ^:\textrm{Lip}(Y)\rightarrow L_q^1(\Omega ), $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mi>φ</mi> <mo>∗</mo> </msup> <mo>:</mo> <mtext>Lip</mtext> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msubsup> <mi>L</mi> <mi>q</mi> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$1\le q\le \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varphi ^u=u\circ \varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>φ</mi> <mo>∗</mo> </msup> <mi>u</mi> <mo>=</mo> <mi>u</mi> <mo>∘</mo> <mi>φ</mi> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u\in \textrm{Lip}(Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>∈</mo> <mtext>Lip</mtext> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We demonstrate the utility of our approach by characterizing the homeomorphisms <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varphi :\Omega \rightarrow \Omega '$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <msup> <mi mathvariant="normal">Ω</mi> <mo>′</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> of domains in some Carnot group <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\mathbb {G}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">G</mi> </math></EquationSource> </InlineEquation> which induce the bounded composition operator <Equation ID="Equ17"> <EquationSource Format="TEX">$ \varphi ^*: L^1_p(\Omega ')\cap \textrm{Lip}_{\textrm{loc}}(\Omega ')\rightarrow L^1_q (\Omega ),\quad 1\le q \le p\le \infty , $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mi>φ</mi> <mo>∗</mo> </msup> <mo>:</mo> <msubsup> <mi>L</mi> <mi>p</mi> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Ω</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <msub> <mtext>Lip</mtext> <mtext>loc</mtext> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Ω</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msubsup> <mi>L</mi> <mi>q</mi> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>on homogeneous Sobolev spaces. The new proof of this known criterion is much shorter than the one already available, requires a minimum of tools, and enables us to obtain new properties of the homeomorphisms in question.</p>

Reshetnyak-class mappings and composition operators

期刊论文
发表日期：

2025年11月13日

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Stepan V. Pavlov,
Sergey K. Vodopyanov

摘要

For the Reshetnyak-class homeomorphisms $\varphi :\Omega \rightarrow Y$ $φ : Ω \to Y$ , where $\Omega $ $Ω$ is a domain in some Carnot group and Y is a metric space, we obtain an equivalent description as the homeomorphisms which induce the bounded composition operator $ \varphi ^*:\textrm{Lip}(Y)\rightarrow L_q^1(\Omega ), $ $φ^{*} : Lip (Y) \to L_{q}^{1} (Ω),$ where $1\le q\le \infty $ $1 \leq q \leq \infty$ , as $\varphi ^*u=u\circ \varphi $ $φ^{*} u = u \circ φ$ for $u\in \textrm{Lip}(Y)$ $u \in Lip (Y)$ . We demonstrate the utility of our approach by characterizing the homeomorphisms $\varphi :\Omega \rightarrow \Omega '$ $φ : Ω \to Ω^{'}$ of domains in some Carnot group ${\mathbb {G}}$ $G$ which induce the bounded composition operator $ \varphi ^*: L^1_p(\Omega ')\cap \textrm{Lip}_{\textrm{loc}}(\Omega ')\rightarrow L^1_q (\Omega ),\quad 1\le q \le p\le \infty , $ $φ^{*} : L_{p}^{1} (Ω^{'}) \cap {Lip}_{loc} (Ω^{'}) \to L_{q}^{1} (Ω), 1 \leq q \leq p \leq \infty,$ on homogeneous Sobolev spaces. The new proof of this known criterion is much shorter than the one already available, requires a minimum of tools, and enables us to obtain new properties of the homeomorphisms in question.