<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ (X,d,\mu ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a metric measure space equipped with a doubling measure and supporting a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ p $</EquationSource> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> </InlineEquation>-Poincaré inequality, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ 1< p < \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. For a function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ u \in N^{1,p}(X) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>∈</mo> <msup> <mi>N</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we establish a sufficient condition for fine continuity at a point in terms of the convergence of a Wiener-type integral involving its minimal <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ p $</EquationSource> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> </InlineEquation>-weak upper gradient. This condition holds outside a set of zero <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ q $</EquationSource> <EquationSource Format="MATHML"><math> <mi>q</mi> </math></EquationSource> </InlineEquation>-capacity for each <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ 1< q < p $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation>, and hence yields fine continuity <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ q $</EquationSource> <EquationSource Format="MATHML"><math> <mi>q</mi> </math></EquationSource> </InlineEquation>-quasieverywhere.</p>

A Wiener-Type Criterion for Fine Continuity in Newton-Sobolev Spaces

期刊论文
发表日期：

2026年7月25日

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M. Ashraf Bhat,
G. Sankara Raju Kosuru

摘要

Let $ (X,d,\mu ) $ $(X, d, μ)$ be a metric measure space equipped with a doubling measure and supporting a $ p $ $p$ -Poincaré inequality, where $ 1< p < \infty $ $1 < p < \infty$ . For a function $ u \in N^{1,p}(X) $ $u \in N^{1, p} (X)$ , we establish a sufficient condition for fine continuity at a point in terms of the convergence of a Wiener-type integral involving its minimal $ p $ $p$ -weak upper gradient. This condition holds outside a set of zero $ q $ $q$ -capacity for each $ 1< q < p $ $1 < q < p$ , and hence yields fine continuity $ q $ $q$ -quasieverywhere.