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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(V\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>V</mi> </math></EquationSource> </InlineEquation> be a smooth vector field on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb{R}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> whose divergence is non-vanishing.Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\{\Delta_j\}_{j \in \mathbb{Z}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>j</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>j</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> be a family of Littlewood–Paley operators.We obtain bilinear estimates for commutators of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(V\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>V</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\Delta_j\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> on weighted Besov and Triebel–Lizorkin spaces with variable smoothness and integrability.</p>

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Commutator estimates for vector fields on weighted Besov and Triebel–Lizorkin spaces with variable smoothness and integrability

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • S. Wang,
  • P. Guo,
  • J. Xu

摘要

Let \(V\) V be a smooth vector field on \(\mathbb{R}^n\) R n whose divergence is non-vanishing.Let \(\{\Delta_j\}_{j \in \mathbb{Z}}\) { Δ j } j ∈ Z be a family of Littlewood–Paley operators.We obtain bilinear estimates for commutators of \(V\) V and \(\Delta_j\) Δ j on weighted Besov and Triebel–Lizorkin spaces with variable smoothness and integrability.