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<p>We continue the&#xa0;study of embeddings between different classes of Sobolev spaces of differential forms started in 2006 in a&#xa0;paper by Gol’dshtein and Troyanov. Like in this paper, our study is based on relations between <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_{q,p}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mrow> <mi>q</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>-cohomology and Sobolev type inequalities. The&#xa0;main results are estimates for the norms of the embedding operators for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(q=p\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p&gt;\frac{n-1}{k-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>&gt;</mo> <mfrac> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> in the&#xa0;Euclidean <i>r</i>-ball <i>B</i>(<i>r</i>) and its bi-Lipschitz images. We also study the&#xa0;compactness of such operators.</p>

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SOME POINCARÉ–SOBOLEV INEQUALITIES FOR DIFFERENTIAL FORMS

  • 期刊论文

  • 发表日期：

    2025年12月3日

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  • V. Gol’dshtein,
  • Ya. Kopylov,
  • R. Panenko

摘要

We continue the study of embeddings between different classes of Sobolev spaces of differential forms started in 2006 in a paper by Gol’dshtein and Troyanov. Like in this paper, our study is based on relations between \(L_{q,p}\) L q , p -cohomology and Sobolev type inequalities. The main results are estimates for the norms of the embedding operators for \(q=p\) q = p and \(p>\frac{n-1}{k-1}\) p > n - 1 k - 1 in the Euclidean r-ball B(r) and its bi-Lipschitz images. We also study the compactness of such operators.