<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(\mathcal {X},d,\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">X</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a metric measure space with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> being doubling. In this article, via Hajłasz gradients, we define the variable Besov–Triebel–Lizorkin spaces on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">X</mi> </math></EquationSource> </InlineEquation> and then establish an approximation property of those spaces in terms of discrete <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>-median. By a new concept of the variable power of variable mixed Lebesgue-sequence (quasi-)norms, we introduce the variable Besov–Triebel–Lizorkin <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p(\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-capacity on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">X</mi> </math></EquationSource> </InlineEquation> and also obtain an equivalent expression of the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p(\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-capacity. Moreover, the lower and upper bound estimates for these <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$p(\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-capacity in terms of a modified version of the generalized Netrusov–Hausdorff content are established</p>

Variable Hajłasz–Besov–Triebel–Lizorkin spaces and capacities in metric measure spaces

期刊论文
发表日期：

2026年4月17日

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Ziwei Li,
Ciqiang Zhuo

摘要

Let $(\mathcal {X},d,\mu )$ $(X, d, μ)$ be a metric measure space with $\mu $ $μ$ being doubling. In this article, via Hajłasz gradients, we define the variable Besov–Triebel–Lizorkin spaces on $\mathcal {X}$ $X$ and then establish an approximation property of those spaces in terms of discrete $\gamma $ $γ$ -median. By a new concept of the variable power of variable mixed Lebesgue-sequence (quasi-)norms, we introduce the variable Besov–Triebel–Lizorkin $p(\cdot )$ $p (\cdot)$ -capacity on $\mathcal {X}$ $X$ and also obtain an equivalent expression of the $p(\cdot )$ $p (\cdot)$ -capacity. Moreover, the lower and upper bound estimates for these $p(\cdot )$ $p (\cdot)$ -capacity in terms of a modified version of the generalized Netrusov–Hausdorff content are established