Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\operatorname {X}=(\operatorname {X},\operatorname {d})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>X</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>X</mo> <mo>,</mo> <mo>d</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be an arbitrary metric space. For each <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p \in [1,\infty ],$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> we prove that a map <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma :[a,b] \rightarrow \operatorname {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo>X</mo> </mrow> </math></EquationSource> </InlineEquation> is p-absolutely continuous if and only if, for every Lipschitz function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$h:\operatorname {X} \rightarrow \mathbb {R},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>:</mo> <mo>X</mo> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the post-composition <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$h \circ \gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>∘</mo> <mi>γ</mi> </mrow> </math></EquationSource> </InlineEquation> is a p-absolutely continuous function. Furthermore, if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\operatorname {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mo>X</mo> </math></EquationSource> </InlineEquation> is complete and separable, then, for each <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p \in (1,\infty ),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> we show that the equivalence class (up to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {L}^{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">L</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-a.e. equality) of a Borel map <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\gamma :[a,b] \rightarrow \operatorname {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo>X</mo> </mrow> </math></EquationSource> </InlineEquation> belongs to the Sobolev <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$W_{p}^{1}([a,b],\operatorname {X})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>W</mi> <mrow> <mi>p</mi> </mrow> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mrow> <mo>,</mo> <mo>X</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-space if and only if, for every Lipschitz function <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$h:\operatorname {X} \rightarrow \mathbb {R},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>:</mo> <mo>X</mo> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the equivalence class (up to <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathcal {L}^{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">L</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-a.e. equality) of the post-composition <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$h \circ \gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>∘</mo> <mi>γ</mi> </mrow> </math></EquationSource> </InlineEquation> belongs to the Sobolev <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$W_{p}^{1}([a,b],\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>W</mi> <mrow> <mi>p</mi> </mrow> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mrow> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-space.

Characterization of AC and Sobolev curves via Lipschitz post-compositions

期刊论文
发表日期：

2026年7月10日

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Roman D. Oleinik,
Alexander I. Tyulenev

摘要

Let $\operatorname {X}=(\operatorname {X},\operatorname {d})$ $X = (X, d)$ be an arbitrary metric space. For each $p \in [1,\infty ],$ $p \in [1, \infty],$ we prove that a map $\gamma :[a,b] \rightarrow \operatorname {X}$ $γ : [a, b] \to X$ is p-absolutely continuous if and only if, for every Lipschitz function $h:\operatorname {X} \rightarrow \mathbb {R},$ $h : X \to R,$ the post-composition $h \circ \gamma $ $h \circ γ$ is a p-absolutely continuous function. Furthermore, if $\operatorname {X}$ $X$ is complete and separable, then, for each $p \in (1,\infty ),$ $p \in (1, \infty),$ we show that the equivalence class (up to $\mathcal {L}^{1}$ $L^{1}$ -a.e. equality) of a Borel map $\gamma :[a,b] \rightarrow \operatorname {X}$ $γ : [a, b] \to X$ belongs to the Sobolev $W_{p}^{1}([a,b],\operatorname {X})$ $W_{p}^{1} ([a, b], X)$ -space if and only if, for every Lipschitz function $h:\operatorname {X} \rightarrow \mathbb {R},$ $h : X \to R,$ the equivalence class (up to $\mathcal {L}^{1}$ $L^{1}$ -a.e. equality) of the post-composition $h \circ \gamma $ $h \circ γ$ belongs to the Sobolev $W_{p}^{1}([a,b],\mathbb {R})$ $W_{p}^{1} ([a, b], R)$ -space.