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<p>The goal of this note is to prove that every real-valued Lipschitz function on a Banach space can be pointwise approximated on a given <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-compact set by smooth cylindrical functions whose asymptotic Lipschitz constants are controlled. This result has applications in the study of metric Sobolev and BV spaces: it implies that smooth cylindrical functions are dense in energy in these kinds of functional spaces defined over any weighted Banach space.</p>

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Smooth approximations preserving asymptotic Lipschitz bounds

  • 期刊论文

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  • 发表日期：

    2025年10月17日

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  • Enrico Pasqualetto

摘要

The goal of this note is to prove that every real-valued Lipschitz function on a Banach space can be pointwise approximated on a given \(\sigma \) σ -compact set by smooth cylindrical functions whose asymptotic Lipschitz constants are controlled. This result has applications in the study of metric Sobolev and BV spaces: it implies that smooth cylindrical functions are dense in energy in these kinds of functional spaces defined over any weighted Banach space.